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Step * 1 1 of Lemma det-equal-rows

.....equality..... 
1. r : CRng
2. n : ℕ
3. M : Matrix(n;n;r)
4. i : ℕn
5. j : ℕn
6. ¬(i = j ∈ ℤ)
7. matrix-swap-rows(M;i;j) = M ∈ Matrix(n;n;r)
⊢ Σ{r} x ∈ filter(λf.(permutation-sign(n;f) =z 1);permutations-list(n)). let k = Π(r) 0 
                                                                                      ≤ i 
                                                                                      < n
                                                                                  M[i,x i] in
                                                                             if permutation-sign(n;x)=1
                                                                             then k
                                                                             else (-r k)
= (-r 
   Σ{r} x ∈ filter(λa.(¬b(permutation-sign(n;a) =z 1));permutations-list(n)). let k = Π(r) 0 
                                                                                           ≤ i 
                                                                                           < n
                                                                                       M[i,x i] in
                                                                                  if permutation-sign(n;x)=1
                                                                                  then k
                                                                                  else (-r k))
∈ |r|
BY
{ ((InstLemma `rng_lsum_map` [⌜r⌝;⌜ℕn ⟶ ℕn⌝;⌜ℕn ⟶ ℕn⌝;⌜λf.(f o (i, j))⌝;⌜λ2f.let k = Π(r) 0 
                                                                                            ≤ i 
                                                                                            < n
                                                                                        M[i,f i] in
                                                                                   if permutation-sign(n;f)=1
                                                                                   then k
                                                                                   else (-r k)⌝;
    ⌜filter(λa.(¬b(permutation-sign(n;a) =z 1));permutations-list(n))⌝]⋅
    THENA Auto
    )
   THEN NthHypEq (-1)
   THEN EqCDA
   THEN (RWW  "rng_minus_lsum" 0 THENA Auto)) }

1
.....subterm..... T:t
2:n
1. r : CRng
2. n : ℕ
3. M : Matrix(n;n;r)
4. i : ℕn
5. j : ℕn
6. ¬(i = j ∈ ℤ)
7. matrix-swap-rows(M;i;j) = M ∈ Matrix(n;n;r)
8. Σ{r} x ∈ map(λf.(f o (i, j));filter(λa.(¬b(permutation-sign(n;a) =z 1));
                                       permutations-list(n))). let k = Π(r) 0 
                                                                            ≤ i 
                                                                            < n
                                                                        M[i,x i] in
                                                                   if permutation-sign(n;x)=1 then k else (-r k)
= Σ{r} x ∈ filter(λa.(¬b(permutation-sign(n;a) =z 1));permutations-list(n)). let k = Π(r) 0 
                                                                                          ≤ i@0 
                                                                                          < n
                                                                                      M[i@0,(λf.(f o (i, j))) x i@0] in
                                                                                 if permutation-sign(n;(λf.(f
                                                                                                            o (i, j))) 
                                                                                                       x)=1
                                                                                 then k
                                                                                 else (-r k)
∈ |r|
⊢ Σ{r} x ∈ filter(λf.(permutation-sign(n;f) =z 1);permutations-list(n)). let k = Π(r) 0 
                                                                                      ≤ i 
                                                                                      < n
                                                                                  M[i,x i] in
                                                                             if permutation-sign(n;x)=1
                                                                             then k
                                                                             else (-r k)
= Σ{r} x ∈ map(λf.(f o (i, j));filter(λa.(¬b(permutation-sign(n;a) =z 1));
                                      permutations-list(n))). let k = Π(r) 0 
                                                                           ≤ i 
                                                                           < n
                                                                       M[i,x i] in
                                                                  if permutation-sign(n;x)=1 then k else (-r k)
∈ |r|

2
1. r : CRng
2. n : ℕ
3. M : Matrix(n;n;r)
4. i : ℕn
5. j : ℕn
6. ¬(i = j ∈ ℤ)
7. matrix-swap-rows(M;i;j) = M ∈ Matrix(n;n;r)
8. Σ{r} x ∈ map(λf.(f o (i, j));filter(λa.(¬b(permutation-sign(n;a) =z 1));
                                       permutations-list(n))). let k = Π(r) 0 
                                                                            ≤ i 
                                                                            < n
                                                                        M[i,x i] in
                                                                   if permutation-sign(n;x)=1 then k else (-r k)
= Σ{r} x ∈ filter(λa.(¬b(permutation-sign(n;a) =z 1));permutations-list(n)). let k = Π(r) 0 
                                                                                          ≤ i@0 
                                                                                          < n
                                                                                      M[i@0,(λf.(f o (i, j))) x i@0] in
                                                                                 if permutation-sign(n;(λf.(f
                                                                                                            o (i, j))) 
                                                                                                       x)=1
                                                                                 then k
                                                                                 else (-r k)
∈ |r|
⊢ Σ{r} x ∈ filter(λa.(¬b(permutation-sign(n;a) =z 1));permutations-list(n)). (-r 
                                                                              let k = Π(r) 0 
                                                                                           ≤ i 
                                                                                           < n
                                                                                       M[i,x i] in
                                                                                  if permutation-sign(n;x)=1
                                                                                  then k
                                                                                  else (-r k))
= Σ{r} x ∈ filter(λa.(¬b(permutation-sign(n;a) =z 1));permutations-list(n)). let k = Π(r) 0 
                                                                                          ≤ i@0 
                                                                                          < n
                                                                                      M[i@0,(λf.(f o (i, j))) x i@0] in
                                                                                 if permutation-sign(n;(λf.(f
                                                                                                            o (i, j))) 
                                                                                                       x)=1
                                                                                 then k
                                                                                 else (-r k)
∈ |r|

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Latex:

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Latex:
.....equality..... 
1.  r  :  CRng
2.  n  :  \mBbbN{}
3.  M  :  Matrix(n;n;r)
4.  i  :  \mBbbN{}n
5.  j  :  \mBbbN{}n
6.  \mneg{}(i  =  j)
7.  matrix-swap-rows(M;i;j)  =  M
\mvdash{}  \mSigma{}\{r\}  x  \mmember{}  filter(\mlambda{}f.(permutation-sign(n;f)  =\msubz{}  1);
                                    permutations-list(n)).  let  k  =  \mPi{}(r)  0 
                                                                                                            \mleq{}  i 
                                                                                                            <  n
                                                                                                    M[i,x  i]  in
                                                                                          if  permutation-sign(n;x)=1  then  k  else  (-r  k)
=  (-r 
      \mSigma{}\{r\}  x  \mmember{}  filter(\mlambda{}a.(\mneg{}\msubb{}(permutation-sign(n;a)  =\msubz{}  1));
                                      permutations-list(n)).  let  k  =  \mPi{}(r)  0 
                                                                                                              \mleq{}  i 
                                                                                                              <  n
                                                                                                      M[i,x  i]  in
                                                                                            if  permutation-sign(n;x)=1  then  k  else  (-r  k))


By

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Latex:
((InstLemma  `rng\_lsum\_map`  [\mkleeneopen{}r\mkleeneclose{};\mkleeneopen{}\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n\mkleeneclose{};\mkleeneopen{}\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n\mkleeneclose{};\mkleeneopen{}\mlambda{}f.(f  o  (i,  j))\mkleeneclose{};
    \mkleeneopen{}\mlambda{}\msubtwo{}f.let  k  =  \mPi{}(r)  0 
                                        \mleq{}  i 
                                        <  n
                                M[i,f  i]  in
                      if  permutation-sign(n;f)=1  then  k  else  (-r  k)\mkleeneclose{};
    \mkleeneopen{}filter(\mlambda{}a.(\mneg{}\msubb{}(permutation-sign(n;a)  =\msubz{}  1));permutations-list(n))\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  NthHypEq  (-1)
  THEN  EqCDA
  THEN  (RWW    "rng\_minus\_lsum"  0  THENA  Auto))

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