Nuprl Lemma : equal-rows-det
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)].
|M| = 0 ∈ |r| supposing ∃i,j:ℕn. ((¬(i = j ∈ ℤ)) ∧ (matrix-swap-rows(M;i;j) = M ∈ Matrix(n;n;r)))
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-swap-rows: matrix-swap-rows(M;i;j),
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
so_apply: x[s],
rng: Rng,
crng: CRng,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
cand: A c∧ B
Lemmas referenced :
crng_wf,
nat_wf,
matrix-swap-rows_wf,
matrix_wf,
equal_wf,
not_wf,
int_seg_wf,
exists_wf,
det-equal-rows
Rules used in proof :
intEquality,
productEquality,
lambdaEquality,
sqequalRule,
because_Cache,
rename,
setElimination,
natural_numberEquality,
productElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
independent_isectElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)].
|M| = 0 supposing \mexists{}i,j:\mBbbN{}n. ((\mneg{}(i = j)) \mwedge{} (matrix-swap-rows(M;i;j) = M))
Date html generated:
2018_05_21-PM-09_36_16
Last ObjectModification:
2017_12_13-PM-11_45_13
Theory : matrices
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