Nuprl Lemma : det-swap-rows
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[i,j:ℕn]. |matrix-swap-rows(M;i;j)| = (-r |M|) ∈ |r| supposing ¬(i = j ∈ ℤ)
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],
not: ¬A,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
crng: CRng,
rng_minus: -r,
rng_car: |r|
Definitions unfolded in proof :
true: True,
rng: Rng,
crng: CRng,
nat: ℕ,
int_seg: {i..j-},
prop: ℙ,
top: Top,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
matrix-det_wf,
rng_minus_wf,
matrix-transpose_wf,
matrix-swap-cols_wf,
rng_car_wf,
crng_wf,
nat_wf,
matrix_wf,
int_seg_wf,
equal_wf,
not_wf,
swap-rows-is-transpose-swap-cols,
squash_wf,
true_wf,
det-transpose,
det-swap-cols,
iff_weakening_equal
Rules used in proof :
applyEquality,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
axiomEquality,
hypothesisEquality,
rename,
setElimination,
intEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaEquality,
imageElimination,
universeEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[i,j:\mBbbN{}n].
|matrix-swap-rows(M;i;j)| = (-r |M|) supposing \mneg{}(i = j)
Date html generated:
2018_05_21-PM-09_36_35
Last ObjectModification:
2017_12_11-PM-04_13_25
Theory : matrices
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