Nuprl Lemma : det-multiple-col-ops
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[a:ℕn]. ∀[k:|r|].
(|matrix(if y=a then M[x,y] else (M[x,y] +r (k * M[x,a])))| = |M| ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
int_eq: if a=b then c else d,
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
true: True,
squash: ↓T,
prop: ℙ,
rng: Rng,
crng: CRng,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
not: ¬A,
implies: P ⇒ Q,
false: False,
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
top: Top,
all: ∀x:A. B[x],
matrix-transpose: M'
Lemmas referenced :
crng_wf,
nat_wf,
matrix_wf,
int_seg_wf,
rng_car_wf,
true_wf,
squash_wf,
equal_wf,
matrix-transpose_wf,
det-multiple-row-ops,
iff_weakening_equal,
det-transpose,
rng_times_wf,
rng_plus_wf,
infix_ap_wf,
matrix-ap_wf,
mx_wf,
matrix-det_wf,
matrix_ap_mx_lemma
Rules used in proof :
axiomEquality,
isect_memberEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
sqequalRule,
equalitySymmetry,
hyp_replacement,
hypothesis,
because_Cache,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyLambdaEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
int_eqEquality,
voidEquality,
voidElimination,
dependent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[a:\mBbbN{}n]. \mforall{}[k:|r|].
(|matrix(if y=a then M[x,y] else (M[x,y] +r (k * M[x,a])))| = |M|)
Date html generated:
2018_05_21-PM-09_37_14
Last ObjectModification:
2018_01_02-PM-04_19_11
Theory : matrices
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