Nuprl Lemma : det-mul-col
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[i:ℕn]. ∀[k:|r|]. (|matrix-mul-col(r;k;i;M)| = (k * |M|) ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-mul-col: matrix-mul-col(r;k;i;M),
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
true: True,
top: Top,
nat: ℕ,
rng: Rng,
crng: CRng,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
infix_ap: x f y,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
matrix-det_wf,
rng_times_wf,
infix_ap_wf,
matrix-transpose_wf,
matrix-mul-row_wf,
crng_wf,
nat_wf,
matrix_wf,
int_seg_wf,
rng_car_wf,
mul-col-is-transpose-mul-row,
equal_wf,
squash_wf,
true_wf,
det-transpose,
det-mul-row,
iff_weakening_equal
Rules used in proof :
voidEquality,
voidElimination,
natural_numberEquality,
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[i:\mBbbN{}n]. \mforall{}[k:|r|]. (|matrix-mul-col(r;k;i;M)| = (k * |M|))
Date html generated:
2018_05_21-PM-09_36_31
Last ObjectModification:
2017_12_11-PM-04_46_39
Theory : matrices
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