Nuprl Lemma : matrix-det-is-determinant
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. (|M| = (determinant(n;r) M) ∈ |r|)
Proof
Definitions occuring in Statement :
determinant: determinant(n;r),
matrix-det: |M|,
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
rng: Rng,
crng: CRng,
nat: ℕ,
and: P ∧ Q,
true: True,
squash: ↓T,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
crng_wf,
nat_wf,
matrix-det-is-det-fun,
det-fun-is-determinant,
matrix_wf,
rng_times_one,
determinant_wf,
rng_times_wf,
rng_car_wf,
equal_wf,
squash_wf,
true_wf,
det-id,
iff_weakening_equal
Rules used in proof :
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
rename,
setElimination,
equalityTransitivity,
sqequalRule,
because_Cache,
functionExtensionality,
applyEquality,
applyLambdaEquality,
productElimination,
natural_numberEquality,
lambdaEquality,
imageElimination,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. (|M| = (determinant(n;r) M))
Date html generated:
2018_05_21-PM-09_38_07
Last ObjectModification:
2017_12_13-PM-01_41_42
Theory : matrices
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