Nuprl Lemma : matrix-det-is-det-fun
∀[r:CRng]. ∀[n:ℕ]. (λM.|M| ∈ det-fun(r;n))
Proof
Definitions occuring in Statement :
det-fun: det-fun(r;n),
matrix-det: |M|,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x],
crng: CRng
Definitions unfolded in proof :
infix_ap: x f y,
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
not: ¬A,
false: False,
so_lambda: λ2x y.t[x; y],
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
cand: A c∧ B,
and: P ∧ Q,
rng: Rng,
nat: ℕ,
crng: CRng,
det-fun: det-fun(r;n),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
crng_wf,
nat_wf,
rng_zero_wf,
matrix-mul-row_wf,
matrix-swap-cols_wf,
all_wf,
identity-matrix_wf,
matrix-swap-rows_wf,
det-equal-rows,
det-swap-rows,
matrix-ap_wf,
mx_wf,
rng_plus_wf,
det-add-row,
rng_times_wf,
infix_ap_wf,
det-mul-row,
int_seg_wf,
not_wf,
rng_minus_wf,
det-swap-cols,
iff_weakening_equal,
rng_one_wf,
det-id,
rng_car_wf,
true_wf,
squash_wf,
equal_wf,
matrix_wf,
matrix-det_wf
Rules used in proof :
isect_memberEquality,
axiomEquality,
productEquality,
functionEquality,
functionExtensionality,
int_eqEquality,
intEquality,
lambdaFormation,
independent_pairFormation,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
sqequalRule,
because_Cache,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaEquality,
dependent_set_memberEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. (\mlambda{}M.|M| \mmember{} det-fun(r;n))
Date html generated:
2018_05_21-PM-09_37_03
Last ObjectModification:
2017_12_13-PM-00_17_45
Theory : matrices
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