Nuprl Lemma : det-transpose
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. (|M'| = |M| ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-transpose: M',
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
matrix-det: |M|,
crng: CRng,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
rng: Rng,
subtype_rel: A ⊆r B,
false: False,
implies: P ⇒ Q,
not: ¬A,
so_apply: x[s],
injection: A →⟶ B,
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
matrix-transpose: M',
matrix-ap: M[i,j],
mx: matrix(M[x; y]),
lelt: i ≤ j < k,
int_seg: {i..j-},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
let: let
Lemmas referenced :
rng_lsum_functionality_wrt_permutation,
int_seg_wf,
inject_wf,
let_wf,
rng_car_wf,
permutation-sign_wf,
equal-wf-base,
int_subtype_base,
rng_minus_wf,
rng_prod_wf,
matrix-ap_wf,
permutations-list_wf,
list_wf,
injection_wf,
no_repeats_wf,
all_wf,
l_member_wf,
map_wf,
funinv_wf2,
funinv-permutes-permutations-list,
equal_wf,
squash_wf,
true_wf,
rng_lsum_map,
funinv_wf3,
iff_weakening_equal,
matrix_wf,
nat_wf,
crng_wf,
rng_lsum_wf,
rng_wf,
matrix-transpose_wf,
sign-inverse,
rng_prod_injection,
rng_sig_wf,
int_seg_properties,
nat_properties,
decidable__le,
lelt_wf,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
funinv-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
setEquality,
functionEquality,
natural_numberEquality,
because_Cache,
functionExtensionality,
applyEquality,
sqequalRule,
lambdaEquality,
lambdaFormation,
int_eqEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
productEquality,
independent_isectElimination,
dependent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
cumulativity,
independent_pairFormation,
dependent_set_memberEquality,
unionElimination,
applyLambdaEquality,
voidElimination,
approximateComputation,
dependent_pairFormation,
voidEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. (|M'| = |M|)
Date html generated:
2018_05_21-PM-09_35_41
Last ObjectModification:
2017_12_11-PM-03_32_59
Theory : matrices
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