Nuprl Lemma : det-times
∀[r:CRng]. ∀[n:ℕ]. ∀[A,B:Matrix(n;n;r)]. (|(A*B)| = (|A| * |B|) ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
rng: Rng,
crng: CRng,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
not: ¬A,
false: False,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
det-fun: det-fun(r;n),
nequal: a ≠ b ∈ T ,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
top: Top,
matrix-times: (M*N),
matrix-mul-row: matrix-mul-row(r;k;i;M),
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix-swap-rows: matrix-swap-rows(M;i;j)
Lemmas referenced :
crng_wf,
nat_wf,
matrix_wf,
rng_zero_wf,
rng_minus_wf,
matrix-ap_wf,
rng_plus_wf,
mx_wf,
rng_times_wf,
infix_ap_wf,
matrix-mul-row_wf,
all_wf,
matrix-swap-rows_wf,
equal_wf,
not_wf,
int_seg_wf,
rng_car_wf,
matrix-times_wf,
matrix-det_wf,
det-mul-row,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
rng_times_assoc,
iff_weakening_equal,
rng_sum_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
rng_times_sum_l,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
rng_sig_wf,
matrix_ap_mx_lemma,
rng_wf,
true_wf,
squash_wf,
lelt_wf,
false_wf,
det-add-row,
rng_sum_plus,
rng_times_over_plus,
det-swap-rows,
det-equal-rows,
det-fun-is-determinant,
crng_times_comm,
matrix-det-is-determinant,
determinant_wf,
matrix-times-id-left
Rules used in proof :
axiomEquality,
isect_memberEquality,
sqequalRule,
because_Cache,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
int_eqEquality,
functionExtensionality,
applyEquality,
productEquality,
intEquality,
functionEquality,
independent_pairFormation,
natural_numberEquality,
lambdaFormation,
lambdaEquality,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
int_eqReduceFalseSq,
cumulativity,
instantiate,
promote_hyp,
baseClosed,
imageMemberEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
universeEquality,
int_eqReduceTrueSq,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
voidEquality,
voidElimination,
dependent_functionElimination,
imageElimination,
hyp_replacement,
levelHypothesis,
equalityUniverse,
applyLambdaEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[A,B:Matrix(n;n;r)]. (|(A*B)| = (|A| * |B|))
Date html generated:
2018_05_21-PM-09_38_18
Last ObjectModification:
2017_12_13-PM-05_32_22
Theory : matrices
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