Nuprl Lemma : real-matrix-times_functionality
∀[n,a,b:ℕ]. ∀[A1,A2:ℝ(a × n)]. ∀[B1,B2:ℝ(n × b)]. ((A1*B1) ≡ (A2*B2)) supposing (A1 ≡ A2 and B1 ≡ B2)
Proof
Definitions occuring in Statement :
real-matrix-times: (A*B),
reqmatrix: X ≡ Y,
rmatrix: ℝ(a × b),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
reqmatrix: X ≡ Y,
rmatrix: ℝ(a × b),
real-matrix-times: (A*B),
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_wf,
req_witness,
real-matrix-times_wf,
subtype_rel_self,
real_wf,
reqmatrix_wf,
rmatrix_wf,
istype-nat,
rsum_wf,
subtract_wf,
rmul_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
istype-le,
istype-less_than,
req_weakening,
req_functionality,
rsum_functionality2,
rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
functionEquality,
imageElimination,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
productIsType,
addEquality
Latex:
\mforall{}[n,a,b:\mBbbN{}]. \mforall{}[A1,A2:\mBbbR{}(a \mtimes{} n)]. \mforall{}[B1,B2:\mBbbR{}(n \mtimes{} b)].
((A1*B1) \mequiv{} (A2*B2)) supposing (A1 \mequiv{} A2 and B1 \mequiv{} B2)
Date html generated:
2019_10_30-AM-08_16_14
Last ObjectModification:
2019_09_19-PM-00_54_11
Theory : reals
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