Nuprl Lemma : real-matrix-times_wf
∀[n,a,b:ℕ]. ∀[A:ℝ(a × n)]. ∀[B:ℝ(n × b)]. ((A*B) ∈ ℝ(a × b))
Proof
Definitions occuring in Statement :
real-matrix-times: (A*B),
rmatrix: ℝ(a × b),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
rmatrix: ℝ(a × b),
uall: ∀[x:A]. B[x],
member: t ∈ T,
real-matrix-times: (A*B),
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_apply: x[s]
Lemmas referenced :
rsum_wf,
subtract_wf,
rmul_wf,
subtract-add-cancel,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_wf,
real_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
closedConclusion,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
hypothesisEquality,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
imageElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
addEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[n,a,b:\mBbbN{}]. \mforall{}[A:\mBbbR{}(a \mtimes{} n)]. \mforall{}[B:\mBbbR{}(n \mtimes{} b)]. ((A*B) \mmember{} \mBbbR{}(a \mtimes{} b))
Date html generated:
2019_10_30-AM-08_15_48
Last ObjectModification:
2019_09_18-PM-06_56_29
Theory : reals
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