Nuprl Lemma : vector-mul-mul
∀[i:Type]. ∀[r:Rng]. ∀[c1,c2:|r|]. ∀[a:i ⟶ |r|]. ((c1*(c2*a)) = (c1 * c2*a) ∈ (i ⟶ |r|))
Proof
Definitions occuring in Statement :
vector-mul: (c*a),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
true: True,
rng: Rng,
vector-mul: (c*a),
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
rng_times_wf,
infix_ap_wf,
rng_wf,
rng_car_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_assoc,
iff_weakening_equal
Rules used in proof :
natural_numberEquality,
applyEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cumulativity,
functionEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
functionExtensionality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[i:Type]. \mforall{}[r:Rng]. \mforall{}[c1,c2:|r|]. \mforall{}[a:i {}\mrightarrow{} |r|]. ((c1*(c2*a)) = (c1 * c2*a))
Date html generated:
2018_05_21-PM-09_40_38
Last ObjectModification:
2018_01_09-PM-00_59_59
Theory : matrices
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