Nuprl Lemma : scalar-product-comm
∀[r:CRng]. ∀[n:ℕ]. ∀[a,b:ℕn ⟶ |r|]. ((a . b) = (b . a) ∈ |r|)
Proof
Definitions occuring in Statement :
scalar-product: (a . b),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
so_apply: x[s],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
nat: ℕ,
crng: CRng,
rng: Rng,
prop: ℙ,
squash: ↓T,
scalar-product: (a . b),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
crng_wf,
nat_wf,
iff_weakening_equal,
rng_times_wf,
infix_ap_wf,
crng_times_comm,
equal_wf,
rng_car_wf,
int_seg_wf,
true_wf,
squash_wf,
rng_sum_wf
Rules used in proof :
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
functionExtensionality,
universeEquality,
sqequalRule,
natural_numberEquality,
because_Cache,
intEquality,
rename,
setElimination,
functionEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbN{}n {}\mrightarrow{} |r|]. ((a . b) = (b . a))
Date html generated:
2018_05_21-PM-09_41_57
Last ObjectModification:
2017_12_18-PM-05_11_08
Theory : matrices
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