Nuprl Lemma : Binet-Cauchy-corollary
∀[r:CRng]. ∀[a,b:ℕ3 ⟶ |r|]. (((a x b) . (a x b)) = (((a . a) * (b . b)) +r (-r ((a . b) * (b . a)))) ∈ |r|)
Proof
Definitions occuring in Statement :
scalar-product: (a . b),
cross-product: (a x b),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_minus: -r,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng,
rng: Rng
Lemmas referenced :
Binet-Cauchy-identity,
int_seg_wf,
rng_car_wf,
crng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionEquality,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache
Latex:
\mforall{}[r:CRng]. \mforall{}[a,b:\mBbbN{}3 {}\mrightarrow{} |r|].
(((a x b) . (a x b)) = (((a . a) * (b . b)) +r (-r ((a . b) * (b . a)))))
Date html generated:
2018_05_21-PM-09_42_26
Last ObjectModification:
2018_05_19-PM-04_34_02
Theory : matrices
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