Nuprl Lemma : Binet-Cauchy-identity
∀[r:CRng]. ∀[a,b,c,d:ℕ3 ⟶ |r|]. (((a x b) . (c x d)) = (((a . c) * (b . d)) +r (-r ((a . d) * (b . c)))) ∈ |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,
scalar-product: (a . b),
cross-product: (a x b),
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
select: L[n],
cons: [a / b],
bfalse: ff,
crng: CRng,
rng: Rng,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
ringeq_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
int_seg_wf,
rng_car_wf,
crng_wf,
infix_ap_wf,
rng_plus_wf,
rng_zero_wf,
rng_times_wf,
false_wf,
lelt_wf,
rng_minus_wf,
itermAdd_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
itermMinus_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
int-to-ring_wf,
ring_term_value_add_lemma,
ring_term_value_const_lemma,
int-to-ring-zero,
ring_term_value_mul_lemma,
ring_term_value_var_lemma,
ring_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
imageMemberEquality,
baseClosed,
productElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[a,b,c,d:\mBbbN{}3 {}\mrightarrow{} |r|].
(((a x b) . (c x d)) = (((a . c) * (b . d)) +r (-r ((a . d) * (b . c)))))
Date html generated:
2018_05_21-PM-09_42_24
Last ObjectModification:
2018_05_19-PM-04_33_57
Theory : matrices
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