Nuprl Lemma : rcp-Binet-Cauchy
∀[a,b,c,d:ℝ^3]. ((a x b)⋅(c x d) = ((a⋅c * b⋅d) - a⋅d * b⋅c))
Proof
Definitions occuring in Statement :
rcp: (a x b),
dot-product: x⋅y,
real-vec: ℝ^n,
rsub: x - y,
req: x = y,
rmul: a * b,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
real-vec: ℝ^n,
rcp: (a x b),
select: L[n],
cons: [a / b],
subtract: n - m,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
req_witness,
dot-product_wf,
false_wf,
le_wf,
rcp_wf,
rsub_wf,
rmul_wf,
real-vec_wf,
radd_wf,
lelt_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
req_functionality,
r3-dot-product,
rsub_functionality,
rmul_functionality,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
hypothesisEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality,
applyEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[a,b,c,d:\mBbbR{}\^{}3]. ((a x b)\mcdot{}(c x d) = ((a\mcdot{}c * b\mcdot{}d) - a\mcdot{}d * b\mcdot{}c))
Date html generated:
2018_05_22-PM-02_42_32
Last ObjectModification:
2018_05_09-PM-01_35_53
Theory : reals
Home
Index