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