Nuprl Lemma : rcp-Binet-Cauchy

[a,b,c,d:ℝ^3].  ((a b)⋅(c d) ((a⋅b⋅d) a⋅b⋅c))


Proof




Definitions occuring in Statement :  rcp: (a b) dot-product: x⋅y real-vec: ^n rsub: y req: y rmul: b uall: [x:A]. B[x] natural_number: $n
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat: le: A ≤ B and: P ∧ Q less_than': less_than'(a;b) false: False not: ¬A implies:  Q prop: subtype_rel: A ⊆B int_seg: {i..j-} lelt: i ≤ j < k less_than: a < b squash: T true: True real-vec: ^n rcp: (a b) select: L[n] cons: [a b] subtract: m uiff: uiff(P;Q) uimplies: 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


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