Nuprl Lemma : rv-congruent_functionality2
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2,d1,d2:ℝ^n. ((¬a1 ≠ a2) ⇒ (¬b1 ≠ b2) ⇒ (¬c1 ≠ c2) ⇒ (¬d1 ≠ d2) ⇒ (a1b1=c1d1 ⇐⇒ a2b2=c2d2))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
rv-congruent_functionality,
not-real-vec-sep-iff-eq,
not_wf,
real-vec-sep_wf,
real-vec_wf,
nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
productElimination,
independent_isectElimination,
because_Cache
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a1,a2,b1,b2,c1,c2,d1,d2:\mBbbR{}\^{}n.
((\mneg{}a1 \mneq{} a2) {}\mRightarrow{} (\mneg{}b1 \mneq{} b2) {}\mRightarrow{} (\mneg{}c1 \mneq{} c2) {}\mRightarrow{} (\mneg{}d1 \mneq{} d2) {}\mRightarrow{} (a1b1=c1d1 \mLeftarrow{}{}\mRightarrow{} a2b2=c2d2))
Date html generated:
2016_10_26-AM-10_30_55
Last ObjectModification:
2016_09_25-PM-01_10_31
Theory : reals
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