Nuprl Lemma : rv-Tsep-alt'
∀n:ℕ. ∀a,b,c,d:ℝ^n. ((¬¬(∃u,v,w:ℝ^n. ((¬(u ≠ v ∧ v ≠ w ∧ (¬u-v-w))) ∧ ab=uv ∧ cd=uw))) ⇒ a ≠ b ⇒ c ≠ d)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rv-Tsep-alt,
rv-T_wf,
not_wf,
real-vec-sep_wf,
rv-between_wf,
exists_wf,
real-vec_wf,
rv-congruent_wf,
rv-T-iff
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
productEquality,
sqequalRule,
lambdaEquality,
because_Cache,
allFunctionality,
lambdaFormation,
independent_functionElimination,
impliesFunctionality,
existsFunctionality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
andLevelFunctionality,
existsLevelFunctionality,
impliesLevelFunctionality,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,d:\mBbbR{}\^{}n.
((\mneg{}\mneg{}(\mexists{}u,v,w:\mBbbR{}\^{}n. ((\mneg{}(u \mneq{} v \mwedge{} v \mneq{} w \mwedge{} (\mneg{}u-v-w))) \mwedge{} ab=uv \mwedge{} cd=uw))) {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} c \mneq{} d)
Date html generated:
2016_10_28-AM-07_30_02
Last ObjectModification:
2016_10_26-PM-02_13_52
Theory : reals
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