Nuprl Lemma : rv-Gsep
∀n:ℕ. ∀a,b,c:ℝ^n. ∀d:{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,
set: {x:A| B[x]}
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
false: False,
not: ¬A,
exists: ∃x:A. B[x],
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rv-T-iff,
rv-T_wf,
nat_wf,
rv-congruent_wf,
rv-between_wf,
real-vec-sep_wf,
exists_wf,
not_wf,
real-vec_wf,
set_wf,
rv-Tsep-alt
Rules used in proof :
levelHypothesis,
impliesLevelFunctionality,
existsLevelFunctionality,
andLevelFunctionality,
independent_pairFormation,
productElimination,
existsFunctionality,
impliesFunctionality,
addLevel,
voidElimination,
productEquality,
because_Cache,
lambdaEquality,
sqequalRule,
isectElimination,
independent_functionElimination,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n. \mforall{}d:\{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))\} .
(a \mneq{} b {}\mRightarrow{} c \mneq{} d)
Date html generated:
2016_10_28-AM-07_30_16
Last ObjectModification:
2016_10_26-PM-06_46_07
Theory : reals
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