Nuprl Lemma : rv-inner-Pasch2
∀n:ℕ. ∀a,b,c,p,q:ℝ^n.
(a ≠ p
⇒ b ≠ c
⇒ rv-T(n;a;p;c)
⇒ rv-T(n;b;q;c)
⇒ (∃x:ℝ^n
((((a ≠ q ∧ p ≠ c ∧ b ≠ q) ⇒ a-x-q) ∧ ((b ≠ p ∧ q ≠ c ∧ b ≠ q) ⇒ b-x-p)) ∧ rv-T(n;a;x;q) ∧ rv-T(n;b;x;p))))
Proof
Definitions occuring in Statement :
rv-T: rv-T(n;a;b;c),
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-inner-Pasch3,
rv-between-iff,
real-vec-sep-symmetry,
real-vec-sep_wf,
rv-between_wf,
rv-T_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,
productElimination,
dependent_pairFormation,
independent_pairFormation,
because_Cache,
productEquality,
isectElimination,
functionEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,p,q:\mBbbR{}\^{}n.
(a \mneq{} p
{}\mRightarrow{} b \mneq{} c
{}\mRightarrow{} rv-T(n;a;p;c)
{}\mRightarrow{} rv-T(n;b;q;c)
{}\mRightarrow{} (\mexists{}x:\mBbbR{}\^{}n
((((a \mneq{} q \mwedge{} p \mneq{} c \mwedge{} b \mneq{} q) {}\mRightarrow{} a-x-q) \mwedge{} ((b \mneq{} p \mwedge{} q \mneq{} c \mwedge{} b \mneq{} q) {}\mRightarrow{} b-x-p))
\mwedge{} rv-T(n;a;x;q)
\mwedge{} rv-T(n;b;x;p))))
Date html generated:
2016_10_26-AM-10_50_15
Last ObjectModification:
2016_10_24-PM-03_49_28
Theory : reals
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