Nuprl Lemma : rv-inner-Pasch''
∀n:ℕ. ∀a,b,c,p,q:ℝ^n.
(a-p-c
⇒ b-q-c
⇒ (∃x:ℝ^n. ((¬(a ≠ x ∧ x ≠ q ∧ (¬a-x-q))) ∧ (¬(b ≠ x ∧ x ≠ p ∧ (¬b-x-p))) ∧ (a ≠ q ⇒ a-x-q) ∧ (b ≠ p ⇒ b-x-p))))
Proof
Definitions occuring in Statement :
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],
not: ¬A,
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,
not: ¬A,
rv-T: rv-T(n;a;b;c),
real-vec-be: real-vec-be(n;a;b;c),
top: Top,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-inner-Pasch',
member_rccint_lemma,
real-vec-sep_wf,
not_wf,
rv-between_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,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
isectElimination,
productEquality,
independent_pairFormation,
because_Cache,
functionEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,p,q:\mBbbR{}\^{}n.
(a-p-c
{}\mRightarrow{} b-q-c
{}\mRightarrow{} (\mexists{}x:\mBbbR{}\^{}n
((\mneg{}(a \mneq{} x \mwedge{} x \mneq{} q \mwedge{} (\mneg{}a-x-q)))
\mwedge{} (\mneg{}(b \mneq{} x \mwedge{} x \mneq{} p \mwedge{} (\mneg{}b-x-p)))
\mwedge{} (a \mneq{} q {}\mRightarrow{} a-x-q)
\mwedge{} (b \mneq{} p {}\mRightarrow{} b-x-p))))
Date html generated:
2016_10_26-AM-10_50_41
Last ObjectModification:
2016_10_21-PM-01_49_49
Theory : reals
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