Nuprl Lemma : rv-line-circle
∀n:ℕ. ∀a,b,p,q:ℝ^n.
(p ≠ q
⇒ (∀x:{x:ℝ^n| ap=ax ∧ (¬(a ≠ x ∧ x ≠ b ∧ (¬a-x-b)))} . ∀y:{y:ℝ^n| aq=ay ∧ (¬(a ≠ b ∧ b ≠ y ∧ (¬a-b-y)))} .
∃u:{u:ℝ^n| ab=au ∧ (¬(q ≠ u ∧ u ≠ p ∧ (¬q-u-p)))}
∃v:{v:ℝ^n| ab=av ∧ (¬(q ≠ p ∧ p ≠ v ∧ (¬q-p-v)))} . (a-x-b ⇒ (q-p-v ∧ (a-b-y ⇒ q-u-p)))))
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 :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rv-congruent: ab=cd,
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
req_int_terms: t1 ≡ t2,
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
rv-between: a-b-c,
real-vec-sep: a ≠ b
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,p,q:\mBbbR{}\^{}n.
(p \mneq{} q
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}\^{}n| ap=ax \mwedge{} (\mneg{}(a \mneq{} x \mwedge{} x \mneq{} b \mwedge{} (\mneg{}a-x-b)))\} . \mforall{}y:\{y:\mBbbR{}\^{}n|
aq=ay \mwedge{} (\mneg{}(a \mneq{} b \mwedge{} b \mneq{} y \mwedge{} (\mneg{}a-b-y)))\} \000C.
\mexists{}u:\{u:\mBbbR{}\^{}n| ab=au \mwedge{} (\mneg{}(q \mneq{} u \mwedge{} u \mneq{} p \mwedge{} (\mneg{}q-u-p)))\}
\mexists{}v:\{v:\mBbbR{}\^{}n| ab=av \mwedge{} (\mneg{}(q \mneq{} p \mwedge{} p \mneq{} v \mwedge{} (\mneg{}q-p-v)))\} . (a-x-b {}\mRightarrow{} (q-p-v \mwedge{} (a-b-y {}\mRightarrow{} q-u-p)))))
Date html generated:
2020_05_20-PM-00_55_20
Last ObjectModification:
2020_03_19-PM-02_21_46
Theory : reals
Home
Index