Nuprl Lemma : rv-inner-Pasch3
∀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
(rv-T(n;a;x;q)
∧ rv-T(n;b;x;p)
∧ (a ≠ q ⇒ x ≠ a)
∧ ((a ≠ q ∧ p ≠ c ∧ b ≠ q) ⇒ x ≠ q)
∧ ((b ≠ p ∧ b ≠ q) ⇒ x ≠ b)
∧ ((b ≠ p ∧ q ≠ c) ⇒ x ≠ p))))
Proof
Definitions occuring in Statement :
rv-T: rv-T(n;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],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
rv-T: rv-T(n;a;b;c),
and: P ∧ Q,
false: False,
or: P ∨ Q,
rv-between: a-b-c,
real-vec-be: real-vec-be(n;a;b;c),
exists: ∃x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
real-vec-sep: a ≠ b,
iff: P ⇐⇒ Q,
real-vec-between: a-b-c,
cand: A c∧ B,
rev_implies: P ⇐ Q,
rge: x ≥ y,
guard: {T},
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
req_int_terms: t1 ≡ t2,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
real-vec-mul: a*X,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
i-member: r ∈ I,
rcoint: [l, u),
rccint: [l, u]
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
(rv-T(n;a;x;q)
\mwedge{} rv-T(n;b;x;p)
\mwedge{} (a \mneq{} q {}\mRightarrow{} x \mneq{} a)
\mwedge{} ((a \mneq{} q \mwedge{} p \mneq{} c \mwedge{} b \mneq{} q) {}\mRightarrow{} x \mneq{} q)
\mwedge{} ((b \mneq{} p \mwedge{} b \mneq{} q) {}\mRightarrow{} x \mneq{} b)
\mwedge{} ((b \mneq{} p \mwedge{} q \mneq{} c) {}\mRightarrow{} x \mneq{} p))))
Date html generated:
2020_05_20-PM-00_52_15
Last ObjectModification:
2020_01_06-PM-00_44_03
Theory : reals
Home
Index