Nuprl Lemma : rv-extend-1
∀n:ℕ. ∀a,b:ℝ^n. (a ≠ b ⇒ (∀s:{s:ℝ| r0 ≤ s} . ∃x:ℝ^n. ((d(b;x) = s) ∧ ((r0 < s) ⇒ a-b-x))))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
real-vec-between: a-b-c,
real-vec: ℝ^n,
rleq: x ≤ y,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-sep: a ≠ b,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
sq_stable: SqStable(P),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rgt: x > y,
real-vec-between: a-b-c,
req-vec: req-vec(n;x;y),
real-vec-mul: a*X,
real-vec-add: X + Y,
nat: ℕ,
real-vec: ℝ^n
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbR{}\^{}n. (a \mneq{} b {}\mRightarrow{} (\mforall{}s:\{s:\mBbbR{}| r0 \mleq{} s\} . \mexists{}x:\mBbbR{}\^{}n. ((d(b;x) = s) \mwedge{} ((r0 < s) {}\mRightarrow{} a-b-x))))
Date html generated:
2020_05_20-PM-00_50_51
Last ObjectModification:
2020_01_03-AM-00_24_17
Theory : reals
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