Nuprl Lemma : rv-congruent-iff2
∀n:ℕ. ∀a,b,c,d:ℝ^n. (ab=cd ⇐⇒ ¬((d(a;b) < d(c;d)) ∨ (d(c;d) < d(a;b))))
Proof
Definitions occuring in Statement :
rv-congruent: ab=cd,
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rless: x < y,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
rv-congruent: ab=cd,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
guard: {T}
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. (ab=cd \mLeftarrow{}{}\mRightarrow{} \mneg{}((d(a;b) < d(c;d)) \mvee{} (d(c;d) < d(a;b))))
Date html generated:
2020_05_20-PM-01_03_27
Last ObjectModification:
2019_12_11-PM-00_54_08
Theory : reals
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