Nuprl Lemma : rv-pos-angle-implies
∀n:ℕ. ∀a,b,c:ℝ^n.
(rv-pos-angle(n;a;b;c)
⇒ (rv-pos-angle(n;c;b;a)
∧ rv-pos-angle(n;c;a;b)
∧ a ≠ c
∧ (¬a-b-c)
∧ (∀z:ℝ^n. (z ≠ b ⇒ (¬rv-pos-angle(n;a;b;z)) ⇒ rv-pos-angle(n;z;b;c)))))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
rv-pos-angle-symmetry,
rv-pos-angle-permute,
rv-pos-angle-implies-separated,
rv-pos-angle-not-between,
rv-pos-angle-shift,
not_wf,
rv-pos-angle_wf,
real-vec-sep_wf,
real-vec_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
because_Cache,
isectElimination,
independent_isectElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n.
(rv-pos-angle(n;a;b;c)
{}\mRightarrow{} (rv-pos-angle(n;c;b;a)
\mwedge{} rv-pos-angle(n;c;a;b)
\mwedge{} a \mneq{} c
\mwedge{} (\mneg{}a-b-c)
\mwedge{} (\mforall{}z:\mBbbR{}\^{}n. (z \mneq{} b {}\mRightarrow{} (\mneg{}rv-pos-angle(n;a;b;z)) {}\mRightarrow{} rv-pos-angle(n;z;b;c)))))
Date html generated:
2017_10_03-AM-11_07_17
Last ObjectModification:
2017_03_02-PM-05_32_36
Theory : reals
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