Nuprl Lemma : not-rv-pos-angle-iff
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. uiff(¬rv-pos-angle(n;a;b;c);¬(a ≠ b ∧ b ≠ c ∧ c ≠ a ∧ (¬a-b-c) ∧ (¬b-c-a) ∧ (¬c-a-b)))
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,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
prop: ℙ,
cand: A c∧ B,
iff: P ⇐⇒ Q
Lemmas referenced :
not-rv-pos-angle-implies,
real-vec-sep_wf,
not_wf,
rv-between_wf,
rv-pos-angle_wf,
real-vec_wf,
nat_wf,
rv-pos-angle-permute,
rv-pos-angle-implies-separated,
real-vec-sep-symmetry,
rv-pos-angle-not-between
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
voidElimination,
productEquality,
sqequalRule,
lambdaEquality,
because_Cache,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b,c:\mBbbR{}\^{}n].
uiff(\mneg{}rv-pos-angle(n;a;b;c);\mneg{}(a \mneq{} b \mwedge{} b \mneq{} c \mwedge{} c \mneq{} a \mwedge{} (\mneg{}a-b-c) \mwedge{} (\mneg{}b-c-a) \mwedge{} (\mneg{}c-a-b)))
Date html generated:
2017_10_03-AM-11_08_43
Last ObjectModification:
2017_03_07-PM-00_11_38
Theory : reals
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