Nuprl Lemma : ip-triangle-implies
∀rv:InnerProductSpace. ∀a,b,c:Point.
(Δ(a;b;c) ⇒ (Δ(c;b;a) ∧ Δ(c;a;b) ∧ a # c ∧ (¬a_b_c) ∧ (∀z:Point. (z # b ⇒ (¬Δ(a;b;z)) ⇒ Δ(z;b;c)))))
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
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,
subtype_rel: A ⊆r B
Lemmas referenced :
ip-triangle-symmetry,
ip-triangle-permute,
ip-triangle-implies-separated,
ip-triangle-not-between,
ip-triangle-shift,
not_wf,
ip-triangle_wf,
ss-sep_wf,
ss-point_wf,
inner-product-space_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,
isectElimination,
independent_isectElimination,
because_Cache,
applyEquality,
sqequalRule
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point.
(\mDelta{}(a;b;c)
{}\mRightarrow{} (\mDelta{}(c;b;a) \mwedge{} \mDelta{}(c;a;b) \mwedge{} a \# c \mwedge{} (\mneg{}a\_b\_c) \mwedge{} (\mforall{}z:Point. (z \# b {}\mRightarrow{} (\mneg{}\mDelta{}(a;b;z)) {}\mRightarrow{} \mDelta{}(z;b;c)))))
Date html generated:
2017_10_04-PM-11_59_06
Last ObjectModification:
2017_08_10-PM-03_38_19
Theory : inner!product!spaces
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