Nuprl Lemma : not-ip-triangle-iff
∀rv:InnerProductSpace. ∀a,b,c:Point. (¬Δ(a;b;c) ⇐⇒ ¬((¬a_b_c) ∧ (¬b_c_a) ∧ (¬c_a_b)))
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
ss-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
cand: A c∧ B
Lemmas referenced :
not_wf,
ip-between_wf,
ip-triangle_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
not-ip-triangle-implies,
ip-triangle-not-between,
ip-triangle-permute
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
productEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. (\mneg{}\mDelta{}(a;b;c) \mLeftarrow{}{}\mRightarrow{} \mneg{}((\mneg{}a\_b\_c) \mwedge{} (\mneg{}b\_c\_a) \mwedge{} (\mneg{}c\_a\_b)))
Date html generated:
2017_10_05-AM-00_00_49
Last ObjectModification:
2017_03_11-AM-01_26_46
Theory : inner!product!spaces
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