Nuprl Lemma : ip-triangle-shift
∀rv:InnerProductSpace. ∀a,b,c:Point. (Δ(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),
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not-ip-triangle,
not_wf,
ip-triangle_wf,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-point_wf,
ip-triangle-implies-separated2,
ip-triangle-linearity,
rv-add_wf,
rv-mul_wf,
rv-sub_wf,
ip-triangle_functionality,
ss-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. (\mDelta{}(a;b;c) {}\mRightarrow{} (\mforall{}z:Point. (z \# b {}\mRightarrow{} (\mneg{}\mDelta{}(a;b;z)) {}\mRightarrow{} \mDelta{}(z;b;c))))
Date html generated:
2017_10_04-PM-11_58_58
Last ObjectModification:
2017_03_10-PM-02_44_48
Theory : inner!product!spaces
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