Nuprl Lemma : ip-triangle_functionality
∀rv:InnerProductSpace. ∀a,b,c,a2,b2,c2:Point. (a ≡ a2 ⇒ b ≡ b2 ⇒ c ≡ c2 ⇒ {Δ(a;b;c) ⇐⇒ Δ(a2;b2;c2)})
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
ip-triangle: Δ(a;b;c),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
ip-triangle_wf,
ss-eq_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,
rabs_wf,
rv-ip_wf,
rv-sub_wf,
rmul_wf,
rv-norm_wf,
rless_functionality,
rabs_functionality,
rv-ip_functionality,
rv-sub_functionality,
rmul_functionality,
rv-norm_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c,a2,b2,c2:Point.
(a \mequiv{} a2 {}\mRightarrow{} b \mequiv{} b2 {}\mRightarrow{} c \mequiv{} c2 {}\mRightarrow{} \{\mDelta{}(a;b;c) \mLeftarrow{}{}\mRightarrow{} \mDelta{}(a2;b2;c2)\})
Date html generated:
2017_10_04-PM-11_58_03
Last ObjectModification:
2017_03_09-PM-05_32_31
Theory : inner!product!spaces
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