Nuprl Lemma : ip-between_functionality
∀[rv:InnerProductSpace]. ∀[a,b,c,a2,b2,c2:Point]. (a ≡ a2 ⇒ b ≡ b2 ⇒ c ≡ c2 ⇒ {a_b_c ⇐⇒ a2_b2_c2})
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
uall: ∀[x:A]. B[x],
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
ip-between: a_b_c,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ip-between_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,
req_witness,
radd_wf,
rmul_wf,
rv-norm_wf,
rv-sub_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rv-ip_wf,
ss-point_wf,
req_functionality,
radd_functionality,
rv-ip_functionality,
rv-sub_functionality,
rmul_functionality,
rv-norm_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
independent_functionElimination,
isect_memberEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c,a2,b2,c2:Point].
(a \mequiv{} a2 {}\mRightarrow{} b \mequiv{} b2 {}\mRightarrow{} c \mequiv{} c2 {}\mRightarrow{} \{a\_b\_c \mLeftarrow{}{}\mRightarrow{} a2\_b2\_c2\})
Date html generated:
2017_10_04-PM-11_57_00
Last ObjectModification:
2017_03_09-PM-05_27_01
Theory : inner!product!spaces
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