Nuprl Lemma : ip-extend
∀rv:InnerProductSpace. ∀a:Point. ∀b:{b:Point| a # b} . ∀c,d:Point. (∃x:{Point| (a_b_x ∧ bx=cd)})
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
cand: A c∧ B,
ip-congruent: ab=cd,
req: x = y,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
ip-extend-lemma,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
ip-between_wf,
ip-congruent_wf,
ss-point_wf,
set_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-sep_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyEquality,
sqequalRule,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
productEquality,
instantiate,
independent_isectElimination,
lambdaEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a:Point. \mforall{}b:\{b:Point| a \# b\} . \mforall{}c,d:Point. (\mexists{}x:\{Point| (a\_b\_x \mwedge{} bx=cd)\})
Date html generated:
2017_10_05-AM-00_12_41
Last ObjectModification:
2017_03_19-PM-01_15_57
Theory : inner!product!spaces
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