Nuprl Lemma : ip-line-circle
∀rv:InnerProductSpace. ∀a:Point(rv). ∀b:{b:Point(rv)| a # b} . ∀p:{p:Point(rv)| ab ≥ ap} . ∀q:{q:Point(rv)|
p # q ∧ aq ≥ ab} .
∃u:{u:Point(rv)| ab=au ∧ q_u_p} . (∃v:Point(rv) [(ab=av ∧ q_p_v)])
Proof
Definitions occuring in Statement :
ip-ge: cd ≥ ab,
ip-between: a_b_c,
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
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,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
ip-ge: cd ≥ ab,
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
ip-line-circle-1,
sq_stable__rv-sep-ext,
Error :ss-sep_wf,
ip-ge_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rv-norm-difference-symmetry,
rv-norm_wf,
rv-sub_wf,
rleq_functionality,
req_weakening,
ip-ge-iff,
sq_stable__not,
not_wf,
ip-between_wf,
ip-congruent_wf,
sq_stable__ip-congruent,
sq_stable__ip-between
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
setIsType,
inhabitedIsType,
productIsType,
universeIsType,
isectElimination,
applyEquality,
because_Cache,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
productEquality,
dependent_pairFormation_alt,
dependent_set_memberFormation_alt,
independent_pairFormation
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a:Point(rv). \mforall{}b:\{b:Point(rv)| a \# b\} . \mforall{}p:\{p:Point(rv)| ab \mgeq{} ap\} .
\mforall{}q:\{q:Point(rv)| p \# q \mwedge{} aq \mgeq{} ab\} .
\mexists{}u:\{u:Point(rv)| ab=au \mwedge{} q\_u\_p\} . (\mexists{}v:Point(rv) [(ab=av \mwedge{} q\_p\_v)])
Date html generated:
2020_05_20-PM-01_15_42
Last ObjectModification:
2019_12_09-PM-11_24_10
Theory : inner!product!spaces
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