Nuprl Lemma : ip-ge-sep
∀rv:InnerProductSpace. ∀a,c:Point. ∀[b:Point]. (a # c) supposing (a # b and ac ≥ ab)
Proof
Definitions occuring in Statement :
ip-ge: cd ≥ ab,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
ip-ge: cd ≥ ab,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not_wf,
exists_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ip-between_wf,
ip-congruent_wf,
sq_stable__rv-sep-ext,
ss-sep_wf,
ip-ge_wf,
ip-ge-iff,
rv-norm-positive,
rv-sub_wf,
rv-sep-iff,
rv-norm-positive-iff,
rless_transitivity1,
int-to-real_wf,
rv-norm_wf,
real_wf,
rleq_wf,
req_wf,
rmul_wf,
rv-ip_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
productEquality,
because_Cache,
rename,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
natural_numberEquality,
setElimination,
setEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,c:Point. \mforall{}[b:Point]. (a \# c) supposing (a \# b and ac \mgeq{} ab)
Date html generated:
2017_10_05-AM-00_12_21
Last ObjectModification:
2017_03_20-PM-02_26_30
Theory : inner!product!spaces
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