Nuprl Lemma : ip-congruent-sep
∀rv:InnerProductSpace. ∀a,b,c:Point. ∀d:{d:Point| ab=cd} . (a # b ⇒ c # d)
Proof
Definitions occuring in Statement :
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
sq_stable: SqStable(P),
req: x = y,
ip-congruent: ab=cd,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
rv-sep-iff-norm,
sq_stable__req,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
rless_transitivity1,
rleq_weakening,
ss-sep_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
set_wf,
ss-point_wf,
ip-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
setElimination,
rename,
isectElimination,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
productEquality,
natural_numberEquality,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
instantiate
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. \mforall{}d:\{d:Point| ab=cd\} . (a \# b {}\mRightarrow{} c \# d)
Date html generated:
2017_10_04-PM-11_56_50
Last ObjectModification:
2017_03_15-PM-04_42_49
Theory : inner!product!spaces
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