Nuprl Lemma : rv-sep-iff
∀rv:InnerProductSpace. ∀x,y:Point. (x # y ⇐⇒ x - y # 0)
Proof
Definitions occuring in Statement :
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rv-0: 0,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rv-sub: x - y,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rv-add-comm,
rv-add-sep2,
rv-add-0,
ss-sep_functionality,
rv-0-add,
rv-add-minus,
rv-add_functionality,
rv-add-assoc,
ss-eq_inversion,
ss-eq_transitivity,
ss-eq_functionality,
uiff_transitivity,
ss-eq_weakening,
rv-minus_wf,
rv-add_wf,
ss-eq_wf,
rv-add-sep1,
ss-point_wf,
rv-0_wf,
rv-sub_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-sep_wf
Rules used in proof :
productElimination,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x,y:Point. (x \# y \mLeftarrow{}{}\mRightarrow{} x - y \# 0)
Date html generated:
2016_11_08-AM-09_15_57
Last ObjectModification:
2016_11_02-PM-03_04_47
Theory : inner!product!spaces
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