Nuprl Lemma : rv-sep-shift
∀rv:InnerProductSpace. ∀a,p,q:Point. (p # q ⇒ p - a # q - a)
Proof
Definitions occuring in Statement :
rv-sub: x - y,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
rv-sub: x - y,
rv-minus: -x,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-point_wf,
rv-sep-iff-norm,
rv-sub_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
int-to-real_wf,
radd_wf,
rmul_wf,
rv-minus_wf,
rv-0_wf,
rv-norm_wf,
real_wf,
rleq_wf,
req_wf,
rv-ip_wf,
uiff_transitivity,
ss-eq_functionality,
rv-add_functionality,
ss-eq_weakening,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
ss-eq_transitivity,
rv-add-swap,
rv-add-comm,
rv-mul-add,
rv-mul_functionality,
req_transitivity,
radd_functionality,
rmul-int,
req_weakening,
radd-int,
rv-mul0,
rv-add-0,
rless_functionality,
rv-norm_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
minusEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,p,q:Point. (p \# q {}\mRightarrow{} p - a \# q - a)
Date html generated:
2017_10_04-PM-11_51_39
Last ObjectModification:
2017_03_13-PM-00_37_44
Theory : inner!product!spaces
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