Nuprl Lemma : rv-ip-rneq-0
∀rv:InnerProductSpace. ∀a,b:Point. (a ⋅ b ≠ r0 ⇒ (a # 0 ∧ b # 0))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
rneq: x ≠ y,
int-to-real: r(n),
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
false: False
Lemmas referenced :
rv-Cauchy-Schwarz',
rneq_wf,
rv-ip_wf,
int-to-real_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,
rabs-neq-zero,
rless_transitivity1,
rabs_wf,
rmul_wf,
rv-norm_wf,
real_wf,
rleq_wf,
req_wf,
rmul-is-positive,
rv-norm-positive-iff,
rv-norm-nonneg,
rless_irreflexivity
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
productElimination,
unionElimination,
because_Cache,
independent_pairFormation,
voidElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b:Point. (a \mcdot{} b \mneq{} r0 {}\mRightarrow{} (a \# 0 \mwedge{} b \# 0))
Date html generated:
2017_10_04-PM-11_52_19
Last ObjectModification:
2017_06_21-PM-11_24_25
Theory : inner!product!spaces
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