Nuprl Lemma : rv-mul-sep-zero
∀rv:InnerProductSpace. ∀t:ℝ. ∀x:Point. (t*x # 0 ⇐⇒ x # 0 ∧ (r0 < |t|))
Proof
Definitions occuring in Statement :
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-0: 0,
rless: x < y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
or: P ∨ Q,
false: False,
cand: A c∧ B
Lemmas referenced :
zero-rleq-rabs,
rv-norm-positive-iff,
rv-mul_wf,
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,
rv-0_wf,
rless_wf,
int-to-real_wf,
rabs_wf,
ss-point_wf,
real_wf,
rv-norm_wf,
rleq_wf,
req_wf,
rmul_wf,
rv-ip_wf,
rmul-is-positive,
rless_transitivity1,
rless_irreflexivity,
rless_transitivity2,
rless_functionality,
req_weakening,
rv-norm-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
applyEquality,
because_Cache,
sqequalRule,
productElimination,
independent_functionElimination,
instantiate,
independent_isectElimination,
productEquality,
natural_numberEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
unionElimination,
voidElimination,
inlFormation
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}t:\mBbbR{}. \mforall{}x:Point. (t*x \# 0 \mLeftarrow{}{}\mRightarrow{} x \# 0 \mwedge{} (r0 < |t|))
Date html generated:
2017_10_04-PM-11_51_51
Last ObjectModification:
2017_06_26-PM-09_10_12
Theory : inner!product!spaces
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