Nuprl Lemma : ip-non-trivial
∀rv:InnerProductSpace. ∀x:{x:Point| r0 < ||x||} . ∃a:Point. (∃b:{Point| a # b})
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
inner-product-space: InnerProductSpace,
rless: x < y,
int-to-real: r(n),
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
true: True,
rge: x ≥ y,
subtract: n - m
Lemmas referenced :
sq_exists_wf,
ss-point_wf,
ss-sep_wf,
set_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rless_wf,
int-to-real_wf,
rv-norm_wf,
real_wf,
rleq_wf,
req_wf,
rmul_wf,
rv-ip_wf,
rv-perp-1,
rv-norm-positive-iff,
sq_stable__rless,
rv-sep-iff-norm,
square-rless-implies,
rv-sub_wf,
rv-norm-nonneg,
rnexp_wf,
false_wf,
le_wf,
radd_wf,
rsub_wf,
less_than_wf,
rless_functionality,
rnexp0,
req_transitivity,
rv-norm-squared,
rv-ip-sub-squared,
radd_functionality,
rsub_functionality,
req_weakening,
rmul_functionality,
rv-ip-nonneg,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rsub_functionality_wrt_rleq,
rless-int,
subtract_wf,
radd-int,
rsub-int,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
because_Cache,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality,
instantiate,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
setEquality,
productEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberFormation,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
addEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x:\{x:Point| r0 < ||x||\} . \mexists{}a:Point. (\mexists{}b:\{Point| a \# b\})
Date html generated:
2017_10_05-AM-00_12_38
Last ObjectModification:
2017_03_15-PM-11_52_33
Theory : inner!product!spaces
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