Nuprl Lemma : ip-extend-lemma
∀rv:InnerProductSpace. ∀a:Point(rv). ∀b:{b:Point(rv)| a # b} . ∀dcd:{d:ℝ| r0 ≤ d} .
(∃x:Point(rv) [(a_b_x ∧ (||b - x|| = dcd))])
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
rneq: x ≠ y,
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
top: Top,
rge: x ≥ y,
exists: ∃x:A. B[x],
true: True
Lemmas referenced :
rv-sep-iff-norm,
sq_stable__rv-sep-ext,
sq_stable__rleq,
int-to-real_wf,
real_wf,
rleq_wf,
Error :ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
Error :ss-point_wf,
rv-add_wf,
rv-mul_wf,
rdiv_wf,
radd_wf,
rv-norm_wf,
rv-sub_wf,
rless_wf,
rminus_wf,
rmul_preserves_req,
rsub_wf,
ip-between_wf,
req_wf,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
req-same,
req_functionality,
req_transitivity,
rminus_functionality,
rmul-rinv3,
radd_functionality,
rmul_functionality,
req_weakening,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
ip-between_functionality,
Error :ss-eq_weakening,
rv-add_functionality,
rv-mul_functionality,
iff_weakening_uiff,
rv-norm_functionality,
rv-sub_functionality,
Error :ss-eq_wf,
ip-between-iff2,
Error :ss-eq_functionality,
req_inversion,
rv-mul-cancel,
trivial-rless-radd,
rmul_preserves_rless,
rless_functionality,
istype-void,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rv-add-cancel-left,
uiff_transitivity,
rv-mul-add-1,
rv-add-assoc,
rv-add-swap,
rv-mul-add,
rv-0_wf,
rv-mul0,
rv-add-0,
Error :ss-eq_inversion,
Error :ss-eq_transitivity,
rinv-mul-as-rdiv,
rv-mul1,
rccint_wf,
i-member_wf,
rv-norm-nonneg,
trivial-rleq-radd,
rmul_preserves_rleq,
member_rccint_lemma,
rleq_functionality,
rv-mul-linear,
rv-mul-mul,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
rabs_wf,
rv-norm-difference-symmetry,
ip-dist-between-1,
rabs-rminus,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesisEquality,
setElimination,
rename,
hypothesis,
productElimination,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
isectElimination,
natural_numberEquality,
setIsType,
universeIsType,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
dependent_set_memberFormation_alt,
inrFormation_alt,
productIsType,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
independent_pairFormation,
approximateComputation,
int_eqEquality,
Error :memTop,
promote_hyp,
equalityIstype,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
universeEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a:Point(rv). \mforall{}b:\{b:Point(rv)| a \# b\} . \mforall{}dcd:\{d:\mBbbR{}| r0 \mleq{} d\} .
(\mexists{}x:Point(rv) [(a\_b\_x \mwedge{} (||b - x|| = dcd))])
Date html generated:
2020_05_20-PM-01_15_22
Last ObjectModification:
2020_01_03-PM-07_33_49
Theory : inner!product!spaces
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