Nuprl Lemma : ip-ge-dist
∀[rv:InnerProductSpace]. ∀[a,b,c,d:Point]. ((||a - b|| ≤ ||c - d||) ⇒ (¬¬(∃w:Point. (a_b_w ∧ cd=aw))))
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
ip-congruent: ab=cd,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
ss-point: Point,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
stable: Stable{P},
or: P ∨ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
ip-congruent: ab=cd,
ss-eq: x ≡ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rv-sub: x - y,
rv-minus: -x
Lemmas referenced :
not_wf,
exists_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,
ip-between_wf,
ip-congruent_wf,
rleq_wf,
rv-norm_wf,
rv-sub_wf,
real_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
stable__not,
false_wf,
or_wf,
ss-sep_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ip-extend-lemma,
rsub_wf,
radd-preserves-rleq,
radd_wf,
rminus_wf,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd_functionality,
req_weakening,
radd-rminus-assoc,
radd-zero-both,
ip-dist-between,
req_functionality,
req_transitivity,
ip-between_functionality,
ss-eq_weakening,
ip-congruent_functionality,
rv-add_wf,
ip-between-trivial,
rv-norm_functionality,
ss-eq_wf,
rv-mul_wf,
rv-minus_wf,
rv-0_wf,
ss-eq_functionality,
rv-add_functionality,
rv-mul-linear,
rv-add-assoc,
rv-mul-1-add,
rv-mul-mul,
rv-add-swap,
rv-mul_functionality,
radd-int,
rmul-int,
rv-mul0,
rv-mul1,
rv-add-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
productEquality,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
setEquality,
natural_numberEquality,
isect_memberEquality,
functionEquality,
unionElimination,
dependent_set_memberEquality,
productElimination,
dependent_pairFormation,
independent_pairFormation,
addLevel,
impliesFunctionality,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality,
impliesLevelFunctionality,
minusEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c,d:Point].
((||a - b|| \mleq{} ||c - d||) {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}w:Point. (a\_b\_w \mwedge{} cd=aw))))
Date html generated:
2017_10_05-AM-00_11_38
Last ObjectModification:
2017_03_19-PM-02_39_25
Theory : inner!product!spaces
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