Nuprl Lemma : ip-ge-iff
∀[rv:InnerProductSpace]. ∀[a,b,c,d:Point(rv)]. uiff(cd ≥ ab;||a - b|| ≤ ||c - d||)
Proof
Definitions occuring in Statement :
ip-ge: cd ≥ ab,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
prop: ℙ,
ip-ge: cd ≥ ab,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
stable: Stable{P},
or: P ∨ Q,
ip-congruent: ab=cd,
rev_uimplies: rev_uimplies(P;Q),
ip-gt: cd > ab,
cand: A c∧ B,
squash: ↓T
Lemmas referenced :
le_witness_for_triv,
ip-ge_wf,
rleq_wf,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
ip-congruent_wf,
ip-between_wf,
exists_wf,
not_wf,
or_wf,
false_wf,
rv-ip_wf,
rmul_wf,
req_wf,
int-to-real_wf,
real_wf,
stable__rleq,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ip-dist-between,
rv-norm-nonneg,
rminus_wf,
radd-preserves-rleq,
radd_wf,
rleq_functionality,
req_weakening,
req_transitivity,
radd_functionality,
uiff_transitivity,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-assoc,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
stable__not,
rless_wf,
istype-void,
ip-gt-iff,
not-rless,
ip-between-trivial2,
req_fake_le_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
voidElimination,
applyEquality,
setElimination,
rename,
because_Cache,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
lambdaFormation,
independent_functionElimination,
functionEquality,
natural_numberEquality,
productEquality,
setEquality,
lambdaEquality,
unionElimination,
addEquality,
minusEquality,
unionEquality,
functionIsType,
productIsType,
unionIsType,
lambdaFormation_alt,
dependent_pairFormation,
imageElimination,
dependent_pairFormation_alt
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c,d:Point(rv)]. uiff(cd \mgeq{} ab;||a - b|| \mleq{} ||c - d||)
Date html generated:
2020_05_20-PM-01_15_36
Last ObjectModification:
2019_12_08-PM-07_01_49
Theory : inner!product!spaces
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