Nuprl Lemma : ip-gt-iff
∀rv:InnerProductSpace. ∀a,b,c,d:Point(rv). uiff(cd > ab;||a - b|| < ||c - d||)
Proof
Definitions occuring in Statement :
ip-gt: cd > ab,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rless: x < y,
uiff: uiff(P;Q),
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
ip-gt: cd > ab,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
ip-congruent: ab=cd,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rneq: x ≠ y,
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
rv-sub: x - y,
rv-minus: -x,
cand: A c∧ B,
ss-eq: Error :ss-eq,
true: True,
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
ip-gt_wf,
rless_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,
sq_stable__rless,
ip-dist-between,
radd_wf,
radd-preserves-rless,
rminus_wf,
int-to-real_wf,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
rless_functionality,
req_weakening,
req_transitivity,
radd_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rv-sep-iff-norm,
rv-norm-nonneg,
rless_transitivity2,
member_rccint_lemma,
rmul_preserves_rleq,
rdiv_wf,
rmul_wf,
itermMultiply_wf,
rinv_wf2,
rleq_weakening_rless,
rleq_functionality,
rmul_functionality,
rmul-rinv,
real_term_value_mul_lemma,
radd-preserves-rleq,
rsub_wf,
trivial-rleq-radd,
Error :ss-eq_wf,
rv-add_wf,
rv-mul_wf,
rv-minus_wf,
uiff_transitivity,
Error :ss-eq_functionality,
rv-add_functionality,
Error :ss-eq_weakening,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
rv-add-swap,
rv-mul-1-add,
rv-mul_functionality,
rsub_functionality,
ip-between-iff2,
i-member_wf,
rccint_wf,
Error :ss-sep_wf,
ip-between_wf,
ip-congruent_wf,
rv-add-comm,
rv-mul-add,
req_wf,
rabs_wf,
uiff_transitivity2,
req_functionality,
rv-norm_functionality,
rv-norm-mul,
squash_wf,
true_wf,
real_wf,
rabs-rminus,
rmul_preserves_req,
rabs-of-nonneg,
rmul-rinv3,
rv-norm-difference-symmetry,
rv-mul-add-alt,
rless-implies-rless,
rminus_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
rename,
universeIsType,
extract_by_obid,
isectElimination,
applyEquality,
lambdaEquality_alt,
setElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
productElimination,
natural_numberEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
inrFormation_alt,
closedConclusion,
equalityIstype,
minusEquality,
dependent_pairFormation_alt,
productIsType
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c,d:Point(rv). uiff(cd > ab;||a - b|| < ||c - d||)
Date html generated:
2020_05_20-PM-01_15_31
Last ObjectModification:
2019_12_10-AM-10_20_39
Theory : inner!product!spaces
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