Nuprl Lemma : ip-circle-circle-lemma3
∀rv:InnerProductSpace. ∀a,b:Point(rv). ∀c:{c:Point(rv)| a # c} . ∀d:Point(rv).
∀[p:{p:Point(rv)| ab=ap ∧ (||c - p|| ≤ ||c - d||)} ]. ∀[q:{q:Point(rv)| cd=cq ∧ (||a - q|| ≤ ||a - b||)} ].
∃u,v:{p:Point(rv)| ab=ap ∧ cd=cp} . (((||a - q|| < ||a - b||) ∧ (||c - p|| < ||c - d||)) ⇒ u # v)
Proof
Definitions occuring in Statement :
ip-congruent: ab=cd,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
rless: x < y,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
cand: A c∧ B,
exists: ∃x:A. B[x],
guard: {T},
false: False,
not: ¬A,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
rnonneg: rnonneg(x),
rleq: x ≤ y,
ip-congruent: ab=cd,
rev_implies: P ⇐ Q,
true: True,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
rsub: x - y,
rv-sub: x - y,
rv-minus: -x,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
rv-sep-iff-norm,
Error :ss-sep-symmetry,
sq_stable__rv-sep-ext,
ip-circle-circle-lemma2,
rv-norm_wf,
rv-sub_wf,
subtype_rel_sets_simple,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
inner-product-space_subtype,
ip-congruent_wf,
Error :ss-sep_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
Error :ss-point_wf,
rsub_wf,
radd_wf,
istype-le,
istype-void,
rnexp_wf,
sq_stable__rleq,
le_witness_for_triv,
sq_stable__and,
rv-norm-nonneg,
rnexp-rleq,
iff_weakening_equal,
subtype_rel_self,
radd_comm_eq,
true_wf,
squash_wf,
sq_stable__req,
rv-norm-triangle-inequality2,
rleq_functionality_wrt_implies,
rleq_weakening,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
rabs-difference-bound-rleq,
zero-rleq-rabs,
rabs_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
radd-preserves-rleq,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rv-norm-difference-symmetry,
req_weakening,
radd_functionality,
rnexp2-nonneg,
req_inversion,
rabs-rnexp,
rabs-of-nonneg,
itermConstant_wf,
itermMinus_wf,
rminus_wf,
real_term_value_minus_lemma,
rminus_functionality,
rnexp_functionality,
itermMultiply_wf,
rnexp2,
real_term_value_mul_lemma,
req_functionality,
rleq-implies-rleq,
rmul-nonneg-case1,
rsub_functionality,
rmul_functionality,
req-same,
req_witness,
rv-add_wf,
rless_wf,
rv-norm_functionality,
Error :ss-eq_wf,
rv-mul_wf,
rv-0_wf,
rv-minus_wf,
iff_weakening_uiff,
uiff_transitivity,
Error :ss-eq_functionality,
rv-mul-1-add-alt,
Error :ss-eq_weakening,
rv-add-comm,
rv-add_functionality,
rv-mul_functionality,
rv-mul0,
rv-add-0,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
Error :ss-eq_transitivity,
rv-add-swap,
rv-mul1,
sq_stable__rless,
rv-sep-shift2,
rnexp-rless,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rless_functionality,
radd-preserves-rless,
rless_functionality_wrt_implies,
rabs-difference-bound-iff,
rless-implies-rless,
radd-assoc,
rmul-is-positive,
exp_wf2,
subtype_base_sq,
nat_plus_wf,
set_subtype_base,
less_than_wf,
int_subtype_base,
req_transitivity,
rnexp-rmul,
rnexp-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesisEquality,
productElimination,
independent_functionElimination,
setElimination,
rename,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
isectElimination,
applyEquality,
lambdaEquality_alt,
productEquality,
natural_numberEquality,
universeIsType,
independent_isectElimination,
productIsType,
setIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
instantiate,
promote_hyp,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
functionIsTypeImplies,
isect_memberEquality_alt,
universeEquality,
approximateComputation,
int_eqEquality,
equalityIstype,
independent_pairEquality,
dependent_pairFormation_alt,
functionIsType,
minusEquality,
Error :memTop,
unionElimination,
inlFormation_alt,
cumulativity,
intEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b:Point(rv). \mforall{}c:\{c:Point(rv)| a \# c\} . \mforall{}d:Point(rv).
\mforall{}[p:\{p:Point(rv)| ab=ap \mwedge{} (||c - p|| \mleq{} ||c - d||)\} ]. \mforall{}[q:\{q:Point(rv)|
cd=cq \mwedge{} (||a - q|| \mleq{} ||a - b||)\} ].
\mexists{}u,v:\{p:Point(rv)| ab=ap \mwedge{} cd=cp\}
(((||a - q|| < ||a - b||) \mwedge{} (||c - p|| < ||c - d||)) {}\mRightarrow{} u \# v)
Date html generated:
2020_05_20-PM-01_15_13
Last ObjectModification:
2020_01_07-AM-10_57_56
Theory : inner!product!spaces
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