Nuprl Lemma : ip-line-circle-1
∀rv:InnerProductSpace. ∀a,b,p,q:Point(rv).
(a # b
⇒ p # q
⇒ (||p - a|| ≤ ||a - b||)
⇒ (||a - b|| ≤ ||q - a||)
⇒ (∃u:{u:Point(rv)| ab=au ∧ q_u_p}
∃v:{v:Point(rv)| ab=av ∧ q_p_v} . ((||a - p|| < ||a - b||) ⇒ (p # v ∧ ((||a - b|| < ||a - q||) ⇒ q # u)))))
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,
rless: x < y,
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],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
let: let,
iff: P ⇐⇒ Q,
and: P ∧ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
rneq: x ≠ y,
or: P ∨ Q,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
cand: A c∧ B,
top: Top,
quadratic1: quadratic1(a;b;c),
true: True,
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
sq_exists: ∃x:A [B[x]],
rless: x < y,
nat_plus: ℕ+,
rge: x ≥ y,
quadratic2: quadratic2(a;b;c),
rv-sub: x - y,
rv-minus: -x,
ip-congruent: ab=cd,
so_apply: x[s],
so_lambda: λ2x.t[x],
pi2: snd(t),
rtermConstant: "const",
rtermSubtract: left "-" right,
rtermDivide: num "/" denom,
pi1: fst(t),
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
rat_term_to_real: rat_term_to_real(f;t),
sq_type: SQType(T),
nequal: a ≠ b ∈ T
Lemmas referenced :
ip-line-circle-lemma,
rv-norm_wf,
rv-sub_wf,
rleq_wf,
inner-product-space_subtype,
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,
rv-sep-shift,
rv-sep-iff-norm,
Error :ss-sep-symmetry,
rnexp-positive,
istype-void,
istype-le,
int-to-real_wf,
rsub_wf,
rmul_wf,
rv-ip_wf,
rnexp_wf,
req_wf,
rv-add_wf,
rv-mul_wf,
quadratic1_wf,
rless_wf,
quadratic2_wf,
sq_stable__rleq,
radd-preserves-rleq,
itermMinus_wf,
itermAdd_wf,
rinv_wf2,
rminus_wf,
radd_wf,
rdiv_wf,
rmul_preserves_rleq,
rsqrt_wf,
member_rccint_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermSubtract_wf,
rless-int,
rmul_preserves_rless,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rleq_functionality,
req_transitivity,
radd_functionality,
rmul_functionality,
req_weakening,
rmul-rinv,
real_term_value_add_lemma,
real_term_value_minus_lemma,
square-rleq-implies,
req_functionality,
req_inversion,
rnexp2,
rsub_functionality,
istype-false,
rleq-int,
rmul_preserves_rleq2,
rnexp2-nonneg,
rmul-assoc,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rv-norm-nonneg,
rnexp-rleq-iff,
squash_wf,
true_wf,
real_wf,
istype-nat,
subtype_rel_self,
iff_weakening_equal,
rv-norm-squared,
rv-ip-sub-squared,
rv-ip-sub2,
rv-ip-symmetry,
rleq_transitivity,
uiff_transitivity3,
trivial-rleq-radd,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rminus_functionality,
uiff_transitivity,
sq_stable__rless,
radd-preserves-rless,
rleq-implies-rleq,
rnexp-nonneg,
rmul-nonneg-case1,
square-rless-implies,
rless-implies-rless,
rv-norm-difference-symmetry,
rnexp-rless,
rmul-is-positive,
Error :ss-eq_wf,
Error :ss-eq_functionality,
Error :ss-eq_transitivity,
rv-add-swap,
rv-add_functionality,
rv-add-comm,
Error :ss-eq_weakening,
i-member_wf,
rccint_wf,
rv-minus_wf,
rv-0_wf,
iff_weakening_uiff,
rv-norm_functionality,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
rv-mul-add,
rv-mul_functionality,
rv-mul0,
rv-add-0,
rv-sub_functionality,
ip-congruent_wf,
ip-between_wf,
ip-between-iff2,
rv-mul-1-add,
rv-mul1,
rv-add-cancel-left,
rv-mul-add-alt,
rv-mul-add-1,
rabs_wf,
Error :ss-eq_inversion,
rv-norm-is-zero,
rv-norm-mul,
rleq_weakening_rless,
rless_transitivity2,
rabs-difference-symmetry,
rabs-of-nonneg,
int_term_value_var_lemma,
int_term_value_add_lemma,
rmul-rinv3,
rmul_preserves_req,
rv-0-add,
rneq-by-function,
rabs-positive,
rpositive-rless,
rv-mul-sep-zero,
rv-sep-iff,
rv-norm-positive,
uimplies_transitivity,
rless_functionality_wrt_implies,
rdiv_functionality,
rtermVar_wf,
rtermConstant_wf,
rtermSubtract_wf,
rtermDivide_wf,
assert-rat-term-eq2,
int_subtype_base,
subtype_base_sq,
int_entire_a,
rneq-int,
rmul-int,
rneq_functionality,
square-nonzero,
rabs-neq-zero,
rv-mul-1-add-alt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
instantiate,
independent_isectElimination,
productElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
voidElimination,
productIsType,
closedConclusion,
inrFormation_alt,
equalityIstype,
imageElimination,
minusEquality,
isect_memberEquality_alt,
baseClosed,
imageMemberEquality,
approximateComputation,
int_eqEquality,
Error :memTop,
dependent_pairFormation_alt,
unionElimination,
promote_hyp,
universeEquality,
inlFormation_alt,
functionIsType,
setIsType,
sqequalBase,
intEquality,
cumulativity,
multiplyEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,p,q:Point(rv).
(a \# b
{}\mRightarrow{} p \# q
{}\mRightarrow{} (||p - a|| \mleq{} ||a - b||)
{}\mRightarrow{} (||a - b|| \mleq{} ||q - a||)
{}\mRightarrow{} (\mexists{}u:\{u:Point(rv)| ab=au \mwedge{} q\_u\_p\}
\mexists{}v:\{v:Point(rv)| ab=av \mwedge{} q\_p\_v\}
((||a - p|| < ||a - b||) {}\mRightarrow{} (p \# v \mwedge{} ((||a - b|| < ||a - q||) {}\mRightarrow{} q \# u)))))
Date html generated:
2020_05_20-PM-01_14_39
Last ObjectModification:
2020_01_06-AM-10_55_14
Theory : inner!product!spaces
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