Nuprl Lemma : rv-midpoint-unique
∀rv:InnerProductSpace. ∀a,b,m:Point(rv).
((||m - a|| = (||b - a||/r(2))) ∧ (||m - b|| = (||b - a||/r(2))) ⇐⇒ m ≡ (r1/r(2))*a + b)
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
nat: ℕ,
le: A ≤ B,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
rv-sub: x - y,
rv-minus: -x,
rat_term_to_real: rat_term_to_real(f;t),
rtermMinus: rtermMinus(num),
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermConstant: "const",
pi1: fst(t),
rtermAdd: left "+" right,
pi2: snd(t)
Lemmas referenced :
req_wf,
rv-norm_wf,
rv-sub_wf,
rdiv_wf,
inner-product-space_subtype,
int-to-real_wf,
rless-int,
rless_wf,
Error :ss-eq_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rv-mul_wf,
rv-add_wf,
Error :ss-point_wf,
rv-norm-difference-symmetry,
req_functionality,
req_weakening,
rv-ip-zero-iff,
radd_wf,
rv-ip_wf,
rmul_wf,
rsub_wf,
radd_functionality,
rv-ip-sub-squared,
rv-ip-add-squared,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
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_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
rnexp_wf,
istype-le,
req_inversion,
rv-norm-squared,
rnexp_functionality,
exp_wf2,
req-int,
rnexp-int,
rless_transitivity1,
rleq_weakening,
rnexp-rdiv,
rdiv_functionality,
rmul_preserves_req,
rinv_wf2,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
nequal_wf,
req_transitivity,
int-rinv-cancel,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
rv-minus_wf,
rv-add-cancel-left,
rv-0_wf,
Error :ss-eq_functionality,
rv-add-comm,
Error :ss-eq_weakening,
Error :ss-eq_inversion,
rv-add-assoc,
uiff_transitivity,
Error :ss-eq_transitivity,
rv-add_functionality,
rv-add-minus,
rv-add-0,
rv-ip_functionality,
radd-preserves-req,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rv-mul-linear,
rv-mul-mul,
rv-add-swap,
rv-mul-1-add-alt,
rv-mul_functionality,
rmul-int,
rv-sub-is-zero,
rmul-rinv,
rv-mul1,
rabs_wf,
rleq-int-fractions2,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
istype-false,
rv-norm_functionality,
rv-sub_functionality,
rv-norm-mul,
rmul_functionality,
rabs-of-nonneg,
rmul-rinv3,
assert-rat-term-eq2,
rtermAdd_wf,
rtermConstant_wf,
rtermDivide_wf,
rtermMinus_wf,
rv-mul-add,
rinv-as-rdiv,
rminus_functionality,
rv-norm-sub,
rv-mul-add-alt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
instantiate,
inhabitedIsType,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
cumulativity,
intEquality,
unionElimination,
dependent_pairFormation_alt,
Error :memTop,
sqequalBase,
imageElimination,
universeEquality,
minusEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,m:Point(rv).
((||m - a|| = (||b - a||/r(2))) \mwedge{} (||m - b|| = (||b - a||/r(2))) \mLeftarrow{}{}\mRightarrow{} m \mequiv{} (r1/r(2))*a + b)
Date html generated:
2020_05_20-PM-01_12_12
Last ObjectModification:
2019_12_09-PM-11_53_15
Theory : inner!product!spaces
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