Nuprl Lemma : implies-isometry-lemma4
∀rv:InnerProductSpace. ∀f:Point(rv) ⟶ Point(rv). ∀r:{r:ℝ| r0 < r} .
((∀x,y:Point(rv). (x ≡ y ⇒ f x ≡ f y))
⇒ (∀x,y:Point(rv). (((||x - y|| = r) ∨ (||x - y|| = (r(2) * r))) ⇒ (||f x - f y|| = ||x - y||)))
⇒ (∀n,m:ℕ+. ∀x,y:Point(rv). ((||x - y|| < (r(n) * r/r(m))) ⇒ (||f x - f y|| ≤ (r(n) * r/r(m))))))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
stable: Stable{P},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
rv-sub: x - y,
rv-minus: -x,
cand: A c∧ B,
ip-congruent: ab=cd,
nat: ℕ,
le: A ≤ B,
sq_stable: SqStable(P),
rge: x ≥ y,
real: ℝ
Lemmas referenced :
implies-isometry-lemma3,
rless_wf,
rv-norm_wf,
rv-sub_wf,
rdiv_wf,
rmul_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
Error :ss-point_wf,
nat_plus_wf,
req_wf,
inner-product-space_subtype,
Error :ss-eq_wf,
real_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
stable__rleq,
false_wf,
not_wf,
rleq_wf,
istype-void,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rless_transitivity2,
rleq_weakening_rless,
rmul_preserves_req,
itermMultiply_wf,
int_term_value_mul_lemma,
rinv_wf2,
itermSubtract_wf,
rneq_functionality,
rmul-int,
req_weakening,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
nequal_wf,
req_functionality,
req_transitivity,
rmul_functionality,
req_inversion,
rmul-rinv3,
rinv_functionality2,
rinv-of-rmul,
int-rinv-cancel,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rless_functionality,
rv-sep-iff-norm,
rmul_preserves_rless,
rv-add_wf,
rv-mul_wf,
radd_wf,
rminus_wf,
rv-minus_wf,
itermMinus_wf,
itermAdd_wf,
uiff_transitivity,
Error :ss-eq_functionality,
Error :ss-eq_weakening,
rv-mul-linear,
rv-add_functionality,
rv-add-assoc,
rv-mul-mul,
rv-mul-1-add,
rv-add-swap,
rv-mul-add,
rv-mul_functionality,
radd_functionality,
real_term_value_minus_lemma,
real_term_value_add_lemma,
rv-add-comm,
rv-mul-add-alt,
ip-circle-circle-lemma3,
Error :ss-sep_wf,
ip-congruent_wf,
rabs_wf,
rleq_functionality,
rv-norm_functionality,
rv-norm-mul,
rabs-of-nonneg,
rsub_wf,
Error :ss-eq_transitivity,
rv-mul-1-add-alt,
rv-norm-difference-symmetry,
rabs_functionality,
rminus_functionality,
rinv-mul-as-rdiv,
square-rleq-implies,
rnexp_wf,
istype-le,
rnexp2-nonneg,
rnexp2,
radd-preserves-rleq,
rnexp-rmul,
rabs-rnexp,
sq_stable__and,
sq_stable__req,
req_witness,
istype-less_than,
rv-norm-triangle-inequality2,
rleq_functionality_wrt_implies,
rleq_transitivity,
rleq_weakening,
rleq_weakening_equal,
radd-rdiv,
rleq-int-fractions,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rmul-is-positive,
sq_stable__rless,
rmul_preserves_rleq2,
not-rless,
rv-norm-nonneg,
rleq_antisymmetry,
rv-sub-is-zero,
rv-norm-is-zero,
rv-sub_functionality,
rv-sub-same,
rv-0_wf,
rv-norm0
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
universeIsType,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
independent_isectElimination,
inrFormation_alt,
productElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
inhabitedIsType,
functionIsType,
unionIsType,
equalityTransitivity,
equalitySymmetry,
setIsType,
instantiate,
unionEquality,
functionEquality,
equalityIstype,
multiplyEquality,
closedConclusion,
imageMemberEquality,
baseClosed,
baseApply,
intEquality,
sqequalBase,
cumulativity,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
minusEquality,
productIsType,
functionIsTypeImplies,
promote_hyp,
imageElimination,
addEquality,
inlFormation_alt
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point(rv) {}\mrightarrow{} Point(rv). \mforall{}r:\{r:\mBbbR{}| r0 < r\} .
((\mforall{}x,y:Point(rv). (x \mequiv{} y {}\mRightarrow{} f x \mequiv{} f y))
{}\mRightarrow{} (\mforall{}x,y:Point(rv).
(((||x - y|| = r) \mvee{} (||x - y|| = (r(2) * r))) {}\mRightarrow{} (||f x - f y|| = ||x - y||)))
{}\mRightarrow{} (\mforall{}n,m:\mBbbN{}\msupplus{}. \mforall{}x,y:Point(rv).
((||x - y|| < (r(n) * r/r(m))) {}\mRightarrow{} (||f x - f y|| \mleq{} (r(n) * r/r(m))))))
Date html generated:
2020_05_20-PM-01_16_13
Last ObjectModification:
2019_12_11-PM-07_59_28
Theory : inner!product!spaces
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