Nuprl Lemma : implies-isometry
∀rv:InnerProductSpace. ∀f:Point ⟶ Point. ∀r:{r:ℝ| r0 < r} . ∀N:{2...}.
((∀x,y:Point. (x ≡ y ⇒ f x ≡ f y))
⇒ (∀x,y:Point. ((||x - y|| = r) ⇒ (||f x - f y|| ≤ r)))
⇒ (∀x,y:Point. ((||x - y|| = (r(N) * r)) ⇒ ((r(N) * r) ≤ ||f x - f y||)))
⇒ is-isometry(rv;f))
Proof
Definitions occuring in Statement :
is-isometry: is-isometry(rv;f),
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
rless: x < y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
int_upper: {i...},
all: ∀x:A. B[x],
implies: 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,
guard: {T},
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
and: P ∧ Q,
int_upper: {i...},
so_apply: x[s],
uimplies: b supposing a,
is-isometry: is-isometry(rv;f),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rv-sub: x - y,
rv-minus: -x,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
rational-approx: (x within 1/n),
int_nzero: ℤ-o,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
satisfiable_int_formula: satisfiable_int_formula(fmla),
real: ℝ,
rneq: x ≠ y,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y,
rdiv: (x/y),
int-to-real: r(n),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_stable: SqStable(P),
le: A ≤ B,
subtract: n - m,
cand: A c∧ B,
rgt: x > y,
rsub: x - y,
rnonneg: rnonneg(x),
rleq: x ≤ y,
rv-isometry: Isometry(f)
Lemmas referenced :
implies-isometry-lemma1,
all_wf,
ss-point_wf,
req_wf,
rv-norm_wf,
rv-sub_wf,
real_wf,
rleq_wf,
int-to-real_wf,
rmul_wf,
rv-ip_wf,
inner-product-space_subtype,
ss-eq_wf,
int_upper_wf,
rless_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
implies-isometry-lemma5,
rv-0_wf,
rv-orthogonal_wf,
rv-add_wf,
exists_wf,
rv-orthogonal-iff,
rv-mul_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
req-iff-rsub-is-0,
uiff_transitivity,
ss-eq_functionality,
rv-mul-1-add,
ss-eq_weakening,
rv-mul_functionality,
rv-mul0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
stable_req,
false_wf,
or_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
req-iff-rabs-rleq,
nat_plus_wf,
equal_wf,
small-reciprocal-real,
rational-approx-property,
rabs-difference-bound-rleq,
int-rdiv_wf,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nequal_wf,
rdiv_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
member_rooint_lemma,
rsub_wf,
subtract_wf,
rmul_preserves_rless,
rinv_wf2,
rneq_functionality,
rmul-int,
req_weakening,
rneq-int,
equal-wf-T-base,
rmul-one,
rminus_wf,
itermMinus_wf,
rmul_preserves_req,
less_than_wf,
int_term_value_add_lemma,
rleq_functionality,
rsub_functionality,
int-rdiv-req,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rless_functionality,
req_transitivity,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rmul-rinv,
rmul-rinv3,
rsub-int,
radd_functionality,
radd-int,
squash_wf,
true_wf,
rminus-int,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req_functionality,
rless-iff4,
int_upper_properties,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf,
iff_weakening_equal,
imax_unfold,
multiply_nat_plus,
imax_nat_plus,
mul_nat_plus,
rleq-int-fractions,
le_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
satisfiable-full-omega-tt,
sq_stable__less_than,
imax_ub,
imax_wf,
decidable__equal_int,
less_than_transitivity1,
le-add-cancel,
add-swap,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-minus,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
subtype_rel_sets,
rooint_wf,
i-member_wf,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
int_term_value_subtract_lemma,
int_term_value_minus_lemma,
rleq-int,
rmul_preserves_rleq,
rmul-rdiv-cancel2,
rmul-distrib,
rmul_over_rminus,
rminus_functionality,
uiff_transitivity3,
sq_stable__rless,
rmul-is-positive,
rmul-zero-both,
less_than'_wf,
rmul_preserves_rleq2,
radd_comm,
rmul-assoc,
rmul_comm,
rmul-ac,
rccint_wf,
set_wf,
member_rccint_lemma,
radd-preserves-rleq,
radd-ac,
radd-rminus-both,
radd-zero-both,
radd-rminus-assoc,
rless_transitivity1,
radd_functionality_wrt_rleq,
rleq_transitivity,
radd_functionality_wrt_rless2,
rleq_antisymmetry,
rv-norm-nonneg,
not-rless,
rv-norm-is-zero,
rv-sub-is-zero,
rv-norm_functionality,
rv-sub_functionality,
rv-sub-same,
rv-norm0,
rv-minus_wf,
req_witness,
rv-add_functionality,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
ss-eq_transitivity,
rv-add-swap,
rv-add-comm,
rv-mul-add,
rv-add-0,
rv-mul-add-1-alt
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
instantiate,
independent_isectElimination,
dependent_pairFormation,
independent_pairFormation,
productElimination,
minusEquality,
approximateComputation,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
allFunctionality,
multiplyEquality,
int_eqEquality,
baseApply,
closedConclusion,
baseClosed,
inrFormation,
promote_hyp,
addEquality,
imageMemberEquality,
imageElimination,
cumulativity,
equalityElimination,
universeEquality,
applyLambdaEquality,
computeAll,
inlFormation,
addLevel,
levelHypothesis,
axiomEquality,
independent_pairEquality,
isect_memberFormation,
functionExtensionality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point {}\mrightarrow{} Point. \mforall{}r:\{r:\mBbbR{}| r0 < r\} . \mforall{}N:\{2...\}.
((\mforall{}x,y:Point. (x \mequiv{} y {}\mRightarrow{} f x \mequiv{} f y))
{}\mRightarrow{} (\mforall{}x,y:Point. ((||x - y|| = r) {}\mRightarrow{} (||f x - f y|| \mleq{} r)))
{}\mRightarrow{} (\mforall{}x,y:Point. ((||x - y|| = (r(N) * r)) {}\mRightarrow{} ((r(N) * r) \mleq{} ||f x - f y||)))
{}\mRightarrow{} is-isometry(rv;f))
Date html generated:
2018_05_22-PM-09_37_59
Last ObjectModification:
2018_05_18-PM-00_39_55
Theory : inner!product!spaces
Home
Index