Nuprl Lemma : implies-isometry-lemma1
∀rv:InnerProductSpace. ∀f:Point(rv) ⟶ Point(rv). ∀r:{r:ℝ| r0 < r} . ∀N:{2...}.
((∀x,y:Point(rv). (x ≡ y ⇒ f x ≡ f y))
⇒ (∀x,y:Point(rv). ((||x - y|| = r) ⇒ (||f x - f y|| ≤ r)))
⇒ (∀x,y:Point(rv). ((||x - y|| = (r(N) * r)) ⇒ ((r(N) * r) ≤ ||f x - f y||)))
⇒ {∀x,y:Point(rv). (((||x - y|| = r) ∨ (||x - y|| = (r(2) * r))) ⇒ (||f x - f y|| = ||x - y||))})
Proof
Definitions occuring in Statement :
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,
int_upper: {i...},
guard: {T},
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],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
int_upper: {i...},
prop: ℙ,
and: P ∧ Q,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
nat: ℕ,
rv-sub: x - y,
rv-minus: -x,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
le: A ≤ B,
decidable: Dec(P),
nat_plus: ℕ+,
rge: x ≥ y,
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
pi1: fst(t),
rtermDivide: num "/" denom,
rtermConstant: "const",
pi2: snd(t),
rnonneg: rnonneg(x),
rleq: x ≤ y,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtract: n - m,
lelt: i ≤ j < k,
int_seg: {i..j-},
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,
cand: A c∧ B,
sq_stable: SqStable(P),
real: ℝ,
sq_exists: ∃x:A [B[x]],
rless: x < y
Lemmas referenced :
Error :ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
req_wf,
rv-norm_wf,
rv-sub_wf,
rmul_wf,
int-to-real_wf,
rleq_wf,
Error :ss-eq_wf,
istype-int_upper,
real_wf,
rless_wf,
rv-mul_wf,
rdiv_wf,
rless-int,
rv-add_wf,
istype-nat,
req_weakening,
nequal_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
istype-int,
intformeq_wf,
full-omega-unsat,
int_upper_properties,
nat_properties,
istype-void,
minus-one-mul-top,
rmul_preserves_req,
itermMinus_wf,
itermMultiply_wf,
rinv_wf2,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
rv-minus_wf,
rminus_wf,
radd_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-add-1-alt,
Error :ss-eq_transitivity,
rv-add-swap,
rv-mul-1-add,
rv-mul-add-alt,
rv-mul-add,
rv-mul_functionality,
req_transitivity,
radd_functionality,
rmul_functionality,
rdiv_functionality,
req_inversion,
radd-int,
rinv-mul-as-rdiv,
rminus_functionality,
rinv-as-rdiv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
req_functionality,
int-rinv-cancel,
rmul-rinv3,
rmul-rinv,
int-rinv-cancel2,
istype-false,
istype-less_than,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
intformless_wf,
intformnot_wf,
decidable__lt,
rleq-int-fractions2,
rabs_wf,
rv-norm_functionality,
rv-norm-mul,
rabs-of-nonneg,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
rtermConstant_wf,
int_term_value_add_lemma,
upper_subtype_nat,
istype-le,
subtract-1-ge-0,
le_witness_for_triv,
ge_wf,
rv-0_wf,
rsum-empty,
rv-mul0,
rleq_functionality,
rv-norm0,
int_term_value_subtract_lemma,
subtract_wf,
rv-norm-triangle-inequality,
int_seg_wf,
int_seg_properties,
int_seg_subtype_nat,
rsum_wf,
rv-0-add,
radd_functionality_wrt_rleq,
neg_assert_of_eq_int,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
assert_of_lt_int,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
lt_int_wf,
subtract-add-cancel,
rsum_unroll,
zero-add,
add-commutes,
add-swap,
add-associates,
decidable__equal_int,
int_subtype_base,
rneq_wf,
req-iff-not-rneq,
sq_stable__less_than,
nat_plus_properties,
rsum-split-first,
ifthenelse_wf,
rsum_functionality_wrt_rleq2,
rsum-constant2,
rsub_wf,
rless_functionality,
rsub-int,
radd-preserves-rless,
rless_functionality_wrt_implies,
rleq_transitivity,
rless_irreflexivity,
rless_transitivity1,
nequal-le-implies,
istype-assert,
not_wf,
bnot_wf,
bool_cases,
iff_transitivity,
assert_of_bnot,
rv-norm-triangle-inequality2,
general_arith_equation2,
radd-preserves-rleq,
rv-mul1,
rv-mul-cancel,
rmul-int,
rv-add-0,
rv-sub_functionality,
Error :ss-eq_inversion,
rleq-int,
squash_wf,
true_wf,
rminus-int,
uiff_transitivity3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
because_Cache,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
setIsType,
productIsType,
closedConclusion,
inrFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
intEquality,
sqequalBase,
equalityIstype,
dependent_pairFormation_alt,
approximateComputation,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
minusEquality,
addEquality,
int_eqEquality,
unionElimination,
Error :memTop,
multiplyEquality,
functionIsTypeImplies,
intWeakElimination,
cumulativity,
promote_hyp,
equalityElimination,
imageElimination,
unionIsType
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point(rv) {}\mrightarrow{} Point(rv). \mforall{}r:\{r:\mBbbR{}| r0 < r\} . \mforall{}N:\{2...\}.
((\mforall{}x,y:Point(rv). (x \mequiv{} y {}\mRightarrow{} f x \mequiv{} f y))
{}\mRightarrow{} (\mforall{}x,y:Point(rv). ((||x - y|| = r) {}\mRightarrow{} (||f x - f y|| \mleq{} r)))
{}\mRightarrow{} (\mforall{}x,y:Point(rv). ((||x - y|| = (r(N) * r)) {}\mRightarrow{} ((r(N) * r) \mleq{} ||f x - f y||)))
{}\mRightarrow{} \{\mforall{}x,y:Point(rv).
(((||x - y|| = r) \mvee{} (||x - y|| = (r(2) * r))) {}\mRightarrow{} (||f x - f y|| = ||x - y||))\})
Date html generated:
2020_05_20-PM-01_13_08
Last ObjectModification:
2019_12_09-PM-07_26_04
Theory : inner!product!spaces
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