Nuprl Lemma : implies-isometry-lemma5
∀rv:InnerProductSpace. ∀f:Point(rv) ⟶ Point(rv). ∀d:{r:ℝ| r0 < r} .
((∀x,y:Point(rv). (x ≡ y ⇒ f x ≡ f y))
⇒ (∀x,y:Point(rv). (((||x - y|| = d) ∨ (||x - y|| = (r(2) * d))) ⇒ (||f x - f y|| = ||x - y||)))
⇒ (∀s,r:ℝ.
((∃n,m:ℕ+. (s = (r(n)/r(m))))
⇒ (∃n,m:ℕ+. (r = (r(n)/r(m))))
⇒ (∀x,y:Point(rv). ((||x - y|| ∈ (r * d, s * d)) ⇒ (||f x - f y|| ∈ [r * d, s * d]))))))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rdiv: (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],
exists: ∃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,
prop: ℙ,
exists: ∃x:A. B[x],
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),
false: False,
uiff: uiff(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
sq_exists: ∃x:A [B[x]],
rless: x < y,
top: Top,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
stable: Stable{P},
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
pi2: snd(t),
cand: A c∧ B,
rge: x ≥ y,
rgt: x > y,
rv-sub: x - y,
rv-minus: -x
Lemmas referenced :
implies-isometry-lemma3,
i-member_wf,
rooint_wf,
rmul_wf,
rv-norm_wf,
rv-sub_wf,
Error :ss-point_wf,
nat_plus_wf,
req_wf,
rdiv_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,
rless_wf,
real_wf,
inner-product-space_subtype,
Error :ss-eq_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rsub_functionality,
mul_bounds_1b,
subtract_wf,
rsub_wf,
req_functionality,
req_weakening,
rsub-int-fractions,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
sq_stable__rless,
rmul_preserves_rless,
rleq_weakening_rless,
rless_transitivity2,
istype-void,
member_rooint_lemma,
rless_functionality,
rmul_functionality,
req_transitivity,
rinv-mul-as-rdiv,
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-int-fractions,
mul_nat_plus,
istype-less_than,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
implies-isometry-lemma4,
rmul_preserves_req,
req-same,
rleq_wf,
rmul-rinv3,
iff_weakening_uiff,
rleq_functionality,
stable_req,
minimal-double-negation-hyp-elim,
false_wf,
not_wf,
minimal-not-not-excluded-middle,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
rmul-is-positive,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rless-int-fractions2,
rv-add_wf,
rv-mul_wf,
radd_wf,
rminus_wf,
rv-minus_wf,
itermAdd_wf,
itermMinus_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,
rv-add-swap,
rv-add-comm,
rv-mul-1-add,
rv-mul_functionality,
radd_functionality,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rabs_wf,
rv-norm_functionality,
rv-norm-mul,
rabs-of-nonneg,
rmul_preserves_rleq,
rleq_functionality_wrt_implies,
rleq_weakening,
rv-norm-difference-symmetry,
rv-mul-add-alt,
radd-preserves-rless,
rmul_comm,
rabs-difference-symmetry,
member_rccint_lemma,
rv-norm-triangle-inequality2,
req_inversion,
radd_functionality_wrt_rleq,
rleq-implies-rleq
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,
setElimination,
rename,
because_Cache,
applyEquality,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productIsType,
independent_isectElimination,
inrFormation_alt,
productElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
functionIsType,
unionIsType,
setIsType,
instantiate,
multiplyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
unionEquality,
functionEquality,
inlFormation_alt,
equalityIstype,
minusEquality,
closedConclusion,
promote_hyp
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point(rv) {}\mrightarrow{} Point(rv). \mforall{}d:\{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|| = d) \mvee{} (||x - y|| = (r(2) * d))) {}\mRightarrow{} (||f x - f y|| = ||x - y||)))
{}\mRightarrow{} (\mforall{}s,r:\mBbbR{}.
((\mexists{}n,m:\mBbbN{}\msupplus{}. (s = (r(n)/r(m))))
{}\mRightarrow{} (\mexists{}n,m:\mBbbN{}\msupplus{}. (r = (r(n)/r(m))))
{}\mRightarrow{} (\mforall{}x,y:Point(rv). ((||x - y|| \mmember{} (r * d, s * d)) {}\mRightarrow{} (||f x - f y|| \mmember{} [r * d, s * d]))))))
Date html generated:
2020_05_20-PM-01_16_32
Last ObjectModification:
2020_01_07-AM-10_10_32
Theory : inner!product!spaces
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