Nuprl Lemma : implies-isometry-lemma2
∀rv:InnerProductSpace. ∀f:Point ⟶ Point. ∀r:{r:ℝ| r0 < r} .
((∀x,y:Point. (x ≡ y ⇒ f x ≡ f y))
⇒ (∀x,y:Point. (((||x - y|| = r) ∨ (||x - y|| = (r(2) * r))) ⇒ (||f x - f y|| = ||x - y||)))
⇒ (∀x,y:Point. ((||x - y|| = r) ⇒ (∀j:ℕ. f x + r(j)*y - x ≡ f x + r(j)*f y - f x))))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: 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,
nat: 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],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
ss-eq: x ≡ y,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rv-minus: -x,
rv-sub: x - y,
iff: P ⇐⇒ Q,
cand: A c∧ B,
squash: ↓T,
rsub: x - y,
true: True,
absval: |i|,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
ss-sep_wf,
rv-add_wf,
less_than_transitivity1,
less_than_irreflexivity,
rv-mul_wf,
int-to-real_wf,
rv-sub_wf,
int_seg_wf,
int_seg_properties,
inner-product-space_subtype,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
subtype_base_sq,
int_subtype_base,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
req_wf,
rv-norm_wf,
real_wf,
rleq_wf,
rmul_wf,
rv-ip_wf,
ss-point_wf,
all_wf,
or_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-eq_wf,
set_wf,
rless_wf,
rv-0-add,
rv-mul0,
ss-eq_weakening,
rv-add_functionality,
ss-eq_functionality,
uiff_transitivity,
rv-0_wf,
rv-add-0,
radd-int,
rv-mul_functionality,
rv-mul-1-add,
rv-add-swap,
rv-add-assoc,
rv-mul1,
rv-minus_wf,
radd_wf,
rv-midpoint-unique,
req_weakening,
rsub-int,
req_inversion,
rv-sub_functionality,
rv-norm_functionality,
req_functionality,
rsub_wf,
equal_wf,
rmul_functionality,
radd-zero-both,
radd-rminus-both,
rmul-distrib2,
rmul-identity1,
radd-ac,
rmul-int,
rminus-radd,
rmul-distrib,
radd-assoc,
rminus-int,
true_wf,
squash_wf,
radd_comm,
rminus-rminus,
rmul-one-both,
rminus_functionality,
rmul_over_rminus,
rmul-minus,
rminus-as-rmul,
radd_functionality,
req_transitivity,
rv-mul-add,
rv-mul-add-alt,
ss-eq_transitivity,
rv-mul-add-1-alt,
rv-mul-mul,
rv-mul-linear,
rminus_wf,
rabs-int,
rv-norm-mul,
uiff_transitivity2,
absval_wf,
rabs_wf,
rdiv_functionality,
rless-int,
rdiv_wf,
rmul_comm,
rmul-rdiv-cancel2,
rmul_preserves_req,
rv-mul-cancel,
rinv_wf2,
req-iff-rsub-is-0,
itermMultiply_wf,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
rv-add-cancel-left,
rv-mul-add-1,
rv-add-comm,
itermMinus_wf,
ss-eq_inversion,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
applyEquality,
functionExtensionality,
productElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
instantiate,
cumulativity,
addEquality,
setEquality,
productEquality,
functionEquality,
minusEquality,
addLevel,
baseClosed,
imageMemberEquality,
imageElimination,
multiplyEquality,
inlFormation,
inrFormation
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point {}\mrightarrow{} Point. \mforall{}r:\{r:\mBbbR{}| r0 < r\} .
((\mforall{}x,y:Point. (x \mequiv{} y {}\mRightarrow{} f x \mequiv{} f y))
{}\mRightarrow{} (\mforall{}x,y:Point. (((||x - y|| = r) \mvee{} (||x - y|| = (r(2) * r))) {}\mRightarrow{} (||f x - f y|| = ||x - y||)))
{}\mRightarrow{} (\mforall{}x,y:Point. ((||x - y|| = r) {}\mRightarrow{} (\mforall{}j:\mBbbN{}. f x + r(j)*y - x \mequiv{} f x + r(j)*f y - f x))))
Date html generated:
2017_10_04-PM-11_56_14
Last ObjectModification:
2017_07_28-AM-08_54_28
Theory : inner!product!spaces
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