Nuprl Lemma : sphere-map-from-ball-map
∀[n:ℕ]. ∀[g:{g:B(n + 1) ⟶ B(n + 1)|
(∀x,y:B(n + 1). (req-vec(n + 1;x;y) ⇒ req-vec(n + 1;g x;g y))) ∧ (∀x:B(n + 1). (||g x|| = r1))} ].
(g ∈ sphere-map(n))
Proof
Definitions occuring in Statement :
sphere-map: sphere-map(n),
real-unit-ball: B(n),
real-vec-norm: ||x||,
req-vec: req-vec(n;x;y),
req: x = y,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sphere-map: sphere-map(n),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
real-unit-sphere: S(n),
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real-unit-ball: B(n),
so_lambda: λ2x.t[x],
so_apply: x[s],
real-ball: B(n;r),
cand: A c∧ B,
rless: x < y,
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
nat_plus_wf,
real-unit-sphere_wf,
rleq_wf,
real-vec-dist_wf,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
rdiv_wf,
int-to-real_wf,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rless_wf,
real-unit-ball_wf,
req-vec_wf,
req_wf,
real-vec-norm_wf,
istype-nat,
real-unit-sphere-subtype-ball,
real-ball-uniform-continuity,
istype-less_than,
subtype_rel_sets,
real-ball_wf,
real-vec_wf,
all_wf,
subtype_rel_set,
subtype_rel_dep_function,
subtype_rel_self,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
subtype_rel_sets_simple,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
hypothesis,
sqequalRule,
functionIsType,
because_Cache,
productIsType,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
applyEquality,
closedConclusion,
inrFormation_alt,
productElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setIsType,
isectIsTypeImplies,
inhabitedIsType,
functionExtensionality,
functionEquality,
productEquality,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[g:\{g:B(n + 1) {}\mrightarrow{} B(n + 1)|
(\mforall{}x,y:B(n + 1). (req-vec(n + 1;x;y) {}\mRightarrow{} req-vec(n + 1;g x;g y)))
\mwedge{} (\mforall{}x:B(n + 1). (||g x|| = r1))\} ].
(g \mmember{} sphere-map(n))
Date html generated:
2019_10_30-AM-11_27_55
Last ObjectModification:
2019_07_30-PM-02_38_53
Theory : real!vectors
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