Nuprl Lemma : continuous-image-m-TB
∀[X:Type]
∀dX:metric(X). ∀[Y:Type]. ∀dY:metric(Y). ∀f:X ⟶ Y. (UC(f:X ⟶ Y) ⇒ m-TB(X;dX) ⇒ m-TB(f[X];image-metric(dY)))
Proof
Definitions occuring in Statement :
m-TB: m-TB(X;d),
m-unif-cont: UC(f:X ⟶ Y),
image-metric: image-metric(d),
image-space: f[X],
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
sq_stable: SqStable(P),
pi1: fst(t),
mdist: mdist(d;x;y),
image-metric: image-metric(d),
image-ap: f[x],
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
rneq: x ≠ y,
image-space: f[X],
sq_exists: ∃x:A [B[x]],
rless: x < y,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
nat_plus: ℕ+,
m-unif-cont: UC(f:X ⟶ Y),
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
req_weakening,
meq_weakening,
mdist_functionality,
rleq_functionality,
sq_stable__rleq,
rleq_weakening_rless,
rless_transitivity2,
rless_wf,
int_seg_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
mdist_wf,
rleq_wf,
int_seg_wf,
image-ap_wf,
subtract-add-cancel,
istype-le,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__le,
nat_plus_properties,
subtract_wf,
small-reciprocal-real,
istype-universe,
metric_wf,
m-unif-cont_wf,
m-TB_wf,
istype-nat,
istype-less_than,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
image-metric_wf,
image-space_wf,
m-TB-iff
Rules used in proof :
baseClosed,
imageMemberEquality,
imageElimination,
inrFormation_alt,
because_Cache,
closedConclusion,
productIsType,
applyEquality,
universeEquality,
instantiate,
inhabitedIsType,
functionIsType,
universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
rename,
setElimination,
addEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]
\mforall{}dX:metric(X)
\mforall{}[Y:Type]
\mforall{}dY:metric(Y). \mforall{}f:X {}\mrightarrow{} Y. (UC(f:X {}\mrightarrow{} Y) {}\mRightarrow{} m-TB(X;dX) {}\mRightarrow{} m-TB(f[X];image-metric(dY)))
Date html generated:
2019_10_30-AM-06_51_59
Last ObjectModification:
2019_10_25-PM-02_03_44
Theory : reals
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