Nuprl Lemma : m-sup-property1
∀[X:Type]
∀d:metric(X). ∀mtb:m-TB(X;d). ∀f:X ⟶ ℝ. ∀mc:UC(f:X ⟶ ℝ). sup(λr.∃x:X. (r = (f x))) = m-sup{i:l}(d;mtb;f;mc)
Proof
Definitions occuring in Statement :
m-sup: m-sup{i:l}(d;mtb;f;mc),
m-TB: m-TB(X;d),
m-unif-cont: UC(f:X ⟶ Y),
rmetric: rmetric(),
metric: metric(X),
sup: sup(A) = b,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
pi1: fst(t),
pi2: snd(t),
uimplies: b supposing a,
guard: {T},
rset: Set(ℝ),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
member: t ∈ T,
m-sup: m-sup{i:l}(d;mtb;f;mc),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
sup_wf,
subtype_rel_self,
req_transitivity,
req_inversion,
req_wf,
inf_wf,
rmetric_wf,
m-unif-cont_wf,
real_wf,
m-TB_wf,
metric_wf,
m-TB-sup-and-inf
Rules used in proof :
universeEquality,
independent_functionElimination,
dependent_functionElimination,
equalityIstype,
functionEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
dependent_pairFormation_alt,
productElimination,
productEquality,
dependent_set_memberEquality_alt,
productIsType,
sqequalHypSubstitution,
introduction,
universeIsType,
functionIsType,
isectIsType,
equalitySymmetry,
equalityTransitivity,
isectElimination,
lambdaEquality_alt,
sqequalRule,
applyEquality,
hypothesisEquality,
inhabitedIsType,
hypothesis,
extract_by_obid,
instantiate,
thin,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}mtb:m-TB(X;d). \mforall{}f:X {}\mrightarrow{} \mBbbR{}. \mforall{}mc:UC(f:X {}\mrightarrow{} \mBbbR{}).
sup(\mlambda{}r.\mexists{}x:X. (r = (f x))) = m-sup\{i:l\}(d;mtb;f;mc)
Date html generated:
2019_10_30-AM-06_54_44
Last ObjectModification:
2019_10_25-PM-02_22_02
Theory : reals
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