Nuprl Lemma : m-sup_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[mtb:m-TB(X;d)]. ∀[f:X ⟶ ℝ]. ∀[mc:UC(f:X ⟶ ℝ)]. (m-sup{i:l}(d;mtb;f;mc) ∈ ℝ)
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),
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
top: Top,
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,
all: ∀x:A. B[x],
m-sup: m-sup{i:l}(d;mtb;f;mc),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
istype-void,
pi1_wf_top,
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,
voidElimination,
isect_memberEquality_alt,
independent_pairEquality,
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,
lambdaFormation_alt,
inhabitedIsType,
hypothesis,
extract_by_obid,
instantiate,
thin,
cut,
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{})].
(m-sup\{i:l\}(d;mtb;f;mc) \mmember{} \mBbbR{})
Date html generated:
2019_10_30-AM-06_54_24
Last ObjectModification:
2019_10_25-PM-02_20_20
Theory : reals
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