Nuprl Lemma : sp-lub-property
∀[A:ℕ ⟶ Sierpinski]. ((∀n:ℕ. A[n] ≤ lub(n.A[n])) ∧ (∀c:Sierpinski. ((∀n:ℕ. A[n] ≤ c) ⇒ lub(n.A[n]) ≤ c)))
Proof
Definitions occuring in Statement :
sp-le: x ≤ y,
sp-lub: lub(n.A[n]),
Sierpinski: Sierpinski,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sp-le: x ≤ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
exists: ∃x:A. B[x],
false: False
Lemmas referenced :
nat_wf,
all_wf,
sp-le_wf,
Sierpinski_wf,
equal-wf-T-base,
sp-lub_wf,
sp-lub-is-top,
not_wf,
exists_wf,
not-Sierpinski-bottom,
Sierpinski-unequal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
hypothesis,
independent_pairFormation,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
productElimination,
independent_pairEquality,
dependent_functionElimination,
axiomEquality,
baseClosed,
because_Cache,
functionEquality,
independent_functionElimination,
dependent_pairFormation,
voidElimination,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}[A:\mBbbN{} {}\mrightarrow{} Sierpinski]
((\mforall{}n:\mBbbN{}. A[n] \mleq{} lub(n.A[n])) \mwedge{} (\mforall{}c:Sierpinski. ((\mforall{}n:\mBbbN{}. A[n] \mleq{} c) {}\mRightarrow{} lub(n.A[n]) \mleq{} c)))
Date html generated:
2019_10_31-AM-06_36_23
Last ObjectModification:
2017_07_28-AM-09_12_11
Theory : synthetic!topology
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