Nuprl Lemma : axiom-choice-1X-quot
∀X:Type. ∀P:(ℕ ⟶ ℕ) ⟶ X ⟶ ℙ. ((∀f:ℕ ⟶ ℕ. ⇃(∃m:X. (P f m))) ⇒ ⇃(∃F:(ℕ ⟶ ℕ) ⟶ X. ∀f:ℕ ⟶ ℕ. (P f (F f))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
equiv_rel_true,
true_wf,
exists_wf,
quotient_wf,
all_wf,
canonicalizable-nat-to-nat,
canonicalizable_wf,
trivial-quotient-true,
nat_wf,
axiom-choice-quot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
functionEquality,
hypothesis,
independent_functionElimination,
isectElimination,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
applyEquality,
independent_isectElimination,
universeEquality
Latex:
\mforall{}X:Type. \mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}.
((\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}m:X. (P f m))) {}\mRightarrow{} \00D9(\mexists{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} X. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (P f (F f))))
Date html generated:
2016_05_14-PM-09_42_33
Last ObjectModification:
2016_01_06-PM-01_29_32
Theory : continuity
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