Nuprl Lemma : axiom-choice-quot-alt-proof
∀T:Type
(⇃(canonicalizable(T)) ⇒ (∀X:Type. ∀P:T ⟶ X ⟶ ℙ. ((∀f:T. ⇃(∃m:X. (P f m))) ⇒ ⇃(∃F:T ⟶ X. ∀f:T. (P f (F f))))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
canonicalizable: canonicalizable(T),
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 :
pi1: fst(t),
guard: {T},
choice-principle: ChoicePrinciple(T),
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
equal_wf,
implies-quotient-true,
canonicalizable_wf,
equiv_rel_true,
true_wf,
exists_wf,
quotient_wf,
all_wf,
choice-iff-canonicalizable
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
rename,
dependent_pairFormation,
promote_hyp,
universeEquality,
functionEquality,
independent_isectElimination,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
cumulativity,
isectElimination,
hypothesis,
independent_functionElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}T:Type
(\00D9(canonicalizable(T))
{}\mRightarrow{} (\mforall{}X:Type. \mforall{}P:T {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}. ((\mforall{}f:T. \00D9(\mexists{}m:X. (P f m))) {}\mRightarrow{} \00D9(\mexists{}F:T {}\mrightarrow{} X. \mforall{}f:T. (P f (F f))))))
Date html generated:
2018_07_25-PM-01_50_26
Last ObjectModification:
2018_07_25-PM-00_19_24
Theory : continuity
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