Nuprl Lemma : equipollent-function-product
∀[A,B,C:Type]. C ⟶ (A × B) ~ C ⟶ A × (C ⟶ B)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
surject: Surj(A;B;f),
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top
Lemmas referenced :
pi1_wf,
pi2_wf,
equal_wf,
biject_wf,
pair-eta,
subtype_rel_product,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_pairFormation,
lambdaEquality,
independent_pairEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
applyEquality,
hypothesis,
functionEquality,
productEquality,
independent_pairFormation,
lambdaFormation,
universeEquality,
functionExtensionality,
equalityUniverse,
levelHypothesis,
because_Cache,
productElimination,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[A,B,C:Type]. C {}\mrightarrow{} (A \mtimes{} B) \msim{} C {}\mrightarrow{} A \mtimes{} (C {}\mrightarrow{} B)
Date html generated:
2016_05_14-PM-04_00_44
Last ObjectModification:
2015_12_26-PM-07_44_02
Theory : equipollence!!cardinality!
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