Nuprl Lemma : equipollent-product-sum
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[C:a:A ⟶ B[a] ⟶ Type]. x:A ⟶ (y:B[x] × C[x;y]) ~ f:a:A ⟶ B[a] × (x:A ⟶ C[x;f x])
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
apply: f a,
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_apply: x[s],
prop: ℙ,
pi2: snd(t),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
and: P ∧ Q,
pi1: fst(t),
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f)
Lemmas referenced :
equal_wf,
subtype_rel-equal,
pi1_wf,
and_wf,
biject_wf,
subtype_rel_self,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_pairFormation,
lambdaEquality,
dependent_pairEquality,
cut,
because_Cache,
thin,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
hypothesisEquality,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
cumulativity,
productElimination,
applyEquality,
functionExtensionality,
productEquality,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
rename,
functionEquality,
universeEquality,
hyp_replacement
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[C:a:A {}\mrightarrow{} B[a] {}\mrightarrow{} Type].
x:A {}\mrightarrow{} (y:B[x] \mtimes{} C[x;y]) \msim{} f:a:A {}\mrightarrow{} B[a] \mtimes{} (x:A {}\mrightarrow{} C[x;f x])
Date html generated:
2017_04_17-AM-09_32_38
Last ObjectModification:
2017_02_27-PM-05_32_12
Theory : equipollence!!cardinality!
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