Nuprl Lemma : product_functionality_wrt_equipollent_left
∀[A,B,C,D:Type]. (A ~ B ⇒ A × C ~ B × D supposing C = D ∈ Type)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
all: ∀x:A. B[x],
pi2: snd(t),
pi1: fst(t),
guard: {T}
Lemmas referenced :
exists_wf,
biject_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
sqequalHypSubstitution,
productElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
extract_by_obid,
isectElimination,
functionEquality,
productEquality,
cumulativity,
hypothesisEquality,
lambdaEquality,
functionExtensionality,
applyEquality,
instantiate,
universeEquality,
dependent_pairFormation,
spreadEquality,
independent_pairEquality,
independent_pairFormation,
promote_hyp,
because_Cache,
equalityUniverse,
levelHypothesis,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A,B,C,D:Type]. (A \msim{} B {}\mRightarrow{} A \mtimes{} C \msim{} B \mtimes{} D supposing C = D)
Date html generated:
2016_10_21-AM-10_51_59
Last ObjectModification:
2016_07_12-AM-05_55_56
Theory : equipollence!!cardinality!
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