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Step * 2 2 of Lemma functor-curry-iso


1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. functor-uncurry(C) ∈ Functor(FUN(A;FUN(B;C));FUN(A × B;C))
5. functor-curry(A;B) ∈ Functor(FUN(A × B;C);FUN(A;FUN(B;C)))
6. ∀x:Functor(A × B;C). ((ob(functor-comp(functor-curry(A;B);functor-uncurry(C))) x) = x ∈ Functor(A × B;C))
7. ∀f:Functor(A;FUN(B;C)). ∀a:cat-ob(A).
     ((ob(ob(functor-comp(functor-uncurry(C);functor-curry(A;B))) f) a) = (ob(f) a) ∈ cat-ob(FUN(B;C)))
8. functor-curry(A;B)functor-uncurry(C)=1
9. ∀x:Functor(A;FUN(B;C)). ((ob(functor-comp(functor-uncurry(C);functor-curry(A;B))) x) = x ∈ Functor(A;FUN(B;C)))
⊢ functor-comp(functor-uncurry(C);functor-curry(A;B)) = 1 ∈ Functor(FUN(A;FUN(B;C));FUN(A;FUN(B;C)))
BY
{ ((BLemma `equal-functors` THENW Auto) THEN RepUR ``id_functor`` 0 THEN Try (Trivial) THEN Intros) }

1
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. functor-uncurry(C) ∈ Functor(FUN(A;FUN(B;C));FUN(A × B;C))
5. functor-curry(A;B) ∈ Functor(FUN(A × B;C);FUN(A;FUN(B;C)))
6. ∀x:Functor(A × B;C). ((ob(functor-comp(functor-curry(A;B);functor-uncurry(C))) x) = x ∈ Functor(A × B;C))
7. ∀f:Functor(A;FUN(B;C)). ∀a:cat-ob(A).
     ((ob(ob(functor-comp(functor-uncurry(C);functor-curry(A;B))) f) a) = (ob(f) a) ∈ cat-ob(FUN(B;C)))
8. functor-curry(A;B)functor-uncurry(C)=1
9. ∀x:Functor(A;FUN(B;C)). ((ob(functor-comp(functor-uncurry(C);functor-curry(A;B))) x) = x ∈ Functor(A;FUN(B;C)))
10. x : Functor(A;FUN(B;C))
11. y : Functor(A;FUN(B;C))
12. f : nat-trans(A;FUN(B;C);x;y)
⊢ (arrow(functor-comp(functor-uncurry(C);functor-curry(A;B))) x y f)
= f
∈ nat-trans(A;FUN(B;C);ob(functor-comp(functor-uncurry(C);functor-curry(A;B))) 
                       x;ob(functor-comp(functor-uncurry(C);functor-curry(A;B))) y)

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Latex:

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Latex:

1.  A  :  SmallCategory
2.  B  :  SmallCategory
3.  C  :  SmallCategory
4.  functor-uncurry(C)  \mmember{}  Functor(FUN(A;FUN(B;C));FUN(A  \mtimes{}  B;C))
5.  functor-curry(A;B)  \mmember{}  Functor(FUN(A  \mtimes{}  B;C);FUN(A;FUN(B;C)))
6.  \mforall{}x:Functor(A  \mtimes{}  B;C).  ((ob(functor-comp(functor-curry(A;B);functor-uncurry(C)))  x)  =  x)
7.  \mforall{}f:Functor(A;FUN(B;C)).  \mforall{}a:cat-ob(A).
          ((ob(ob(functor-comp(functor-uncurry(C);functor-curry(A;B)))  f)  a)  =  (ob(f)  a))
8.  functor-curry(A;B)functor-uncurry(C)=1
9.  \mforall{}x:Functor(A;FUN(B;C)).  ((ob(functor-comp(functor-uncurry(C);functor-curry(A;B)))  x)  =  x)
\mvdash{}  functor-comp(functor-uncurry(C);functor-curry(A;B))  =  1


By

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Latex:
((BLemma  `equal-functors`  THENW  Auto)  THEN  RepUR  ``id\_functor``  0  THEN  Try  (Trivial)  THEN  Intros)

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