Step
*
1
1
2
1
of Lemma
adjunction-monad_wf
1. A : SmallCategory
2. B : SmallCategory
3. F : Functor(A;B)
4. G : Functor(B;A)
5. a1 : A:cat-ob(B) ⟶ (cat-arrow(B) (ob(F) (ob(G) A)) A)
6. ∀A,B@0:cat-ob(B). ∀g:cat-arrow(B) A B@0.
((cat-comp(B) (ob(F) (ob(G) A)) A B@0 (a1 A) g)
= (cat-comp(B) (ob(F) (ob(G) A)) (ob(F) (ob(G) B@0)) B@0 (arrow(F) (ob(G) A) (ob(G) B@0) (arrow(G) A B@0 g))
(a1 B@0))
∈ (cat-arrow(B) (ob(F) (ob(G) A)) B@0))
7. a2 : A@0:cat-ob(A) ⟶ (cat-arrow(A) A@0 (ob(G) (ob(F) A@0)))
8. ∀A@0,B:cat-ob(A). ∀g:cat-arrow(A) A@0 B.
((cat-comp(A) A@0 (ob(G) (ob(F) A@0)) (ob(G) (ob(F) B)) (a2 A@0)
(arrow(G) (ob(F) A@0) (ob(F) B) (arrow(F) A@0 B g)))
= (cat-comp(A) A@0 B (ob(G) (ob(F) B)) g (a2 B))
∈ (cat-arrow(A) A@0 (ob(G) (ob(F) B))))
9. ∀d:cat-ob(A)
((cat-comp(B) (ob(F) d) (ob(F) (ob(G) (ob(F) d))) (ob(F) d) (arrow(F) d (ob(G) (ob(F) d)) ((snd(<a1, a2>)) d))
((fst(<a1, a2>)) (ob(F) d)))
= (cat-id(B) (ob(F) d))
∈ (cat-arrow(B) (ob(F) d) (ob(F) d)))
10. ∀c:cat-ob(B)
((cat-comp(A) (ob(G) c) (ob(G) (ob(F) (ob(G) c))) (ob(G) c) ((snd(<a1, a2>)) (ob(G) c))
(arrow(G) (ob(F) (ob(G) c)) c ((fst(<a1, a2>)) c)))
= (cat-id(A) (ob(G) c))
∈ (cat-arrow(A) (ob(G) c) (ob(G) c)))
11. A1 : cat-ob(A)
12. B1 : cat-ob(A)
13. g : cat-arrow(A) A1 B1
⊢ (cat-comp(A) (ob(G) (ob(F) (ob(G) (ob(F) A1)))) (ob(G) (ob(F) A1)) (ob(G) (ob(F) B1))
(arrow(G) (ob(F) (ob(G) (ob(F) A1))) (ob(F) A1) (a1 (ob(F) A1)))
(arrow(G) (ob(F) A1) (ob(F) B1) (arrow(F) A1 B1 g)))
= (cat-comp(A) (ob(G) (ob(F) (ob(G) (ob(F) A1)))) (ob(G) (ob(F) (ob(G) (ob(F) B1)))) (ob(G) (ob(F) B1))
(arrow(G) (ob(F) (ob(G) (ob(F) A1))) (ob(F) (ob(G) (ob(F) B1)))
(arrow(F) (ob(G) (ob(F) A1)) (ob(G) (ob(F) B1)) (arrow(G) (ob(F) A1) (ob(F) B1) (arrow(F) A1 B1 g))))
(arrow(G) (ob(F) (ob(G) (ob(F) B1))) (ob(F) B1) (a1 (ob(F) B1))))
∈ (cat-arrow(A) (ob(G) (ob(F) (ob(G) (ob(F) A1)))) (ob(G) (ob(F) B1)))
BY
{ ((RWO "functor-arrow-comp<" 0 THENA Auto) THEN RWO "6<" 0 THEN Auto) }
Latex:
Latex:
1. A : SmallCategory
2. B : SmallCategory
3. F : Functor(A;B)
4. G : Functor(B;A)
5. a1 : A:cat-ob(B) {}\mrightarrow{} (cat-arrow(B) (ob(F) (ob(G) A)) A)
6. \mforall{}A,B@0:cat-ob(B). \mforall{}g:cat-arrow(B) A B@0.
((cat-comp(B) (ob(F) (ob(G) A)) A B@0 (a1 A) g)
= (cat-comp(B) (ob(F) (ob(G) A)) (ob(F) (ob(G) B@0)) B@0
(arrow(F) (ob(G) A) (ob(G) B@0) (arrow(G) A B@0 g))
(a1 B@0)))
7. a2 : A@0:cat-ob(A) {}\mrightarrow{} (cat-arrow(A) A@0 (ob(G) (ob(F) A@0)))
8. \mforall{}A@0,B:cat-ob(A). \mforall{}g:cat-arrow(A) A@0 B.
((cat-comp(A) A@0 (ob(G) (ob(F) A@0)) (ob(G) (ob(F) B)) (a2 A@0)
(arrow(G) (ob(F) A@0) (ob(F) B) (arrow(F) A@0 B g)))
= (cat-comp(A) A@0 B (ob(G) (ob(F) B)) g (a2 B)))
9. \mforall{}d:cat-ob(A)
((cat-comp(B) (ob(F) d) (ob(F) (ob(G) (ob(F) d))) (ob(F) d)
(arrow(F) d (ob(G) (ob(F) d)) ((snd(<a1, a2>)) d))
((fst(<a1, a2>)) (ob(F) d)))
= (cat-id(B) (ob(F) d)))
10. \mforall{}c:cat-ob(B)
((cat-comp(A) (ob(G) c) (ob(G) (ob(F) (ob(G) c))) (ob(G) c) ((snd(<a1, a2>)) (ob(G) c))
(arrow(G) (ob(F) (ob(G) c)) c ((fst(<a1, a2>)) c)))
= (cat-id(A) (ob(G) c)))
11. A1 : cat-ob(A)
12. B1 : cat-ob(A)
13. g : cat-arrow(A) A1 B1
\mvdash{} (cat-comp(A) (ob(G) (ob(F) (ob(G) (ob(F) A1)))) (ob(G) (ob(F) A1)) (ob(G) (ob(F) B1))
(arrow(G) (ob(F) (ob(G) (ob(F) A1))) (ob(F) A1) (a1 (ob(F) A1)))
(arrow(G) (ob(F) A1) (ob(F) B1) (arrow(F) A1 B1 g)))
= (cat-comp(A) (ob(G) (ob(F) (ob(G) (ob(F) A1)))) (ob(G) (ob(F) (ob(G) (ob(F) B1))))
(ob(G) (ob(F) B1))
(arrow(G) (ob(F) (ob(G) (ob(F) A1))) (ob(F) (ob(G) (ob(F) B1)))
(arrow(F) (ob(G) (ob(F) A1)) (ob(G) (ob(F) B1))
(arrow(G) (ob(F) A1) (ob(F) B1) (arrow(F) A1 B1 g))))
(arrow(G) (ob(F) (ob(G) (ob(F) B1))) (ob(F) B1) (a1 (ob(F) B1))))
By
Latex:
((RWO "functor-arrow-comp<" 0 THENA Auto) THEN RWO "6<" 0 THEN Auto)
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