Step | Hyp | Ref
| Expression |
1 | | catcocl.c |
. . 3
⊢ (𝜑 → 𝐶 ∈ Cat) |
2 | | catcocl.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐶) |
3 | | catcocl.h |
. . . . 5
⊢ 𝐻 = (Hom ‘𝐶) |
4 | | catcocl.o |
. . . . 5
⊢ · =
(comp‘𝐶) |
5 | 2, 3, 4 | iscat 16156 |
. . . 4
⊢ (𝐶 ∈ Cat → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) |
6 | 5 | ibi 255 |
. . 3
⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))))) |
7 | | simpl 472 |
. . . . . . . . 9
⊢ (((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
8 | 7 | ralimi 2936 |
. . . . . . . 8
⊢
(∀𝑔 ∈
(𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) → ∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
9 | 8 | ralimi 2936 |
. . . . . . 7
⊢
(∀𝑓 ∈
(𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
10 | 9 | ralimi 2936 |
. . . . . 6
⊢
(∀𝑧 ∈
𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) → ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
11 | 10 | ralimi 2936 |
. . . . 5
⊢
(∀𝑦 ∈
𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
12 | 11 | adantl 481 |
. . . 4
⊢
((∃𝑔 ∈
(𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
13 | 12 | ralimi 2936 |
. . 3
⊢
(∀𝑥 ∈
𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ (𝑧𝐻𝑤)((𝑣(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑣(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
14 | 1, 6, 13 | 3syl 18 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
15 | | catcocl.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
16 | | catcocl.y |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
17 | 16 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵) |
18 | | catcocl.z |
. . . . . 6
⊢ (𝜑 → 𝑍 ∈ 𝐵) |
19 | 18 | ad2antrr 758 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵) |
20 | | catcocl.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
21 | 20 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑋𝐻𝑌)) |
22 | | simpllr 795 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋) |
23 | | simplr 788 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌) |
24 | 22, 23 | oveq12d 6567 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
25 | 21, 24 | eleqtrrd 2691 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑥𝐻𝑦)) |
26 | | catcocl.g |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) |
27 | 26 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑌𝐻𝑍)) |
28 | | simpr 476 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍) |
29 | 23, 28 | oveq12d 6567 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑦𝐻𝑧) = (𝑌𝐻𝑍)) |
30 | 27, 29 | eleqtrrd 2691 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑦𝐻𝑧)) |
31 | 30 | adantr 480 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑦𝐻𝑧)) |
32 | | simp-5r 805 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑥 = 𝑋) |
33 | | simp-4r 803 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑌) |
34 | 32, 33 | opeq12d 4348 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 〈𝑥, 𝑦〉 = 〈𝑋, 𝑌〉) |
35 | | simpllr 795 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑧 = 𝑍) |
36 | 34, 35 | oveq12d 6567 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (〈𝑥, 𝑦〉 · 𝑧) = (〈𝑋, 𝑌〉 · 𝑍)) |
37 | | simpr 476 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺) |
38 | | simplr 788 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹) |
39 | 36, 37, 38 | oveq123d 6570 |
. . . . . . . 8
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
40 | 32, 35 | oveq12d 6567 |
. . . . . . . 8
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐻𝑧) = (𝑋𝐻𝑍)) |
41 | 39, 40 | eleq12d 2682 |
. . . . . . 7
⊢
((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))) |
42 | 31, 41 | rspcdv 3285 |
. . . . . 6
⊢
(((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))) |
43 | 25, 42 | rspcimdv 3283 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))) |
44 | 19, 43 | rspcimdv 3283 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))) |
45 | 17, 44 | rspcimdv 3283 |
. . 3
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))) |
46 | 15, 45 | rspcimdv 3283 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))) |
47 | 14, 46 | mpd 15 |
1
⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍)) |