Theorem catass 17626

Description: Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
catcocl.b	⊢ 𝐵 = (Base‘𝐶)
catcocl.h	⊢ 𝐻 = (Hom ‘𝐶)
catcocl.o	⊢ · = (comp‘𝐶)
catcocl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
catcocl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
catcocl.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
catcocl.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
catcocl.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
catcocl.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
catass.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
catass.g	⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))

Assertion

Ref	Expression
catass	⊢ (𝜑 → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)))

Proof of Theorem catass

Dummy variables 𝑓 𝑔 𝑘 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

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 17612	. . . 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 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))
6	5	ibi 266	. . 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 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))))
7	1, 6	syl 17	. 2 ⊢ (𝜑 → ∀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 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))))
8		catcocl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catcocl.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵)
11		catcocl.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
12	11	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵)
13		catcocl.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
14	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑋𝐻𝑌))
15		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
16		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
17	15, 16	oveq12d 7423	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
18	14, 17	eleqtrrd 2836	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑥𝐻𝑦))
19		catcocl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
20	19	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑌𝐻𝑍))
21		simpllr 774	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑦 = 𝑌)
22		simplr 767	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑧 = 𝑍)
23	21, 22	oveq12d 7423	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (𝑦𝐻𝑧) = (𝑌𝐻𝑍))
24	20, 23	eleqtrrd 2836	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑦𝐻𝑧))
25		catass.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
26	25	ad5antr 732	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑊 ∈ 𝐵)
27		catass.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))
28	27	ad6antr 734	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) → 𝐾 ∈ (𝑍𝐻𝑊))
29		simp-4r 782	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) → 𝑧 = 𝑍)
30		simpr 485	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊)
31	29, 30	oveq12d 7423	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) → (𝑧𝐻𝑤) = (𝑍𝐻𝑊))
32	28, 31	eleqtrrd 2836	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) → 𝐾 ∈ (𝑧𝐻𝑤))
33		simp-7r 788	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑥 = 𝑋)
34		simp-6r 786	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑦 = 𝑌)
35	33, 34	opeq12d 4880	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
36		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑤 = 𝑊)
37	35, 36	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (⟨𝑥, 𝑦⟩ · 𝑤) = (⟨𝑋, 𝑌⟩ · 𝑊))
38		simp-5r 784	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑧 = 𝑍)
39	34, 38	opeq12d 4880	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨𝑦, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
40	39, 36	oveq12d 7423	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (⟨𝑦, 𝑧⟩ · 𝑤) = (⟨𝑌, 𝑍⟩ · 𝑊))
41		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
42		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑔 = 𝐺)
43	40, 41, 42	oveq123d 7426	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔) = (𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺))
44		simp-4r 782	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑓 = 𝐹)
45	37, 43, 44	oveq123d 7426	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹))
46	33, 38	opeq12d 4880	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
47	46, 36	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (⟨𝑥, 𝑧⟩ · 𝑤) = (⟨𝑋, 𝑍⟩ · 𝑊))
48	35, 38	oveq12d 7423	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
49	48, 42, 44	oveq123d 7426	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
50	47, 41, 49	oveq123d 7426	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)))
51	45, 50	eqeq12d 2748	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) ↔ ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
52	32, 51	rspcdv 3604	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) ∧ 𝑤 = 𝑊) → (∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
53	26, 52	rspcimdv 3602	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
54	53	adantld 491	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
55	24, 54	rspcimdv 3602	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
56	18, 55	rspcimdv 3602	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
57	12, 56	rspcimdv 3602	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
58	10, 57	rspcimdv 3602	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
59	58	adantld 491	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((∃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 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
60	8, 59	rspcimdv 3602	. 2 ⊢ (𝜑 → (∀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 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))) → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
61	7, 60	mpd 15	1 ⊢ (𝜑 → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  ⟨cop 4633  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-cat 17608

This theorem is referenced by:  oppccatid  17661  sectcan  17698  sectco  17699  sectmon  17725  monsect  17726  rcaninv  17737  subccatid  17792  fuccocl  17913  fucass  17917  invfuc  17923  arwass  18020  xpccatid  18136  evlfcllem  18170  hofcllem  18207  bj-endmnd  36187  endmndlem  47588