Theorem fucidcobi 50552

Description: Componentwise functor laws for a correctly typed hom-family into a functor category. The source category 𝑇 and object assignment 𝑂 are arbitrary: the identity and composition laws are equivalent to their component equations. No functor law is assumed in the hypotheses. Functor composition is an application in fucoisfunc2 50695. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
fucidcobi.w	⊢ 𝑊 = (Base‘𝑇)
fucidcobi.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucidcobi.b	⊢ 𝐵 = (Base‘𝐶)
fucidcobi.h	⊢ 𝐻 = (Hom ‘𝑇)
fucidcobi.i	⊢ 𝐼 = (Id‘𝑇)
fucidcobi.j	⊢ 𝐽 = (Id‘𝑄)
fucidcobi.k	⊢ 𝐾 = (Id‘𝐸)
fucidcobi.x	⊢ · = (comp‘𝑇)
fucidcobi.y	⊢ ⚬ = (comp‘𝑄)
fucidcobi.v	⊢ ∙ = (comp‘𝐸)
fucidcobi.c	⊢ (𝜑 → 𝑇 ∈ Cat)
fucidcobi.p	⊢ (𝜑 → 𝑃 ∈ X𝑣 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑣))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑣))) ↑m (𝐻‘𝑣)))

Assertion

Ref	Expression
fucidcobi	⊢ (𝜑 → (∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))) ↔ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))))

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝐶,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝐸,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝐻,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝐼,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝐽,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝐾,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝑂,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝑃,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝑄,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝑇,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥,𝑊,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝜑,𝑥,𝑦,𝑧,𝑚,𝑛,𝑡   𝑥, · ,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥, ⚬ ,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣   𝑥, ∙ ,𝑦,𝑧,𝑚,𝑛,𝑡,𝑣

Allowed substitution hint:   𝜑(𝑣)

Proof of Theorem fucidcobi

Step	Hyp	Ref	Expression
1		fucidcobi.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
2		eqid 2763	. . . 4 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
3		fucidcobi.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		fucidcobi.j	. . . 4 ⊢ 𝐽 = (Id‘𝑄)
5		fucidcobi.k	. . . 4 ⊢ 𝐾 = (Id‘𝐸)
6		fucidcobi.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ X𝑣 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑣))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑣))) ↑m (𝐻‘𝑣)))
7	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑃 ∈ X𝑣 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑣))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑣))) ↑m (𝐻‘𝑣)))
8		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ 𝑊)
9	7, 8, 8	ixpovf 50402	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (𝑥𝑃𝑥):(𝑥𝐻𝑥)⟶((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑥)))
10		fucidcobi.w	. . . . . 6 ⊢ 𝑊 = (Base‘𝑇)
11		fucidcobi.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝑇)
12		fucidcobi.i	. . . . . 6 ⊢ 𝐼 = (Id‘𝑇)
13		fucidcobi.c	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ Cat)
14	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑇 ∈ Cat)
15	10, 11, 12, 14, 8	catidcl 17885	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (𝐼‘𝑥) ∈ (𝑥𝐻𝑥))
16	9, 15	ffvelcdmd 7109	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((𝑥𝑃𝑥)‘(𝐼‘𝑥)) ∈ ((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑥)))
17	1, 2, 3, 4, 5, 16	natideq 50550	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ↔ ∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡))))
18		fucidcobi.v	. . . . . . . 8 ⊢ ∙ = (comp‘𝐸)
19		fucidcobi.y	. . . . . . . 8 ⊢ ⚬ = (comp‘𝑄)
20		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)))
21	20	simpld 500	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊))
22	21	simpld 500	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊))
23	22	simpld 500	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (𝜑 ∧ 𝑥 ∈ 𝑊))
24	23	simpld 500	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝜑)
25	24, 6	syl 18	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝑃 ∈ X𝑣 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑣))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑣))) ↑m (𝐻‘𝑣)))
26	23	simprd 501	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝑥 ∈ 𝑊)
27	22	simprd 501	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝑦 ∈ 𝑊)
28	25, 26, 27	ixpovf 50402	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (𝑥𝑃𝑦):(𝑥𝐻𝑦)⟶((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑦)))
29	20	simprd 501	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝑚 ∈ (𝑥𝐻𝑦))
30	28, 29	ffvelcdmd 7109	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝑃𝑦)‘𝑚) ∈ ((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑦)))
31	21	simprd 501	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝑧 ∈ 𝑊)
32	25, 27, 31	ixpovf 50402	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (𝑦𝑃𝑧):(𝑦𝐻𝑧)⟶((𝑂‘𝑦)(𝐶 Nat 𝐸)(𝑂‘𝑧)))
33		simpr 490	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝑛 ∈ (𝑦𝐻𝑧))
34	32, 33	ffvelcdmd 7109	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑦𝑃𝑧)‘𝑛) ∈ ((𝑂‘𝑦)(𝐶 Nat 𝐸)(𝑂‘𝑧)))
35	25, 26, 31	ixpovf 50402	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (𝑥𝑃𝑧):(𝑥𝐻𝑧)⟶((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑧)))
36		fucidcobi.x	. . . . . . . . . 10 ⊢ · = (comp‘𝑇)
37	24, 13	syl 18	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → 𝑇 ∈ Cat)
38	10, 11, 36, 37, 26, 27, 31, 29, 33	catcocl 17888	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚) ∈ (𝑥𝐻𝑧))
39	35, 38	ffvelcdmd 7109	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) ∈ ((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑧)))
40	1, 2, 3, 18, 19, 30, 34, 39	natcoeq 50551	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)) ↔ ∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))
41	40	ralbidva 3187	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) → (∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)) ↔ ∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))
42	41	ralbidva 3187	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) → (∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)) ↔ ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))
43	42	ralbidva 3187	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) → (∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)) ↔ ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))
44	43	ralbidva 3187	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)) ↔ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))
45	17, 44	anbi12d 644	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))) ↔ (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))))
46	45	ralbidva 3187	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))) ↔ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟨cop 4838   × cxp 5653  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024   ↑_m cmap 8875  Xcixp 8953  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868   Nat cnat 18148   FuncCat cfuc 18149

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-func 18062  df-nat 18150  df-fuc 18151

This theorem is used by:  fucoisfunc2lem2  50694