Theorem fucidcobi 50438

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 50581. (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 2760	. . . 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 50288	. . . . 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 17858	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (𝐼‘𝑥) ∈ (𝑥𝐻𝑥))
16	9, 15	ffvelcdmd 7085	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((𝑥𝑃𝑥)‘(𝐼‘𝑥)) ∈ ((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑥)))
17	1, 2, 3, 4, 5, 16	natideq 50436	. . 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 50288	. . . . . . . . 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 7085	. . . . . . . 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 50288	. . . . . . . . 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 7085	. . . . . . . 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 50288	. . . . . . . . 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 17861	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ 𝑚 ∈ (𝑥𝐻𝑦)) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → (𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚) ∈ (𝑥𝐻𝑧))
39	35, 38	ffvelcdmd 7085	. . . . . . . 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 50437	. . . . . . 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 3183	. . . . . 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 3183	. . . . 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 3183	. . . 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 3183	. . 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 3183	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 1569   ∈ wcel 2144  ∀wral 3076  ⟨cop 4589   × cxp 5648  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8849  Xcixp 8927  Basecbs 17389  Hom chom 17441  compcco 17442  Catccat 17840  Idccid 17841   Nat cnat 18121   FuncCat cfuc 18122

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11258  ax-resscn 11259  ax-1cn 11260  ax-icn 11261  ax-addcl 11262  ax-addrcl 11263  ax-mulcl 11264  ax-mulrcl 11265  ax-mulcom 11266  ax-addass 11267  ax-mulass 11268  ax-distr 11269  ax-i2m1 11270  ax-1ne0 11271  ax-1rid 11272  ax-rnegex 11273  ax-rrecex 11274  ax-cnre 11275  ax-pre-lttri 11276  ax-pre-lttrn 11277  ax-pre-ltadd 11278  ax-pre-mulgt0 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8301  df-wrecs 8332  df-recs 8381  df-rdg 8420  df-1o 8478  df-er 8719  df-map 8851  df-ixp 8928  df-en 8976  df-dom 8977  df-sdom 8978  df-fin 8979  df-pnf 11347  df-mnf 11348  df-xr 11349  df-ltxr 11350  df-le 11351  df-sub 11545  df-neg 11546  df-nn 12338  df-2 12407  df-3 12408  df-4 12409  df-5 12410  df-6 12411  df-7 12412  df-8 12413  df-9 12414  df-n0 12609  df-z 12696  df-dec 12817  df-uz 12968  df-fz 13642  df-struct 17327  df-slot 17362  df-ndx 17374  df-base 17390  df-hom 17454  df-cco 17455  df-cat 17844  df-cid 17845  df-func 18035  df-nat 18123  df-fuc 18124

This theorem is used by:  fucoisfunc2lem2  50580