fucocolem1 - Mathbox for Zhi Wang

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Theorem fucocolem1 49008

Description: Lemma for fucoco 49012. Associativity for morphisms in category 𝐸. To simply put, ((𝑎 · 𝑏) · (𝑐 · 𝑑)) = (𝑎 · ((𝑏 · 𝑐) · 𝑑)) for morphism compositions. (Contributed by Zhi Wang, 2-Oct-2025.)

**Hypotheses**
Ref	Expression
fucoco.r	⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
fucoco.s	⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
fucoco.u	⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
fucoco.v	⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
fucocolem1.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
fucocolem1.p	⊢ (𝜑 → 𝑃 ∈ (𝐷 Func 𝐸))
fucocolem1.q	⊢ (𝜑 → 𝑄 ∈ (𝐶 Func 𝐷))
fucocolem1.a	⊢ (𝜑 → 𝐴 ∈ (((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))(Hom ‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))))
fucocolem1.b	⊢ (𝜑 → 𝐵 ∈ (((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))(Hom ‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))))

**Assertion**
Ref	Expression
fucocolem1	⊢ (𝜑 → (((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))𝐴)(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))(𝐵(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))) = ((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))((𝐴(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))𝐵)(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))))

**Proof of Theorem fucocolem1**
Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
2		eqid 2734	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
3		eqid 2734	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
4		relfunc 17879	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
5		fucoco.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
6		eqid 2734	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
7	6	natrcl 17970	. . . . . . 7 ⊢ (𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
8	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
9	8	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
10		1st2ndbr 8050	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
11	4, 9, 10	sylancr 587	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
12	11	funcrcl3 48866	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		eqid 2734	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
14	13, 1, 11	funcf1 17883	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐷)⟶(Base‘𝐸))
15		eqid 2734	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
16		relfunc 17879	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
17		fucoco.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
18		eqid 2734	. . . . . . . . . 10 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
19	18	natrcl 17970	. . . . . . . . 9 ⊢ (𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
20	17, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
21	20	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
22		1st2ndbr 8050	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
23	16, 21, 22	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
24	15, 13, 23	funcf1 17883	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
25		fucocolem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
26	24, 25	ffvelcdmd 7086	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (Base‘𝐷))
27	14, 26	ffvelcdmd 7086	. . 3 ⊢ (𝜑 → ((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)) ∈ (Base‘𝐸))
28		fucocolem1.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ (𝐷 Func 𝐸))
29		1st2ndbr 8050	. . . . . 6 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑃 ∈ (𝐷 Func 𝐸)) → (1^st ‘𝑃)(𝐷 Func 𝐸)(2^nd ‘𝑃))
30	4, 28, 29	sylancr 587	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑃)(𝐷 Func 𝐸)(2^nd ‘𝑃))
31	13, 1, 30	funcf1 17883	. . . 4 ⊢ (𝜑 → (1^st ‘𝑃):(Base‘𝐷)⟶(Base‘𝐸))
32		fucocolem1.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (𝐶 Func 𝐷))
33		1st2ndbr 8050	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑄 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝑄)(𝐶 Func 𝐷)(2^nd ‘𝑄))
34	16, 32, 33	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑄)(𝐶 Func 𝐷)(2^nd ‘𝑄))
35	15, 13, 34	funcf1 17883	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑄):(Base‘𝐶)⟶(Base‘𝐷))
36	35, 25	ffvelcdmd 7086	. . . 4 ⊢ (𝜑 → ((1^st ‘𝑄)‘𝑋) ∈ (Base‘𝐷))
37	31, 36	ffvelcdmd 7086	. . 3 ⊢ (𝜑 → ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)) ∈ (Base‘𝐸))
38	8	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
39		1st2ndbr 8050	. . . . . 6 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
40	4, 38, 39	sylancr 587	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
41	13, 1, 40	funcf1 17883	. . . 4 ⊢ (𝜑 → (1^st ‘𝐾):(Base‘𝐷)⟶(Base‘𝐸))
42		fucoco.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
43	18	natrcl 17970	. . . . . . . . 9 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
44	42, 43	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
45	44	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
46		1st2ndbr 8050	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
47	16, 45, 46	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
48	15, 13, 47	funcf1 17883	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
49	48, 25	ffvelcdmd 7086	. . . 4 ⊢ (𝜑 → ((1^st ‘𝑁)‘𝑋) ∈ (Base‘𝐷))
50	41, 49	ffvelcdmd 7086	. . 3 ⊢ (𝜑 → ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)) ∈ (Base‘𝐸))
51	20	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
52		1st2ndbr 8050	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
53	16, 51, 52	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
54	15, 13, 53	funcf1 17883	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
55	54, 25	ffvelcdmd 7086	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐿)‘𝑋) ∈ (Base‘𝐷))
56	14, 55	ffvelcdmd 7086	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋)) ∈ (Base‘𝐸))
57		eqid 2734	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
58	13, 57, 2, 11, 26, 55	funcf2 17885	. . . . 5 ⊢ (𝜑 → (((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋)):(((1^st ‘𝐺)‘𝑋)(Hom ‘𝐷)((1^st ‘𝐿)‘𝑋))⟶(((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋))(Hom ‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))))
59	18, 17	nat1st2nd 17971	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩))
60	18, 59, 15, 57, 25	natcl 17973	. . . . 5 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (((1^st ‘𝐺)‘𝑋)(Hom ‘𝐷)((1^st ‘𝐿)‘𝑋)))
61	58, 60	ffvelcdmd 7086	. . . 4 ⊢ (𝜑 → ((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)) ∈ (((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋))(Hom ‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))))
62		fucocolem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))(Hom ‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))))
63	1, 2, 3, 12, 27, 56, 37, 61, 62	catcocl 17700	. . 3 ⊢ (𝜑 → (𝐵(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋))) ∈ (((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋))(Hom ‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))))
64		fucocolem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))(Hom ‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))))
65		fucoco.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
66	6	natrcl 17970	. . . . . . . 8 ⊢ (𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
67	65, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
68	67	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐷 Func 𝐸))
69		1st2ndbr 8050	. . . . . 6 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)) → (1^st ‘𝑀)(𝐷 Func 𝐸)(2^nd ‘𝑀))
70	4, 68, 69	sylancr 587	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑀)(𝐷 Func 𝐸)(2^nd ‘𝑀))
71	13, 1, 70	funcf1 17883	. . . 4 ⊢ (𝜑 → (1^st ‘𝑀):(Base‘𝐷)⟶(Base‘𝐸))
72	71, 49	ffvelcdmd 7086	. . 3 ⊢ (𝜑 → ((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)) ∈ (Base‘𝐸))
73	6, 65	nat1st2nd 17971	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝑀), (2^nd ‘𝑀)⟩))
74	6, 73, 13, 2, 49	natcl 17973	. . 3 ⊢ (𝜑 → (𝑈‘((1^st ‘𝑁)‘𝑋)) ∈ (((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))(Hom ‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋))))
75	1, 2, 3, 12, 27, 37, 50, 63, 64, 72, 74	catass 17701	. 2 ⊢ (𝜑 → (((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))𝐴)(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))(𝐵(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))) = ((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))(𝐴(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))(𝐵(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋))))))
76	1, 2, 3, 12, 27, 56, 37, 61, 62, 50, 64	catass 17701	. . 3 ⊢ (𝜑 → ((𝐴(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))𝐵)(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋))) = (𝐴(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))(𝐵(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))))
77	76	oveq2d 7430	. 2 ⊢ (𝜑 → ((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))((𝐴(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))𝐵)(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))) = ((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))(𝐴(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))(𝐵(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋))))))
78	75, 77	eqtr4d 2772	1 ⊢ (𝜑 → (((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))𝐴)(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))(𝐵(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))) = ((𝑈‘((1^st ‘𝑁)‘𝑋))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)))((𝐴(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋)), ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))𝐵)(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)))((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⟨cop 4614 class class class wbr 5125 Rel wrel 5672 ‘cfv 6542 (class class class)co 7414 1^st c1st 7995 2^nd c2nd 7996 Basecbs 17230 Hom chom 17285 compcco 17286 Func cfunc 17871 Nat cnat 17961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-id 5560 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7997 df-2nd 7998 df-map 8851 df-ixp 8921 df-cat 17683 df-func 17875 df-nat 17963

This theorem is referenced by: fucocolem3 49010 fucoco 49012

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