fucocolem1 - Mathbox for Zhi Wang

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Theorem fucocolem1 50568

Description: Lemma for fucoco 50572. 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 2760	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
2		eqid 2760	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
3		eqid 2760	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
4		fucoco.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
5		eqid 2760	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6	5	natrcl 18133	. . . . . . 7 ⊢ (𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
7	4, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
8	7	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
9	8	func1st2nd 50283	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
10	9	funcrcl3 50287	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
11		eqid 2760	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
12	11, 1, 9	funcf1 18046	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐷)⟶(Base‘𝐸))
13		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		fucoco.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
15		eqid 2760	. . . . . . . . . 10 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
16	15	natrcl 18133	. . . . . . . . 9 ⊢ (𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
17	14, 16	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
18	17	simpld 500	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
19	18	func1st2nd 50283	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
20	13, 11, 19	funcf1 18046	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
21		fucocolem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
22	20, 21	ffvelcdmd 7085	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (Base‘𝐷))
23	12, 22	ffvelcdmd 7085	. . 3 ⊢ (𝜑 → ((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑋)) ∈ (Base‘𝐸))
24		fucocolem1.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ (𝐷 Func 𝐸))
25	24	func1st2nd 50283	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑃)(𝐷 Func 𝐸)(2^nd ‘𝑃))
26	11, 1, 25	funcf1 18046	. . . 4 ⊢ (𝜑 → (1^st ‘𝑃):(Base‘𝐷)⟶(Base‘𝐸))
27		fucocolem1.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (𝐶 Func 𝐷))
28	27	func1st2nd 50283	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑄)(𝐶 Func 𝐷)(2^nd ‘𝑄))
29	13, 11, 28	funcf1 18046	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑄):(Base‘𝐶)⟶(Base‘𝐷))
30	29, 21	ffvelcdmd 7085	. . . 4 ⊢ (𝜑 → ((1^st ‘𝑄)‘𝑋) ∈ (Base‘𝐷))
31	26, 30	ffvelcdmd 7085	. . 3 ⊢ (𝜑 → ((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋)) ∈ (Base‘𝐸))
32	7	simprd 501	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
33	32	func1st2nd 50283	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
34	11, 1, 33	funcf1 18046	. . . 4 ⊢ (𝜑 → (1^st ‘𝐾):(Base‘𝐷)⟶(Base‘𝐸))
35		fucoco.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
36	15	natrcl 18133	. . . . . . . . 9 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
37	35, 36	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
38	37	simprd 501	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
39	38	func1st2nd 50283	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
40	13, 11, 39	funcf1 18046	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
41	40, 21	ffvelcdmd 7085	. . . 4 ⊢ (𝜑 → ((1^st ‘𝑁)‘𝑋) ∈ (Base‘𝐷))
42	34, 41	ffvelcdmd 7085	. . 3 ⊢ (𝜑 → ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋)) ∈ (Base‘𝐸))
43	17	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
44	43	func1st2nd 50283	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
45	13, 11, 44	funcf1 18046	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
46	45, 21	ffvelcdmd 7085	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐿)‘𝑋) ∈ (Base‘𝐷))
47	12, 46	ffvelcdmd 7085	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋)) ∈ (Base‘𝐸))
48		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
49	11, 48, 2, 9, 22, 46	funcf2 18048	. . . . 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))))
50	15, 14	nat1st2nd 18134	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩))
51	15, 50, 13, 48, 21	natcl 18136	. . . . 5 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (((1^st ‘𝐺)‘𝑋)(Hom ‘𝐷)((1^st ‘𝐿)‘𝑋)))
52	49, 51	ffvelcdmd 7085	. . . 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))))
53		fucocolem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑋))(Hom ‘𝐸)((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))))
54	1, 2, 3, 10, 23, 47, 31, 52, 53	catcocl 17864	. . 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))))
55		fucocolem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1^st ‘𝑃)‘((1^st ‘𝑄)‘𝑋))(Hom ‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))))
56		fucoco.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
57	5	natrcl 18133	. . . . . . . 8 ⊢ (𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
58	56, 57	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
59	58	simprd 501	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐷 Func 𝐸))
60	59	func1st2nd 50283	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑀)(𝐷 Func 𝐸)(2^nd ‘𝑀))
61	11, 1, 60	funcf1 18046	. . . 4 ⊢ (𝜑 → (1^st ‘𝑀):(Base‘𝐷)⟶(Base‘𝐸))
62	61, 41	ffvelcdmd 7085	. . 3 ⊢ (𝜑 → ((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋)) ∈ (Base‘𝐸))
63	5, 56	nat1st2nd 18134	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝑀), (2^nd ‘𝑀)⟩))
64	5, 63, 11, 2, 41	natcl 18136	. . 3 ⊢ (𝜑 → (𝑈‘((1^st ‘𝑁)‘𝑋)) ∈ (((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑋))(Hom ‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑋))))
65	1, 2, 3, 10, 23, 31, 42, 54, 55, 62, 64	catass 17865	. 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))))))
66	1, 2, 3, 10, 23, 47, 31, 52, 53, 42, 55	catass 17865	. . 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)))))
67	66	oveq2d 7436	. 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))))))
68	65, 67	eqtr4d 2798	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 Basecbs 17392 Hom chom 17444 compcco 17445 Func cfunc 18034 Nat cnat 18124

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

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 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-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 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-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-map 8852 df-ixp 8929 df-cat 17847 df-func 18038 df-nat 18126

This theorem is used by: fucocolem3 50570 fucoco 50572

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