Theorem fucco 18190

Description: Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fucco.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucco.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucco.a	⊢ 𝐴 = (Base‘𝐶)
fucco.o	⊢ · = (comp‘𝐷)
fucco.x	⊢ ∙ = (comp‘𝑄)
fucco.f	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
fucco.g	⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))

Assertion

Ref	Expression
fucco	⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑥,𝐶   𝑥,𝐷   𝑥, ·   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻

Allowed substitution hints:   𝑄(𝑥)   ∙ (𝑥)   𝑁(𝑥)

Proof of Theorem fucco

Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucco.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		eqid 2763	. . . 4 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷)
3		fucco.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		fucco.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
5		fucco.o	. . . 4 ⊢ · = (comp‘𝐷)
6		fucco.f	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
7	3	natrcl 18178	. . . . . . . 8 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
9	8	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		funcrcl 18088	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	9, 10	syl 18	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
12	11	simpld 500	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	11	simprd 501	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		fucco.x	. . . 4 ⊢ ∙ = (comp‘𝑄)
15	1, 2, 3, 4, 5, 12, 13, 14	fuccofval 18187	. . 3 ⊢ (𝜑 → ∙ = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ℎ ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
16		fvexd 6931	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) ∈ V)
17		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
18	17	fveq2d 6920	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = (1st ‘⟨𝐹, 𝐺⟩))
19		op1stg 8047	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
20	8, 19	syl 18	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
22	18, 21	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = 𝐹)
23		fvexd 6931	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) ∈ V)
24	17	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
25	24	fveq2d 6920	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = (2nd ‘⟨𝐹, 𝐺⟩))
26		op2ndg 8048	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
27	8, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
28	27	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	25, 28	eqtrd 2798	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = 𝐺)
30		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ℎ = 𝐻)
32	31	ad2antrr 739	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ℎ = 𝐻)
33	30, 32	oveq12d 7469	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁ℎ) = (𝐺𝑁𝐻))
34		simplr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
35	34, 30	oveq12d 7469	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
36	34	fveq2d 6920	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑓) = (1st ‘𝐹))
37	36	fveq1d 6918	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
38	30	fveq2d 6920	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑔) = (1st ‘𝐺))
39	38	fveq1d 6918	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
40	37, 39	opeq12d 4853	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ = ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)
41	32	fveq2d 6920	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘ℎ) = (1st ‘𝐻))
42	41	fveq1d 6918	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘ℎ)‘𝑥) = ((1st ‘𝐻)‘𝑥))
43	40, 42	oveq12d 7469	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥)) = (⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥)))
44	43	oveqd 7468	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) = ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))
45	44	mpteq2dv 5205	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥))))
46	33, 35, 45	mpoeq123dv 7526	. . . . 5 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
47	23, 29, 46	csbied2 3888	. . . 4 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
48	16, 22, 47	csbied2 3888	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
49		opelxpi 5695	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
50	8, 49	syl 18	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
51		fucco.g	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
52	3	natrcl 18178	. . . . 5 ⊢ (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
53	51, 52	syl 18	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
54	53	simprd 501	. . 3 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
55		ovex 7484	. . . . 5 ⊢ (𝐺𝑁𝐻) ∈ V
56		ovex 7484	. . . . 5 ⊢ (𝐹𝑁𝐺) ∈ V
57	55, 56	mpoex 8127	. . . 4 ⊢ (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))) ∈ V
58	57	a1i 11	. . 3 ⊢ (𝜑 → (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))) ∈ V)
59	15, 48, 50, 54, 58	ovmpod 7604	. 2 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∙ 𝐻) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
60		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑏 = 𝑆)
61	60	fveq1d 6918	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑏‘𝑥) = (𝑆‘𝑥))
62		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑎 = 𝑅)
63	62	fveq1d 6918	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑎‘𝑥) = (𝑅‘𝑥))
64	61, 63	oveq12d 7469	. . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)) = ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)))
65	64	mpteq2dv 5205	. 2 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
66	4	fvexi 6930	. . . 4 ⊢ 𝐴 ∈ V
67	66	mptex 7260	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ V
68	67	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ V)
69	59, 65, 51, 6, 68	ovmpod 7604	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⦋csb 3851  ⟨cop 4840   ↦ cmpt 5192   × cxp 5656  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034  Basecbs 17431  compcco 17484  Catccat 17888   Func cfunc 18079   Nat cnat 18169   FuncCat cfuc 18170

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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-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 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-func 18083  df-nat 18171  df-fuc 18172

This theorem is used by:  fuccoval  18191  fuccocl  18192  fuclid  18194  fucrid  18195  fucass  18196  fucsect  18200  curfcl  18456  xpcfucco3  50711  fucocolem4  50809  fucoppcco  50875  islmd  51179  iscmd  51180