Theorem fucco 17750

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 2737	. . . 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 17736	. . . . . . . 8 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
9	8	simpld 495	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		funcrcl 17648	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
12	11	simpld 495	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	11	simprd 496	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		fucco.x	. . . 4 ⊢ ∙ = (comp‘𝑄)
15	1, 2, 3, 4, 5, 12, 13, 14	fuccofval 17746	. . 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 6826	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) ∈ V)
17		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
18	17	fveq2d 6815	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = (1st ‘⟨𝐹, 𝐺⟩))
19		op1stg 7888	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
20	8, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
22	18, 21	eqtrd 2777	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = 𝐹)
23		fvexd 6826	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) ∈ V)
24	17	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
25	24	fveq2d 6815	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = (2nd ‘⟨𝐹, 𝐺⟩))
26		op2ndg 7889	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
27	8, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
28	27	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	25, 28	eqtrd 2777	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = 𝐺)
30		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31		simprr 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ℎ = 𝐻)
32	31	ad2antrr 723	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ℎ = 𝐻)
33	30, 32	oveq12d 7333	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁ℎ) = (𝐺𝑁𝐻))
34		simplr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
35	34, 30	oveq12d 7333	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
36	34	fveq2d 6815	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑓) = (1st ‘𝐹))
37	36	fveq1d 6813	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
38	30	fveq2d 6815	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑔) = (1st ‘𝐺))
39	38	fveq1d 6813	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
40	37, 39	opeq12d 4823	. . . . . . . . 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 6815	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘ℎ) = (1st ‘𝐻))
42	41	fveq1d 6813	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘ℎ)‘𝑥) = ((1st ‘𝐻)‘𝑥))
43	40, 42	oveq12d 7333	. . . . . . . 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 7332	. . . . . . 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 5189	. . . . . 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 7390	. . . . 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 3882	. . . 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 3882	. . 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 5644	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
50	8, 49	syl 17	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
51		fucco.g	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
52	3	natrcl 17736	. . . . 5 ⊢ (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
53	51, 52	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
54	53	simprd 496	. . 3 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
55		ovex 7348	. . . . 5 ⊢ (𝐺𝑁𝐻) ∈ V
56		ovex 7348	. . . . 5 ⊢ (𝐹𝑁𝐺) ∈ V
57	55, 56	mpoex 7965	. . . 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 7465	. 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 768	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑏 = 𝑆)
61	60	fveq1d 6813	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑏‘𝑥) = (𝑆‘𝑥))
62		simprr 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑎 = 𝑅)
63	62	fveq1d 6813	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑎‘𝑥) = (𝑅‘𝑥))
64	61, 63	oveq12d 7333	. . 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 5189	. 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 6825	. . . 4 ⊢ 𝐴 ∈ V
67	66	mptex 7138	. . 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 7465	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441  ⦋csb 3842  ⟨cop 4577   ↦ cmpt 5170   × cxp 5605  ‘cfv 6465  (class class class)co 7315   ∈ cmpo 7317  1^st c1st 7874  2^nd c2nd 7875  Basecbs 16982  compcco 17044  Catccat 17443   Func cfunc 17639   Nat cnat 17727   FuncCat cfuc 17728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628  ax-cnex 11000  ax-resscn 11001  ax-1cn 11002  ax-icn 11003  ax-addcl 11004  ax-addrcl 11005  ax-mulcl 11006  ax-mulrcl 11007  ax-mulcom 11008  ax-addass 11009  ax-mulass 11010  ax-distr 11011  ax-i2m1 11012  ax-1ne0 11013  ax-1rid 11014  ax-rnegex 11015  ax-rrecex 11016  ax-cnre 11017  ax-pre-lttri 11018  ax-pre-lttrn 11019  ax-pre-ltadd 11020  ax-pre-mulgt0 11021

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7272  df-ov 7318  df-oprab 7319  df-mpo 7320  df-om 7758  df-1st 7876  df-2nd 7877  df-frecs 8144  df-wrecs 8175  df-recs 8249  df-rdg 8288  df-1o 8344  df-er 8546  df-ixp 8734  df-en 8782  df-dom 8783  df-sdom 8784  df-fin 8785  df-pnf 11084  df-mnf 11085  df-xr 11086  df-ltxr 11087  df-le 11088  df-sub 11280  df-neg 11281  df-nn 12047  df-2 12109  df-3 12110  df-4 12111  df-5 12112  df-6 12113  df-7 12114  df-8 12115  df-9 12116  df-n0 12307  df-z 12393  df-dec 12511  df-uz 12656  df-fz 13313  df-struct 16918  df-slot 16953  df-ndx 16965  df-base 16983  df-hom 17056  df-cco 17057  df-func 17643  df-nat 17729  df-fuc 17730

This theorem is referenced by:  fuccoval  17751  fuccocl  17752  fuclid  17754  fucrid  17755  fucass  17756  fucsect  17760  curfcl  18020