Theorem fucco 17934

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 2730	. . . 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 17922	. . . . . . . 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 494	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		funcrcl 17832	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
12	11	simpld 494	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	11	simprd 495	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		fucco.x	. . . 4 ⊢ ∙ = (comp‘𝑄)
15	1, 2, 3, 4, 5, 12, 13, 14	fuccofval 17931	. . 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 6876	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) ∈ V)
17		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
18	17	fveq2d 6865	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = (1st ‘⟨𝐹, 𝐺⟩))
19		op1stg 7983	. . . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
22	18, 21	eqtrd 2765	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = 𝐹)
23		fvexd 6876	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) ∈ V)
24	17	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
25	24	fveq2d 6865	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = (2nd ‘⟨𝐹, 𝐺⟩))
26		op2ndg 7984	. . . . . . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	25, 28	eqtrd 2765	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = 𝐺)
30		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ℎ = 𝐻)
32	31	ad2antrr 726	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ℎ = 𝐻)
33	30, 32	oveq12d 7408	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁ℎ) = (𝐺𝑁𝐻))
34		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
35	34, 30	oveq12d 7408	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
36	34	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑓) = (1st ‘𝐹))
37	36	fveq1d 6863	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
38	30	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑔) = (1st ‘𝐺))
39	38	fveq1d 6863	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
40	37, 39	opeq12d 4848	. . . . . . . . 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 6865	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘ℎ) = (1st ‘𝐻))
42	41	fveq1d 6863	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘ℎ)‘𝑥) = ((1st ‘𝐻)‘𝑥))
43	40, 42	oveq12d 7408	. . . . . . . 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 7407	. . . . . . 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 5204	. . . . . 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 7467	. . . . 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 3902	. . . 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 3902	. . 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 5678	. . . 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 17922	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
55		ovex 7423	. . . . 5 ⊢ (𝐺𝑁𝐻) ∈ V
56		ovex 7423	. . . . 5 ⊢ (𝐹𝑁𝐺) ∈ V
57	55, 56	mpoex 8061	. . . 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 7544	. 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 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑏 = 𝑆)
61	60	fveq1d 6863	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑏‘𝑥) = (𝑆‘𝑥))
62		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑎 = 𝑅)
63	62	fveq1d 6863	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑎‘𝑥) = (𝑅‘𝑥))
64	61, 63	oveq12d 7408	. . 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 5204	. 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 6875	. . . 4 ⊢ 𝐴 ∈ V
67	66	mptex 7200	. . 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 7544	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 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⦋csb 3865  ⟨cop 4598   ↦ cmpt 5191   × cxp 5639  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17186  compcco 17239  Catccat 17632   Func cfunc 17823   Nat cnat 17913   FuncCat cfuc 17914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-hom 17251  df-cco 17252  df-func 17827  df-nat 17915  df-fuc 17916

This theorem is referenced by:  fuccoval  17935  fuccocl  17936  fuclid  17938  fucrid  17939  fucass  17940  fucsect  17944  curfcl  18200  xpcfucco3  49251  fucocolem4  49349  fucoppcco  49402  islmd  49658  iscmd  49659