Theorem setsstruct 15898

Description: An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)

Assertion

Ref	Expression
setsstruct	⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩)

Proof of Theorem setsstruct

Step	Hyp	Ref	Expression
1		isstruct 15870	. . . . . 6 ⊢ (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2		simp2 1062	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3		simp3l 1089	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐸 ∈ 𝑉)
4		1z 11407	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
5		nnge1 11046	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6		eluzuzle 11696	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
7	4, 5, 6	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
8		elnnuz 11724	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ≥‘1))
9	7, 8	syl6ibr 242	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℕ))
10	9	adantld 483	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ))
11	10	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ))
12	11	a1d 25	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ)))
13	12	3imp 1256	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐼 ∈ ℕ)
14	2, 3, 13	3jca 1242	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → (𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ))
15		op1stg 7180	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
16	15	breq2d 4665	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼 ≤ 𝑀))
17		eqidd 2623	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
18	16, 17, 15	ifbieq12d 4113	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
19	18	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
20	19	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
21		eluz2 11693	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
22		zre 11381	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23	22	rexrd 10089	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24	23	3ad2ant2 1083	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝐼 ∈ ℝ*)
25		zre 11381	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
26	25	rexrd 10089	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27	26	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ∈ ℝ*)
28		simp3 1063	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ≤ 𝐼)
29	24, 27, 28	3jca 1242	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼))
30	29	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
31	21, 30	sylbi 207	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
32	31	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
33	32	impcom 446	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼))
34		xrmineq 12011	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀)
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀)
36	20, 35	eqtr2d 2657	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
37	36	3adant2 1080	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
38		op2ndg 7181	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
39	38	eqcomd 2628	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
40	39	breq2d 4665	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ 𝑁 ↔ 𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
41	40, 39, 17	ifbieq12d 4113	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42	41	3adant3 1081	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43	42	3ad2ant1 1082	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
44	37, 43	opeq12d 4410	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩)
45	14, 44	jca 554	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))
46	45	3exp 1264	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
47	46	3ad2ant1 1082	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
48	1, 47	sylbi 207	. . . . 5 ⊢ (𝐺 Struct ⟨𝑀, 𝑁⟩ → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
49	48	pm2.43i 52	. . . 4 ⊢ (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩)))
50	49	expdcom 455	. . 3 ⊢ (𝐸 ∈ 𝑉 → (𝐼 ∈ (ℤ≥‘𝑀) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
51	50	3imp 1256	. 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))
52		setsstruct2 15896	. 2 ⊢ (((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩)
53	51, 52	syl 17	1 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  1c1 9937  ℝ^*cxr 10073   ≤ cle 10075  ℕcn 11020  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326   Struct cstr 15853   sSet csts 15855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-sets 15864

This theorem is referenced by: (None)