Theorem setsstruct 17240

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 17216	. . . . . 6 ⊢ (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2		simp2 1155	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3		simp3l 1220	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐸 ∈ 𝑉)
4		1z 12628	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
5		nnge1 12268	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6		eluzuzle 12875	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
7	4, 5, 6	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
8		elnnuz 12906	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ≥‘1))
9	7, 8	imbitrrdi 255	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℕ))
10	9	adantld 495	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ))
11	10	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ))
12	11	a1d 26	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ)))
13	12	3imp 1128	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐼 ∈ ℕ)
14	2, 3, 13	3jca 1146	. . . . . . . . 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 7994	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
16	15	breq2d 5121	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼 ≤ 𝑀))
17		eqidd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
18	16, 17, 15	ifbieq12d 4516	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
19	18	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
20	19	adantr 485	. . . . . . . . . . . 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 12872	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
22		zre 12599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23	22	rexrd 11263	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24	23	3ad2ant2 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝐼 ∈ ℝ*)
25		zre 12599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
26	25	rexrd 11263	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27	26	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ∈ ℝ*)
28		simp3 1156	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ≤ 𝐼)
29	24, 27, 28	3jca 1146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼))
30	29	a1d 26	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
31	21, 30	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
32	31	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
33	32	impcom 412	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼))
34		xrmineq 13210	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀)
35	33, 34	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀)
36	20, 35	eqtr2d 2799	. . . . . . . . . . 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 1149	. . . . . . . . . 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 7995	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
39	38	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
40	39	breq2d 5121	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ 𝑁 ↔ 𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
41	40, 39, 17	ifbieq12d 4516	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42	41	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43	42	3ad2ant1 1151	. . . . . . . . . 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 4846	. . . . . . . . 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 520	. . . . . . . 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 1137	. . . . . . 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 1151	. . . . . 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 220	. . . . 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 53	. . . 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 419	. . 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 1128	. 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 17238	. 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 18	1 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  ifcif 4487  {csn 4589  ⟨cop 4595   class class class wbr 5109  dom cdm 5661  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  1c1 11105  ℝ^*cxr 11246   ≤ cle 11248  ℕcn 12237  ℤcz 12595  ℤ_≥cuz 12866  ...cfz 13539   Struct cstr 17210   sSet csts 17227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-n0 12509  df-z 12596  df-uz 12867  df-fz 13540  df-struct 17211  df-sets 17228

This theorem is used by: (None)