Theorem setsstruct 17087

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 17063	. . . . . 6 ⊢ (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2		simp2 1137	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3		simp3l 1202	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐸 ∈ 𝑉)
4		1z 12502	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
5		nnge1 12153	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6		eluzuzle 12741	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
7	4, 5, 6	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
8		elnnuz 12776	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ≥‘1))
9	7, 8	imbitrrdi 252	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℕ))
10	9	adantld 490	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ))
11	10	3ad2ant1 1133	. . . . . . . . . . . 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 1110	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐼 ∈ ℕ)
14	2, 3, 13	3jca 1128	. . . . . . . . 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 7933	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
16	15	breq2d 5101	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼 ≤ 𝑀))
17		eqidd 2732	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
18	16, 17, 15	ifbieq12d 4501	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
19	18	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
20	19	adantr 480	. . . . . . . . . . . 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 12738	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
22		zre 12472	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23	22	rexrd 11162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24	23	3ad2ant2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝐼 ∈ ℝ*)
25		zre 12472	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
26	25	rexrd 11162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27	26	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ∈ ℝ*)
28		simp3 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ≤ 𝐼)
29	24, 27, 28	3jca 1128	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
32	31	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
33	32	impcom 407	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼))
34		xrmineq 13079	. . . . . . . . . . . . 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 2767	. . . . . . . . . . 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 1131	. . . . . . . . . 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 7934	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
39	38	eqcomd 2737	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
40	39	breq2d 5101	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ 𝑁 ↔ 𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
41	40, 39, 17	ifbieq12d 4501	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42	41	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43	42	3ad2ant1 1133	. . . . . . . . . 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 4830	. . . . . . . . 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 511	. . . . . . . 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 1119	. . . . . . 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 1133	. . . . . 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 217	. . . . 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 414	. . 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 1110	. 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 17085	. 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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∖ cdif 3894   ⊆ wss 3897  ∅c0 4280  ifcif 4472  {csn 4573  ⟨cop 4579   class class class wbr 5089  dom cdm 5614  Fun wfun 6475  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  1c1 11007  ℝ^*cxr 11145   ≤ cle 11147  ℕcn 12125  ℤcz 12468  ℤ_≥cuz 12732  ...cfz 13407   Struct cstr 17057   sSet csts 17074

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-struct 17058  df-sets 17075

This theorem is referenced by: (None)