Theorem setsstruct 16729

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 16705	. . . . . 6 ⊢ (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2		simp2 1139	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3		simp3l 1203	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐸 ∈ 𝑉)
4		1z 12207	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
5		nnge1 11858	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6		eluzuzle 12447	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
7	4, 5, 6	sylancr 590	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
8		elnnuz 12478	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ≥‘1))
9	7, 8	syl6ibr 255	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℕ))
10	9	adantld 494	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℕ))
11	10	3ad2ant1 1135	. . . . . . . . . . . 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 1113	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐼 ∈ ℕ)
14	2, 3, 13	3jca 1130	. . . . . . . . 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 7773	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
16	15	breq2d 5065	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼 ≤ 𝑀))
17		eqidd 2738	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
18	16, 17, 15	ifbieq12d 4467	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
19	18	3adant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
20	19	adantr 484	. . . . . . . . . . . 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 12444	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
22		zre 12180	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23	22	rexrd 10883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24	23	3ad2ant2 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝐼 ∈ ℝ*)
25		zre 12180	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
26	25	rexrd 10883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27	26	3ad2ant1 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ∈ ℝ*)
28		simp3 1140	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ≤ 𝐼)
29	24, 27, 28	3jca 1130	. . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
32	31	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼)))
33	32	impcom 411	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → (𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼))
34		xrmineq 12770	. . . . . . . . . . . . 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 2778	. . . . . . . . . . 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 1133	. . . . . . . . . 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 7774	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
39	38	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
40	39	breq2d 5065	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ 𝑁 ↔ 𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
41	40, 39, 17	ifbieq12d 4467	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42	41	3adant3 1134	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43	42	3ad2ant1 1135	. . . . . . . . . 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 4792	. . . . . . . . 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 515	. . . . . . . 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 1121	. . . . . . 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 1135	. . . . . 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 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 418	. . 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 1113	. 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 16727	. 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 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ∖ cdif 3863   ⊆ wss 3866  ∅c0 4237  ifcif 4439  {csn 4541  ⟨cop 4547   class class class wbr 5053  dom cdm 5551  Fun wfun 6374  ‘cfv 6380  (class class class)co 7213  1^st c1st 7759  2^nd c2nd 7760  1c1 10730  ℝ^*cxr 10866   ≤ cle 10868  ℕcn 11830  ℤcz 12176  ℤ_≥cuz 12438  ...cfz 13095   Struct cstr 16699   sSet csts 16716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717

This theorem is referenced by: (None)