Theorem setsstruct 16805

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 16781	. . . . . 6 ⊢ (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2		simp2 1135	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3		simp3l 1199	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐸 ∈ 𝑉)
4		1z 12280	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
5		nnge1 11931	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6		eluzuzle 12520	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
7	4, 5, 6	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ (ℤ≥‘1)))
8		elnnuz 12551	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ≥‘1))
9	7, 8	syl6ibr 251	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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 1109	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀))) → 𝐼 ∈ ℕ)
14	2, 3, 13	3jca 1126	. . . . . . . . 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 7816	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
16	15	breq2d 5082	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼 ≤ 𝑀))
17		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
18	16, 17, 15	ifbieq12d 4484	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼 ≤ 𝑀, 𝐼, 𝑀))
19	18	3adant3 1130	. . . . . . . . . . . . 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 12517	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
22		zre 12253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23	22	rexrd 10956	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24	23	3ad2ant2 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝐼 ∈ ℝ*)
25		zre 12253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
26	25	rexrd 10956	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27	26	3ad2ant1 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ∈ ℝ*)
28		simp3 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ≤ 𝐼)
29	24, 27, 28	3jca 1126	. . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . 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 12843	. . . . . . . . . . . . 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 2779	. . . . . . . . . . 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 1129	. . . . . . . . . 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 7817	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
39	38	eqcomd 2744	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
40	39	breq2d 5082	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ 𝑁 ↔ 𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
41	40, 39, 17	ifbieq12d 4484	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42	41	3adant3 1130	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43	42	3ad2ant1 1131	. . . . . . . . . 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 4809	. . . . . . . . 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 1117	. . . . . . 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 1131	. . . . . 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 216	. . . . 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 1109	. 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 16803	. 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 1085   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  ifcif 4456  {csn 4558  ⟨cop 4564   class class class wbr 5070  dom cdm 5580  Fun wfun 6412  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  1c1 10803  ℝ^*cxr 10939   ≤ cle 10941  ℕcn 11903  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168   Struct cstr 16775   sSet csts 16792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793

This theorem is referenced by: (None)