Theorem fiunelros 31437

Description: A ring of sets is closed under finite union. (Contributed by Thierry Arnoux, 19-Jul-2020.)

Hypotheses

Ref	Expression
isros.1	⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
fiunelros.1	⊢ (𝜑 → 𝑆 ∈ 𝑄)
fiunelros.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
fiunelros.3	⊢ ((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
fiunelros	⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)

Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦   𝑘,𝑁   𝑆,𝑘   𝜑,𝑘

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑠)   𝐵(𝑥,𝑦,𝑘,𝑠)   𝑄(𝑥,𝑦,𝑘,𝑠)   𝑁(𝑥,𝑦,𝑠)   𝑂(𝑥,𝑦,𝑘)

Proof of Theorem fiunelros

Dummy variables 𝑖 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fiunelros.2	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
3	2	nnred 11656	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
4	3	leidd 11209	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ 𝑁)
5		breq1 5072	. . . . 5 ⊢ (𝑛 = 1 → (𝑛 ≤ 𝑁 ↔ 1 ≤ 𝑁))
6		oveq2 7167	. . . . . . 7 ⊢ (𝑛 = 1 → (1..^𝑛) = (1..^1))
7	6	iuneq1d 4949	. . . . . 6 ⊢ (𝑛 = 1 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^1)𝐵)
8	7	eleq1d 2900	. . . . 5 ⊢ (𝑛 = 1 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆))
9	5, 8	imbi12d 347	. . . 4 ⊢ (𝑛 = 1 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (1 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆)))
10		breq1 5072	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
11		oveq2 7167	. . . . . . 7 ⊢ (𝑛 = 𝑖 → (1..^𝑛) = (1..^𝑖))
12	11	iuneq1d 4949	. . . . . 6 ⊢ (𝑛 = 𝑖 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑖)𝐵)
13	12	eleq1d 2900	. . . . 5 ⊢ (𝑛 = 𝑖 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆))
14	10, 13	imbi12d 347	. . . 4 ⊢ (𝑛 = 𝑖 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)))
15		breq1 5072	. . . . 5 ⊢ (𝑛 = (𝑖 + 1) → (𝑛 ≤ 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
16		oveq2 7167	. . . . . . 7 ⊢ (𝑛 = (𝑖 + 1) → (1..^𝑛) = (1..^(𝑖 + 1)))
17	16	iuneq1d 4949	. . . . . 6 ⊢ (𝑛 = (𝑖 + 1) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵)
18	17	eleq1d 2900	. . . . 5 ⊢ (𝑛 = (𝑖 + 1) → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆))
19	15, 18	imbi12d 347	. . . 4 ⊢ (𝑛 = (𝑖 + 1) → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ ((𝑖 + 1) ≤ 𝑁 → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆)))
20		breq1 5072	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁))
21		oveq2 7167	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (1..^𝑛) = (1..^𝑁))
22	21	iuneq1d 4949	. . . . . 6 ⊢ (𝑛 = 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑁)𝐵)
23	22	eleq1d 2900	. . . . 5 ⊢ (𝑛 = 𝑁 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆))
24	20, 23	imbi12d 347	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (𝑁 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)))
25		fzo0 13064	. . . . . . . 8 ⊢ (1..^1) = ∅
26		iuneq1 4938	. . . . . . . 8 ⊢ ((1..^1) = ∅ → ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵
28		0iun 4989	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅
29	27, 28	eqtri 2847	. . . . . 6 ⊢ ∪ 𝑘 ∈ (1..^1)𝐵 = ∅
30		fiunelros.1	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑄)
31		isros.1	. . . . . . . 8 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
32	31	0elros 31433	. . . . . . 7 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
33	30, 32	syl 17	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑆)
34	29, 33	eqeltrid 2920	. . . . 5 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆)
35	34	a1d 25	. . . 4 ⊢ (𝜑 → (1 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆))
36		simpllr 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℕ)
37		fzosplitsn 13148	. . . . . . . . . 10 ⊢ (𝑖 ∈ (ℤ≥‘1) → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
38		nnuz 12284	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
39	37, 38	eleq2s 2934	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
40	39	iuneq1d 4949	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 = ∪ 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵)
41	36, 40	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 = ∪ 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵)
42		iunxun 5019	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵 = (∪ 𝑘 ∈ (1..^𝑖)𝐵 ∪ ∪ 𝑘 ∈ {𝑖}𝐵)
43	41, 42	syl6eq 2875	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 = (∪ 𝑘 ∈ (1..^𝑖)𝐵 ∪ ∪ 𝑘 ∈ {𝑖}𝐵))
44	30	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑆 ∈ 𝑄)
45	36	nnred 11656	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℝ)
46	1	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
47	46	nnred 11656	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
48		simpr 487	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 + 1) ≤ 𝑁)
49		nnltp1le 12041	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
50	36, 46, 49	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
51	48, 50	mpbird 259	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 < 𝑁)
52	45, 47, 51	ltled 10791	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ≤ 𝑁)
53		simplr 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆))
54	52, 53	mpd 15	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)
55		nfcsb1v 3910	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
56		csbeq1a 3900	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
57	55, 56	iunxsngf 5017	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → ∪ 𝑘 ∈ {𝑖}𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
58	36, 57	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ {𝑖}𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
59		simplll 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝜑)
60		elfzo1 13090	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑁) ↔ (𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁))
61	36, 46, 51, 60	syl3anbrc 1339	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ (1..^𝑁))
62		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ (1..^𝑁))
63		nfcv 2980	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑆
64	55, 63	nfel 2995	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆
65	62, 64	nfim 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
66		eleq1w 2898	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ (1..^𝑁) ↔ 𝑖 ∈ (1..^𝑁)))
67	66	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) ↔ (𝜑 ∧ 𝑖 ∈ (1..^𝑁))))
68	56	eleq1d 2900	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝐵 ∈ 𝑆 ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆))
69	67, 68	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) → 𝐵 ∈ 𝑆) ↔ ((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)))
70		fiunelros.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) → 𝐵 ∈ 𝑆)
71	65, 69, 70	chvarfv 2241	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
72	59, 61, 71	syl2anc 586	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
73	58, 72	eqeltrd 2916	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ {𝑖}𝐵 ∈ 𝑆)
74	31	unelros 31434	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆 ∧ ∪ 𝑘 ∈ {𝑖}𝐵 ∈ 𝑆) → (∪ 𝑘 ∈ (1..^𝑖)𝐵 ∪ ∪ 𝑘 ∈ {𝑖}𝐵) ∈ 𝑆)
75	44, 54, 73, 74	syl3anc 1367	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (∪ 𝑘 ∈ (1..^𝑖)𝐵 ∪ ∪ 𝑘 ∈ {𝑖}𝐵) ∈ 𝑆)
76	43, 75	eqeltrd 2916	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆)
77	76	ex 415	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) → ((𝑖 + 1) ≤ 𝑁 → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆))
78	9, 14, 19, 24, 35, 77	nnindd 30539	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝑁 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆))
79	4, 78	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)
80	1, 79	mpdan 685	1 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  {crab 3145  ⦋csb 3886   ∖ cdif 3936   ∪ cun 3937  ∅c0 4294  𝒫 cpw 4542  {csn 4570  ∪ ciun 4922   class class class wbr 5069  ‘cfv 6358  (class class class)co 7159  1c1 10541   + caddc 10543   < clt 10678   ≤ cle 10679  ℕcn 11641  ℤ_≥cuz 12246  ..^cfzo 13036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037

This theorem is referenced by: (None)