Theorem fiunelros 34171

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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
3	2	nnred 12208	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
4	3	leidd 11751	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ 𝑁)
5		breq1 5113	. . . . 5 ⊢ (𝑛 = 1 → (𝑛 ≤ 𝑁 ↔ 1 ≤ 𝑁))
6		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = 1 → (1..^𝑛) = (1..^1))
7	6	iuneq1d 4986	. . . . . 6 ⊢ (𝑛 = 1 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^1)𝐵)
8	7	eleq1d 2814	. . . . 5 ⊢ (𝑛 = 1 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆))
9	5, 8	imbi12d 344	. . . 4 ⊢ (𝑛 = 1 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (1 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆)))
10		breq1 5113	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
11		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = 𝑖 → (1..^𝑛) = (1..^𝑖))
12	11	iuneq1d 4986	. . . . . 6 ⊢ (𝑛 = 𝑖 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑖)𝐵)
13	12	eleq1d 2814	. . . . 5 ⊢ (𝑛 = 𝑖 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆))
14	10, 13	imbi12d 344	. . . 4 ⊢ (𝑛 = 𝑖 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)))
15		breq1 5113	. . . . 5 ⊢ (𝑛 = (𝑖 + 1) → (𝑛 ≤ 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
16		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = (𝑖 + 1) → (1..^𝑛) = (1..^(𝑖 + 1)))
17	16	iuneq1d 4986	. . . . . 6 ⊢ (𝑛 = (𝑖 + 1) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵)
18	17	eleq1d 2814	. . . . 5 ⊢ (𝑛 = (𝑖 + 1) → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆))
19	15, 18	imbi12d 344	. . . 4 ⊢ (𝑛 = (𝑖 + 1) → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ ((𝑖 + 1) ≤ 𝑁 → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆)))
20		breq1 5113	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁))
21		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (1..^𝑛) = (1..^𝑁))
22	21	iuneq1d 4986	. . . . . 6 ⊢ (𝑛 = 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑁)𝐵)
23	22	eleq1d 2814	. . . . 5 ⊢ (𝑛 = 𝑁 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆))
24	20, 23	imbi12d 344	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (𝑁 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)))
25		fzo0 13651	. . . . . . . 8 ⊢ (1..^1) = ∅
26		iuneq1 4975	. . . . . . . 8 ⊢ ((1..^1) = ∅ → ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵
28		0iun 5030	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅
29	27, 28	eqtri 2753	. . . . . 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 34167	. . . . . . 7 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
33	30, 32	syl 17	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑆)
34	29, 33	eqeltrid 2833	. . . . 5 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆)
35	34	a1d 25	. . . 4 ⊢ (𝜑 → (1 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆))
36		simpllr 775	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℕ)
37		fzosplitsn 13743	. . . . . . . . . 10 ⊢ (𝑖 ∈ (ℤ≥‘1) → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
38		nnuz 12843	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
39	37, 38	eleq2s 2847	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
40	39	iuneq1d 4986	. . . . . . . 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 5061	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵 = (∪ 𝑘 ∈ (1..^𝑖)𝐵 ∪ ∪ 𝑘 ∈ {𝑖}𝐵)
43	41, 42	eqtrdi 2781	. . . . . 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 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑆 ∈ 𝑄)
45	36	nnred 12208	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℝ)
46	1	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
47	46	nnred 12208	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
48		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 + 1) ≤ 𝑁)
49		nnltp1le 12597	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
50	36, 46, 49	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
51	48, 50	mpbird 257	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 < 𝑁)
52	45, 47, 51	ltled 11329	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ≤ 𝑁)
53		simplr 768	. . . . . . . 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 3889	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
56		csbeq1a 3879	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
57	55, 56	iunxsngf 5059	. . . . . . . . 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 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝜑)
60		elfzo1 13680	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑁) ↔ (𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁))
61	36, 46, 51, 60	syl3anbrc 1344	. . . . . . . . 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 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑆
64	55, 63	nfel 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆
65	62, 64	nfim 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
66		eleq1w 2812	. . . . . . . . . . . 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 2814	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝐵 ∈ 𝑆 ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆))
69	67, 68	imbi12d 344	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
73	58, 72	eqeltrd 2829	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ {𝑖}𝐵 ∈ 𝑆)
74	31	unelros 34168	. . . . . . 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 1373	. . . . . 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 2829	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆)
77	76	ex 412	. . . 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 12213	. . 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 687	1 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  ⦋csb 3865   ∖ cdif 3914   ∪ cun 3915  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ∪ ciun 4958   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  1c1 11076   + caddc 11078   < clt 11215   ≤ cle 11216  ℕcn 12193  ℤ_≥cuz 12800  ..^cfzo 13622

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623

This theorem is referenced by: (None)