Theorem fiunelros 34672

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 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
3	2	nnred 12275	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
4	3	leidd 11807	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ 𝑁)
5		breq1 5110	. . . . 5 ⊢ (𝑛 = 1 → (𝑛 ≤ 𝑁 ↔ 1 ≤ 𝑁))
6		oveq2 7424	. . . . . . 7 ⊢ (𝑛 = 1 → (1..^𝑛) = (1..^1))
7	6	iuneq1d 4982	. . . . . 6 ⊢ (𝑛 = 1 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^1)𝐵)
8	7	eleq1d 2847	. . . . 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 5110	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
11		oveq2 7424	. . . . . . 7 ⊢ (𝑛 = 𝑖 → (1..^𝑛) = (1..^𝑖))
12	11	iuneq1d 4982	. . . . . 6 ⊢ (𝑛 = 𝑖 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑖)𝐵)
13	12	eleq1d 2847	. . . . 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 5110	. . . . 5 ⊢ (𝑛 = (𝑖 + 1) → (𝑛 ≤ 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
16		oveq2 7424	. . . . . . 7 ⊢ (𝑛 = (𝑖 + 1) → (1..^𝑛) = (1..^(𝑖 + 1)))
17	16	iuneq1d 4982	. . . . . 6 ⊢ (𝑛 = (𝑖 + 1) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵)
18	17	eleq1d 2847	. . . . 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 5110	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁))
21		oveq2 7424	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (1..^𝑛) = (1..^𝑁))
22	21	iuneq1d 4982	. . . . . 6 ⊢ (𝑛 = 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑁)𝐵)
23	22	eleq1d 2847	. . . . 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 13741	. . . . . . . 8 ⊢ (1..^1) = ∅
26		iuneq1 4971	. . . . . . . 8 ⊢ ((1..^1) = ∅ → ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵
28		0iun 5025	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅
29	27, 28	eqtri 2785	. . . . . 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 34668	. . . . . . 7 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
33	30, 32	syl 18	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑆)
34	29, 33	eqeltrid 2866	. . . . 5 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆)
35	34	a1d 26	. . . 4 ⊢ (𝜑 → (1 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆))
36		simpllr 788	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℕ)
37		fzosplitsn 13834	. . . . . . . . . 10 ⊢ (𝑖 ∈ (ℤ≥‘1) → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
38		nnuz 12929	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
39	37, 38	eleq2s 2880	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
40	39	iuneq1d 4982	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 = ∪ 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵)
41	36, 40	syl 18	. . . . . . 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 5058	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵 = (∪ 𝑘 ∈ (1..^𝑖)𝐵 ∪ ∪ 𝑘 ∈ {𝑖}𝐵)
43	41, 42	eqtrdi 2813	. . . . . 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 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑆 ∈ 𝑄)
45	36	nnred 12275	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℝ)
46	1	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
47	46	nnred 12275	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
48		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 + 1) ≤ 𝑁)
49		nnltp1le 12680	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
50	36, 46, 49	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
51	48, 50	mpbird 260	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 < 𝑁)
52	45, 47, 51	ltled 11385	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ≤ 𝑁)
53		simplr 781	. . . . . . . 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 16	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)
55		nfcsb1v 3874	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
56		csbeq1a 3864	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
57	55, 56	iunxsngf 5056	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → ∪ 𝑘 ∈ {𝑖}𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
58	36, 57	syl 18	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ {𝑖}𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
59		simplll 787	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝜑)
60		elfzo1 13770	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑁) ↔ (𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁))
61	36, 46, 51, 60	syl3anbrc 1362	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ (1..^𝑁))
62		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ (1..^𝑁))
63		nfcv 2924	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑆
64	55, 63	nfel 2938	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆
65	62, 64	nfim 1929	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
66		eleq1w 2845	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ (1..^𝑁) ↔ 𝑖 ∈ (1..^𝑁)))
67	66	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) ↔ (𝜑 ∧ 𝑖 ∈ (1..^𝑁))))
68	56	eleq1d 2847	. . . . . . . . . . 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 2278	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
72	59, 61, 71	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
73	58, 72	eqeltrd 2862	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ {𝑖}𝐵 ∈ 𝑆)
74	31	unelros 34669	. . . . . . 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 1398	. . . . . 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 2862	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆)
77	76	ex 418	. . . 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 12280	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝑁 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆))
79	4, 78	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)
80	1, 79	mpdan 700	1 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  ⦋csb 3850   ∖ cdif 3899   ∪ cun 3900  ∅c0 4282  𝒫 cpw 4560  {csn 4587  ∪ ciun 4954   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  1c1 11128   + caddc 11130   < clt 11270   ≤ cle 11271  ℕcn 12260  ℤ_≥cuz 12890  ..^cfzo 13711

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-fzo 13712

This theorem is used by: (None)