Theorem fiunelros 29564

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 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
3	2	nnred 10912	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
4	3	leidd 10473	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ 𝑁)
5		breq1 4586	. . . . 5 ⊢ (𝑛 = 1 → (𝑛 ≤ 𝑁 ↔ 1 ≤ 𝑁))
6		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 1 → (1..^𝑛) = (1..^1))
7	6	iuneq1d 4481	. . . . . 6 ⊢ (𝑛 = 1 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^1)𝐵)
8	7	eleq1d 2672	. . . . 5 ⊢ (𝑛 = 1 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆))
9	5, 8	imbi12d 333	. . . 4 ⊢ (𝑛 = 1 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (1 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆)))
10		breq1 4586	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
11		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 𝑖 → (1..^𝑛) = (1..^𝑖))
12	11	iuneq1d 4481	. . . . . 6 ⊢ (𝑛 = 𝑖 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑖)𝐵)
13	12	eleq1d 2672	. . . . 5 ⊢ (𝑛 = 𝑖 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆))
14	10, 13	imbi12d 333	. . . 4 ⊢ (𝑛 = 𝑖 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)))
15		breq1 4586	. . . . 5 ⊢ (𝑛 = (𝑖 + 1) → (𝑛 ≤ 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
16		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = (𝑖 + 1) → (1..^𝑛) = (1..^(𝑖 + 1)))
17	16	iuneq1d 4481	. . . . . 6 ⊢ (𝑛 = (𝑖 + 1) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵)
18	17	eleq1d 2672	. . . . 5 ⊢ (𝑛 = (𝑖 + 1) → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆))
19	15, 18	imbi12d 333	. . . 4 ⊢ (𝑛 = (𝑖 + 1) → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ ((𝑖 + 1) ≤ 𝑁 → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆)))
20		breq1 4586	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁))
21		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (1..^𝑛) = (1..^𝑁))
22	21	iuneq1d 4481	. . . . . 6 ⊢ (𝑛 = 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑁)𝐵)
23	22	eleq1d 2672	. . . . 5 ⊢ (𝑛 = 𝑁 → (∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆 ↔ ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆))
24	20, 23	imbi12d 333	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) ↔ (𝑁 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)))
25		fzo0 12361	. . . . . . . 8 ⊢ (1..^1) = ∅
26		iuneq1 4470	. . . . . . . 8 ⊢ ((1..^1) = ∅ → ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑘 ∈ (1..^1)𝐵 = ∪ 𝑘 ∈ ∅ 𝐵
28		0iun 4513	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅
29	27, 28	eqtri 2632	. . . . . 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 29560	. . . . . . 7 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
33	30, 32	syl 17	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑆)
34	29, 33	syl5eqel 2692	. . . . 5 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆)
35	34	a1d 25	. . . 4 ⊢ (𝜑 → (1 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^1)𝐵 ∈ 𝑆))
36		simpllr 795	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℕ)
37		fzosplitsn 12442	. . . . . . . . . 10 ⊢ (𝑖 ∈ (ℤ≥‘1) → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
38		nnuz 11599	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
39	37, 38	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
40	39	iuneq1d 4481	. . . . . . . 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 4541	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵 = (∪ 𝑘 ∈ (1..^𝑖)𝐵 ∪ ∪ 𝑘 ∈ {𝑖}𝐵)
43	41, 42	syl6eq 2660	. . . . . 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 762	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑆 ∈ 𝑄)
45	36	nnred 10912	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℝ)
46	1	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
47	46	nnred 10912	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
48		simpr 476	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 + 1) ≤ 𝑁)
49		nnltp1le 11310	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
50	36, 46, 49	syl2anc 691	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
51	48, 50	mpbird 246	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 < 𝑁)
52	45, 47, 51	ltled 10064	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ≤ 𝑁)
53		simplr 788	. . . . . . . 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 3515	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
56		csbeq1a 3508	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
57	55, 56	iunxsngf 28758	. . . . . . . . 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 794	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝜑)
60		elfzo1 12385	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑁) ↔ (𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁))
61	36, 46, 51, 60	syl3anbrc 1239	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ (1..^𝑁))
62		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ (1..^𝑁))
63		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑆
64	55, 63	nfel 2763	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆
65	62, 64	nfim 1813	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
66		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ (1..^𝑁) ↔ 𝑖 ∈ (1..^𝑁)))
67	66	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) ↔ (𝜑 ∧ 𝑖 ∈ (1..^𝑁))))
68	56	eleq1d 2672	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝐵 ∈ 𝑆 ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆))
69	67, 68	imbi12d 333	. . . . . . . . . 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	chvar 2250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑁)) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
72	59, 61, 71	syl2anc 691	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ⦋𝑖 / 𝑘⦌𝐵 ∈ 𝑆)
73	58, 72	eqeltrd 2688	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ {𝑖}𝐵 ∈ 𝑆)
74	31	unelros 29561	. . . . . . 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 1318	. . . . . 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 2688	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ≤ 𝑁 → ∪ 𝑘 ∈ (1..^𝑖)𝐵 ∈ 𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ∪ 𝑘 ∈ (1..^(𝑖 + 1))𝐵 ∈ 𝑆)
77	76	ex 449	. . . 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 28953	. . 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 699	1 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  ⦋csb 3499   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ ciun 4455   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕcn 10897  ℤ_≥cuz 11563  ..^cfzo 12334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335

This theorem is referenced by: (None)