Theorem oaabsb 43307

Description: The right addend absorbs the sum with an ordinal iff that ordinal times omega is less than or equal to the right addend. (Contributed by RP, 19-Feb-2025.)

Assertion

Ref	Expression
oaabsb	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ (𝐴 +o 𝐵) = 𝐵))

Proof of Theorem oaabsb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 9686	. . . . 5 ⊢ ω ∈ On
2		omcl 8574	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ·o ω) ∈ On)
3	1, 2	mpan2 691	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·o ω) ∈ On)
4		oawordex 8595	. . . 4 ⊢ (((𝐴 ·o ω) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ ∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵))
5	3, 4	sylan 580	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ ∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵))
6		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
7	6	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
8	3	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 ·o ω) ∈ On)
9		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
10		oaass 8599	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐴 ·o ω) ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = (𝐴 +o ((𝐴 ·o ω) +o 𝑥)))
11	7, 8, 9, 10	syl3anc 1373	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = (𝐴 +o ((𝐴 ·o ω) +o 𝑥)))
12		1on 8518	. . . . . . . . . 10 ⊢ 1o ∈ On
13		odi 8617	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On ∧ ω ∈ On) → (𝐴 ·o (1o +o ω)) = ((𝐴 ·o 1o) +o (𝐴 ·o ω)))
14	12, 1, 13	mp3an23 1455	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o (1o +o ω)) = ((𝐴 ·o 1o) +o (𝐴 ·o ω)))
15		1oaomeqom 43306	. . . . . . . . . . 11 ⊢ (1o +o ω) = ω
16	15	oveq2i 7442	. . . . . . . . . 10 ⊢ (𝐴 ·o (1o +o ω)) = (𝐴 ·o ω)
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o (1o +o ω)) = (𝐴 ·o ω))
18		om1 8580	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
19	18	oveq1d 7446	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((𝐴 ·o 1o) +o (𝐴 ·o ω)) = (𝐴 +o (𝐴 ·o ω)))
20	14, 17, 19	3eqtr3rd 2786	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +o (𝐴 ·o ω)) = (𝐴 ·o ω))
21	20	oveq1d 7446	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = ((𝐴 ·o ω) +o 𝑥))
22	21	ad2antrr 726	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = ((𝐴 ·o ω) +o 𝑥))
23	11, 22	eqtr3d 2779	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = ((𝐴 ·o ω) +o 𝑥))
24		oveq2 7439	. . . . . 6 ⊢ (((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = (𝐴 +o 𝐵))
25		id 22	. . . . . 6 ⊢ (((𝐴 ·o ω) +o 𝑥) = 𝐵 → ((𝐴 ·o ω) +o 𝑥) = 𝐵)
26	24, 25	eqeq12d 2753	. . . . 5 ⊢ (((𝐴 ·o ω) +o 𝑥) = 𝐵 → ((𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = ((𝐴 ·o ω) +o 𝑥) ↔ (𝐴 +o 𝐵) = 𝐵))
27	23, 26	syl5ibcom 245	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
28	27	rexlimdva 3155	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
29	5, 28	sylbid 240	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 → (𝐴 +o 𝐵) = 𝐵))
30		limom 7903	. . . . . 6 ⊢ Lim ω
31		omlim 8571	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
32	1, 30, 31	mpanr12 705	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
33	32	ad2antrr 726	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
34		oveq2 7439	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
35	34	sseq1d 4015	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o ∅) ⊆ 𝐵))
36		oveq2 7439	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
37	36	sseq1d 4015	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o 𝑦) ⊆ 𝐵))
38		oveq2 7439	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
39	38	sseq1d 4015	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o suc 𝑦) ⊆ 𝐵))
40		om0 8555	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
41		0ss 4400	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐵
42	40, 41	eqsstrdi 4028	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ 𝐵)
43	42	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ∅) ⊆ 𝐵)
44		nnon 7893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
45		omcl 8574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
46	6, 44, 45	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ On)
47		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
48	47	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → 𝐵 ∈ On)
49	6	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → 𝐴 ∈ On)
50	46, 48, 49	3jca 1129	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On))
51	50	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On)))
52	51	adantrd 491	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On)))
53	52	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On))
54		oaword 8587	. . . . . . . . . . . 12 ⊢ (((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑦) ⊆ 𝐵 ↔ (𝐴 +o (𝐴 ·o 𝑦)) ⊆ (𝐴 +o 𝐵)))
55	53, 54	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → ((𝐴 ·o 𝑦) ⊆ 𝐵 ↔ (𝐴 +o (𝐴 ·o 𝑦)) ⊆ (𝐴 +o 𝐵)))
56	55	biimpa 476	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 +o (𝐴 ·o 𝑦)) ⊆ (𝐴 +o 𝐵))
57		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 𝐴 ∈ On)
58	12	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 1o ∈ On)
59	44	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 𝑦 ∈ On)
60		odi 8617	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (1o +o 𝑦)) = ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)))
61	57, 58, 59, 60	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o (1o +o 𝑦)) = ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)))
62		1onn 8678	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ ω
63		nnacom 8655	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1o ∈ ω ∧ 𝑦 ∈ ω) → (1o +o 𝑦) = (𝑦 +o 1o))
64	62, 63	mpan 690	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → (1o +o 𝑦) = (𝑦 +o 1o))
65		oa1suc 8569	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
66	44, 65	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → (𝑦 +o 1o) = suc 𝑦)
67	64, 66	eqtrd 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → (1o +o 𝑦) = suc 𝑦)
68	67	oveq2d 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → (𝐴 ·o (1o +o 𝑦)) = (𝐴 ·o suc 𝑦))
69	68	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o (1o +o 𝑦)) = (𝐴 ·o suc 𝑦))
70	18	oveq1d 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)) = (𝐴 +o (𝐴 ·o 𝑦)))
71	70	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)) = (𝐴 +o (𝐴 ·o 𝑦)))
72	61, 69, 71	3eqtr3rd 2786	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
73	72	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (𝐴 ∈ On → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
74	73	adantrd 491	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
75	74	adantrd 491	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
76	75	imp 406	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
77	76	adantr 480	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
78		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 +o 𝐵) = 𝐵)
79	78	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → (𝐴 +o 𝐵) = 𝐵)
80	79	adantr 480	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)
81	56, 77, 80	3sstr3d 4038	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 ·o suc 𝑦) ⊆ 𝐵)
82	81	exp31 419	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ((𝐴 ·o 𝑦) ⊆ 𝐵 → (𝐴 ·o suc 𝑦) ⊆ 𝐵)))
83	35, 37, 39, 43, 82	finds2 7920	. . . . . . 7 ⊢ (𝑥 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o 𝑥) ⊆ 𝐵))
84	83	com12 32	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝑥 ∈ ω → (𝐴 ·o 𝑥) ⊆ 𝐵))
85	84	ralrimiv 3145	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
86		iunss 5045	. . . . 5 ⊢ (∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵 ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
87	85, 86	sylibr 234	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
88	33, 87	eqsstrd 4018	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ω) ⊆ 𝐵)
89	88	ex 412	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = 𝐵 → (𝐴 ·o ω) ⊆ 𝐵))
90	29, 89	impbid 212	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ (𝐴 +o 𝐵) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333  ∪ ciun 4991  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431  ωcom 7887  1_oc1o 8499   +_o coa 8503   ·_o comu 8504

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511

This theorem is referenced by: (None)