Theorem oaabsb 43819

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 9591	. . . . 5 ⊢ ω ∈ On
2		omcl 8493	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ·o ω) ∈ On)
3	1, 2	mpan2 699	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·o ω) ∈ On)
4		oawordex 8514	. . . 4 ⊢ (((𝐴 ·o ω) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ ∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵))
5	3, 4	sylan 588	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ ∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵))
6		simpl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
7	6	adantr 483	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
8	3	ad2antrr 734	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 ·o ω) ∈ On)
9		simpr 487	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
10		oaass 8518	. . . . . . 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 1386	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = (𝐴 +o ((𝐴 ·o ω) +o 𝑥)))
12		1on 8438	. . . . . . . . . 10 ⊢ 1o ∈ On
13		odi 8536	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On ∧ ω ∈ On) → (𝐴 ·o (1o +o ω)) = ((𝐴 ·o 1o) +o (𝐴 ·o ω)))
14	12, 1, 13	mp3an23 1468	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o (1o +o ω)) = ((𝐴 ·o 1o) +o (𝐴 ·o ω)))
15		1oaomeqom 43818	. . . . . . . . . . 11 ⊢ (1o +o ω) = ω
16	15	oveq2i 7396	. . . . . . . . . 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 8499	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
19	18	oveq1d 7400	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((𝐴 ·o 1o) +o (𝐴 ·o ω)) = (𝐴 +o (𝐴 ·o ω)))
20	14, 17, 19	3eqtr3rd 2800	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +o (𝐴 ·o ω)) = (𝐴 ·o ω))
21	20	oveq1d 7400	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = ((𝐴 ·o ω) +o 𝑥))
22	21	ad2antrr 734	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = ((𝐴 ·o ω) +o 𝑥))
23	11, 22	eqtr3d 2793	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = ((𝐴 ·o ω) +o 𝑥))
24		oveq2 7393	. . . . . 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 2772	. . . . 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 247	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
28	27	rexlimdva 3157	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
29	5, 28	sylbid 242	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 → (𝐴 +o 𝐵) = 𝐵))
30		limom 7851	. . . . . 6 ⊢ Lim ω
31		omlim 8490	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
32	1, 30, 31	mpanr12 713	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
33	32	ad2antrr 734	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
34		oveq2 7393	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
35	34	sseq1d 3962	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o ∅) ⊆ 𝐵))
36		oveq2 7393	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
37	36	sseq1d 3962	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o 𝑦) ⊆ 𝐵))
38		oveq2 7393	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
39	38	sseq1d 3962	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o suc 𝑦) ⊆ 𝐵))
40		om0 8474	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
41		0ss 4348	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐵
42	40, 41	eqsstrdi 3975	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ 𝐵)
43	42	ad2antrr 734	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ∅) ⊆ 𝐵)
44		nnon 7841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
45		omcl 8493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
46	6, 44, 45	syl2an 604	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ On)
47		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
48	47	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → 𝐵 ∈ On)
49	6	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → 𝐴 ∈ On)
50	46, 48, 49	3jca 1137	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On))
51	50	expcom 416	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On)))
52	51	adantrd 494	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On)))
53	52	imp 409	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On))
54		oaword 8506	. . . . . . . . . . . 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 479	. . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 𝐴 ∈ On)
58	12	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 1o ∈ On)
59	44	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 𝑦 ∈ On)
60		odi 8536	. . . . . . . . . . . . . . . . 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 1386	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o (1o +o 𝑦)) = ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)))
62		1onn 8598	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ ω
63		nnacom 8575	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1o ∈ ω ∧ 𝑦 ∈ ω) → (1o +o 𝑦) = (𝑦 +o 1o))
64	62, 63	mpan 698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → (1o +o 𝑦) = (𝑦 +o 1o))
65		oa1suc 8488	. . . . . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → (1o +o 𝑦) = suc 𝑦)
68	67	oveq2d 7401	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → (𝐴 ·o (1o +o 𝑦)) = (𝐴 ·o suc 𝑦))
69	68	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o (1o +o 𝑦)) = (𝐴 ·o suc 𝑦))
70	18	oveq1d 7400	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)) = (𝐴 +o (𝐴 ·o 𝑦)))
71	70	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)) = (𝐴 +o (𝐴 ·o 𝑦)))
72	61, 69, 71	3eqtr3rd 2800	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
73	72	expcom 416	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (𝐴 ∈ On → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
74	73	adantrd 494	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
75	74	adantrd 494	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
76	75	imp 409	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
77	76	adantr 483	. . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 +o 𝐵) = 𝐵)
79	78	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → (𝐴 +o 𝐵) = 𝐵)
80	79	adantr 483	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)
81	56, 77, 80	3sstr3d 3985	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 ·o suc 𝑦) ⊆ 𝐵)
82	81	exp31 422	. . . . . . . 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 7868	. . . . . . 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 3147	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
86		iunss 4996	. . . . 5 ⊢ (∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵 ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
87	85, 86	sylibr 236	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
88	33, 87	eqsstrd 3965	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ω) ⊆ 𝐵)
89	88	ex 415	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = 𝐵 → (𝐴 ·o ω) ⊆ 𝐵))
90	29, 89	impbid 214	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ (𝐴 +o 𝐵) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∀wral 3070  ∃wrex 3080   ⊆ wss 3899  ∅c0 4280  ∪ ciun 4943  Oncon0 6335  Lim wlim 6336  suc csuc 6337  (class class class)co 7385  ωcom 7835  1_oc1o 8418   +_o coa 8422   ·_o comu 8423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707  ax-inf2 9586

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-oadd 8429  df-omul 8430

This theorem is referenced by: (None)