Theorem oaabsb 43318

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 9660	. . . . 5 ⊢ ω ∈ On
2		omcl 8548	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ·o ω) ∈ On)
3	1, 2	mpan2 691	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·o ω) ∈ On)
4		oawordex 8569	. . . 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 8573	. . . . . . 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 8492	. . . . . . . . . 10 ⊢ 1o ∈ On
13		odi 8591	. . . . . . . . . 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 43317	. . . . . . . . . . 11 ⊢ (1o +o ω) = ω
16	15	oveq2i 7416	. . . . . . . . . 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 8554	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
19	18	oveq1d 7420	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((𝐴 ·o 1o) +o (𝐴 ·o ω)) = (𝐴 +o (𝐴 ·o ω)))
20	14, 17, 19	3eqtr3rd 2779	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +o (𝐴 ·o ω)) = (𝐴 ·o ω))
21	20	oveq1d 7420	. . . . . . 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 2772	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = ((𝐴 ·o ω) +o 𝑥))
24		oveq2 7413	. . . . . 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 2751	. . . . 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 3141	. . 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 7877	. . . . . 6 ⊢ Lim ω
31		omlim 8545	. . . . . 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 7413	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
35	34	sseq1d 3990	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o ∅) ⊆ 𝐵))
36		oveq2 7413	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
37	36	sseq1d 3990	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o 𝑦) ⊆ 𝐵))
38		oveq2 7413	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
39	38	sseq1d 3990	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o suc 𝑦) ⊆ 𝐵))
40		om0 8529	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
41		0ss 4375	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐵
42	40, 41	eqsstrdi 4003	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ 𝐵)
43	42	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ∅) ⊆ 𝐵)
44		nnon 7867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
45		omcl 8548	. . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . 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 8561	. . . . . . . . . . . 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 8591	. . . . . . . . . . . . . . . . 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 8652	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ ω
63		nnacom 8629	. . . . . . . . . . . . . . . . . . . 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 8543	. . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → (1o +o 𝑦) = suc 𝑦)
68	67	oveq2d 7421	. . . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . 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 4013	. . . . . . . . 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 7894	. . . . . . 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 3131	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
86		iunss 5021	. . . . 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 3993	. . 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 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∅c0 4308  ∪ ciun 4967  Oncon0 6352  Lim wlim 6353  suc csuc 6354  (class class class)co 7405  ωcom 7861  1_oc1o 8473   +_o coa 8477   ·_o comu 8478

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729  ax-inf2 9655

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485

This theorem is referenced by: (None)