Theorem odi 8285

Description: Distributive law for ordinal arithmetic (left-distributivity). Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)

Assertion

Ref	Expression
odi	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Proof of Theorem odi

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

Step	Hyp	Ref	Expression
1		oveq2 7199	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
2	1	oveq2d 7207	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o ∅)))
3		oveq2 7199	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
4	3	oveq2d 7207	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
5	2, 4	eqeq12d 2752	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅))))
6		oveq2 7199	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
7	6	oveq2d 7207	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝑦)))
8		oveq2 7199	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
9	8	oveq2d 7207	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))
10	7, 9	eqeq12d 2752	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))))
11		oveq2 7199	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
12	11	oveq2d 7207	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o suc 𝑦)))
13		oveq2 7199	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
14	13	oveq2d 7207	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))
15	12, 14	eqeq12d 2752	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))
16		oveq2 7199	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
17	16	oveq2d 7207	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝐶)))
18		oveq2 7199	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
19	18	oveq2d 7207	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
20	17, 19	eqeq12d 2752	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
21		omcl 8241	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
22		oa0 8221	. . . . . 6 ⊢ ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
23	21, 22	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
24		om0 8222	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
25	24	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) = ∅)
26	25	oveq2d 7207	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
27		oa0 8221	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
28	27	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
29	28	oveq2d 7207	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 +o ∅)) = (𝐴 ·o 𝐵))
30	23, 26, 29	3eqtr4rd 2782	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
31		oveq1 7198	. . . . . . . 8 ⊢ ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴))
32		oasuc 8229	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
33	32	3adant1 1132	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
34	33	oveq2d 7207	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 +o suc 𝑦)) = (𝐴 ·o suc (𝐵 +o 𝑦)))
35		oacl 8240	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
36		omsuc 8231	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
37	35, 36	sylan2 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
38	37	3impb 1117	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
39	34, 38	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
40		omsuc 8231	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
41	40	3adant2 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
42	41	oveq2d 7207	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
43		omcl 8241	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
44		oaass 8267	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ (𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
45	21, 44	syl3an1 1165	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
46	43, 45	syl3an2 1166	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
47	46	3exp 1121	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))
48	47	exp4b 434	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (𝑦 ∈ On → (𝐴 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))))
49	48	pm2.43a 54	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))))
50	49	com4r 94	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))))
51	50	pm2.43i 52	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))
52	51	3imp 1113	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
53	42, 52	eqtr4d 2774	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴))
54	39, 53	eqeq12d 2752	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) ↔ ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴)))
55	31, 54	syl5ibr 249	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))
56	55	3exp 1121	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))))
57	56	com3r 87	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))))
58	57	impd 414	. . . 4 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))))
59		vex 3402	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
60		limelon 6254	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
61	59, 60	mpan 690	. . . . . . . . . . . . 13 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
62		oacl 8240	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
63		om0r 8244	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 +o 𝑥) ∈ On → (∅ ·o (𝐵 +o 𝑥)) = ∅)
64	62, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∅ ·o (𝐵 +o 𝑥)) = ∅)
65		om0r 8244	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
66		om0r 8244	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ On → (∅ ·o 𝑥) = ∅)
67	65, 66	oveqan12d 7210	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∅ ·o 𝐵) +o (∅ ·o 𝑥)) = (∅ +o ∅))
68		0elon 6244	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ On
69		oa0 8221	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
70	68, 69	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∅ +o ∅) = ∅
71	67, 70	eqtr2di 2788	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ∅ = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
72	64, 71	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
73	61, 72	sylan2 596	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
74	73	ancoms 462	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
75		oveq1 7198	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (𝐴 ·o (𝐵 +o 𝑥)) = (∅ ·o (𝐵 +o 𝑥)))
76		oveq1 7198	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
77		oveq1 7198	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝑥) = (∅ ·o 𝑥))
78	76, 77	oveq12d 7209	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
79	75, 78	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥))))
80	74, 79	syl5ibr 249	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))
81	80	expd 419	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
82	81	com3r 87	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐴 = ∅ → (Lim 𝑥 → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
83	82	imp 410	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))
84	83	a1dd 50	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
85		simplr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝐵 ∈ On)
86	62	ancoms 462	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
87		onelon 6216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 +o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
88	86, 87	sylan 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
89		ontri1 6225	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝐵))
90		oawordex 8263	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵 ⊆ 𝑧 ↔ ∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧))
91	89, 90	bitr3d 284	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧 ∈ 𝐵 ↔ ∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧))
92	85, 88, 91	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (¬ 𝑧 ∈ 𝐵 ↔ ∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧))
93		oaord 8253	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣 ∈ 𝑥 ↔ (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
94	93	3expb 1122	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣 ∈ 𝑥 ↔ (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
95		eleq1 2818	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 +o 𝑣) = 𝑧 → ((𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥) ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
96	94, 95	sylan9bb 513	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑣) = 𝑧) → (𝑣 ∈ 𝑥 ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
97		iba 531	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 +o 𝑣) = 𝑧 → (𝑣 ∈ 𝑥 ↔ (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
98	97	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑣) = 𝑧) → (𝑣 ∈ 𝑥 ↔ (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
99	96, 98	bitr3d 284	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑣) = 𝑧) → (𝑧 ∈ (𝐵 +o 𝑥) ↔ (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
100	99	an32s 652	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑣 ∈ On ∧ (𝐵 +o 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +o 𝑥) ↔ (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
101	100	biimpcd 252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝐵 +o 𝑥) → (((𝑣 ∈ On ∧ (𝐵 +o 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
102	101	exp4c 436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝐵 +o 𝑥) → (𝑣 ∈ On → ((𝐵 +o 𝑣) = 𝑧 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))))
103	102	com4r 94	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑣 ∈ On → ((𝐵 +o 𝑣) = 𝑧 → (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))))
104	103	imp 410	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑣 ∈ On → ((𝐵 +o 𝑣) = 𝑧 → (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧))))
105	104	reximdvai 3181	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧 → ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
106	92, 105	sylbid 243	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (¬ 𝑧 ∈ 𝐵 → ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
107	106	orrd 863	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ 𝐵 ∨ ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
108	61, 107	sylanl1 680	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ 𝐵 ∨ ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
109	108	adantlrl 720	. . . . . . . . . . . . . . 15 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ 𝐵 ∨ ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
110	109	adantlr 715	. . . . . . . . . . . . . 14 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ 𝐵 ∨ ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
111		0ellim 6253	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
112		om00el 8282	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·o 𝑥) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥)))
113	112	biimprd 251	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥) → ∅ ∈ (𝐴 ·o 𝑥)))
114	111, 113	sylan2i 609	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((∅ ∈ 𝐴 ∧ Lim 𝑥) → ∅ ∈ (𝐴 ·o 𝑥)))
115	61, 114	sylan2 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((∅ ∈ 𝐴 ∧ Lim 𝑥) → ∅ ∈ (𝐴 ·o 𝑥)))
116	115	exp4b 434	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐴 → (Lim 𝑥 → ∅ ∈ (𝐴 ·o 𝑥)))))
117	116	com4r 94	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ·o 𝑥)))))
118	117	pm2.43a 54	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ·o 𝑥))))
119	118	imp31 421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ·o 𝑥))
120	119	a1d 25	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → ∅ ∈ (𝐴 ·o 𝑥)))
121	120	adantlrr 721	. . . . . . . . . . . . . . . . . . 19 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → ∅ ∈ (𝐴 ·o 𝑥)))
122		omordi 8272	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → (𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵)))
123	122	ancom1s 653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → (𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵)))
124		onelss 6233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o 𝐵)))
125	22	sseq2d 3919	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅) ↔ (𝐴 ·o 𝑧) ⊆ (𝐴 ·o 𝐵)))
126	124, 125	sylibrd 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
127	21, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
128	127	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
129	123, 128	syld 47	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
130	129	adantll 714	. . . . . . . . . . . . . . . . . . 19 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
131	121, 130	jcad 516	. . . . . . . . . . . . . . . . . 18 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → (∅ ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅))))
132		oveq2 7199	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ∅ → ((𝐴 ·o 𝐵) +o 𝑤) = ((𝐴 ·o 𝐵) +o ∅))
133	132	sseq2d 3919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ∅ → ((𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤) ↔ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
134	133	rspcev 3527	. . . . . . . . . . . . . . . . . 18 ⊢ ((∅ ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
135	131, 134	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ 𝐵 → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
136	135	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (𝑧 ∈ 𝐵 → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
137		omordi 8272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑣 ∈ 𝑥 → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
138	61, 137	sylanl1 680	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑣 ∈ 𝑥 → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
139	138	adantrd 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
140	139	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
141		oveq2 7199	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑣 → (𝐵 +o 𝑦) = (𝐵 +o 𝑣))
142	141	oveq2d 7207	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑣 → (𝐴 ·o (𝐵 +o 𝑦)) = (𝐴 ·o (𝐵 +o 𝑣)))
143		oveq2 7199	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑣 → (𝐴 ·o 𝑦) = (𝐴 ·o 𝑣))
144	143	oveq2d 7207	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑣 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
145	142, 144	eqeq12d 2752	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑣 → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) ↔ (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
146	145	rspccv 3524	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝑣 ∈ 𝑥 → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
147		oveq2 7199	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐵 +o 𝑣) = 𝑧 → (𝐴 ·o (𝐵 +o 𝑣)) = (𝐴 ·o 𝑧))
148		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) → ((𝐴 ·o (𝐵 +o 𝑣)) = (𝐴 ·o 𝑧) ↔ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) = (𝐴 ·o 𝑧)))
149	147, 148	syl5ib 247	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) → ((𝐵 +o 𝑣) = 𝑧 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) = (𝐴 ·o 𝑧)))
150		eqimss2 3944	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) = (𝐴 ·o 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
151	149, 150	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) → ((𝐵 +o 𝑣) = 𝑧 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
152	151	imim2i 16	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ∈ 𝑥 → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))) → (𝑣 ∈ 𝑥 → ((𝐵 +o 𝑣) = 𝑧 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))))
153	152	impd 414	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ 𝑥 → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))) → ((𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
154	146, 153	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → ((𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
155	154	ad2antll 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
156	140, 155	jcad 516	. . . . . . . . . . . . . . . . . . 19 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ((𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))))
157		oveq2 7199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
158	157	sseq2d 3919	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = (𝐴 ·o 𝑣) → ((𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤) ↔ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
159	158	rspcev 3527	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
160	156, 159	syl6 35	. . . . . . . . . . . . . . . . . 18 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
161	160	rexlimdvw 3199	. . . . . . . . . . . . . . . . 17 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
162	161	adantlrr 721	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
163	136, 162	jaod 859	. . . . . . . . . . . . . . 15 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑧 ∈ 𝐵 ∨ ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
164	163	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ((𝑧 ∈ 𝐵 ∨ ∃𝑣 ∈ On (𝑣 ∈ 𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
165	110, 164	mpd 15	. . . . . . . . . . . . 13 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
166	165	ralrimiva 3095	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
167		iunss2 4944	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤) → ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧) ⊆ ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
168	166, 167	syl 17	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧) ⊆ ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
169		omordlim 8283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
170	169	ex 416	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑤 ∈ (𝐴 ·o 𝑥) → ∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣)))
171	59, 170	mpanr1 703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑥) → ∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣)))
172	171	ancoms 462	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝑤 ∈ (𝐴 ·o 𝑥) → ∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣)))
173	172	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
174	173	adantlrr 721	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
175	174	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
176		oaordi 8252	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣 ∈ 𝑥 → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
177	61, 176	sylan 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝑣 ∈ 𝑥 → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
178	177	imp 410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝑣 ∈ 𝑥) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥))
179	178	adantlrl 720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣 ∈ 𝑥) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥))
180	179	a1d 25	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
181	180	adantlr 715	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
182		limord 6250	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (Lim 𝑥 → Ord 𝑥)
183		ordelon 6215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Ord 𝑥 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ On)
184	182, 183	sylan 583	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Lim 𝑥 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ On)
185		omcl 8241	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ On ∧ 𝑣 ∈ On) → (𝐴 ·o 𝑣) ∈ On)
186	185	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝑣) ∈ On)
187	186	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o 𝑣) ∈ On)
188	21	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o 𝐵) ∈ On)
189		oaordi 8252	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐴 ·o 𝑣) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
190	187, 188, 189	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
191	184, 190	sylan 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((Lim 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
192	191	an32s 652	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
193	192	adantlr 715	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
194	145	rspccva 3526	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) ∧ 𝑣 ∈ 𝑥) → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
195	194	eleq2d 2816	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) ∧ 𝑣 ∈ 𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) ↔ ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
196	195	adantll 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) ↔ ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
197	193, 196	sylibrd 262	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣))))
198		oacl 8240	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 ∈ On ∧ 𝑣 ∈ On) → (𝐵 +o 𝑣) ∈ On)
199	198	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑣) ∈ On)
200		omcl 8241	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ On ∧ (𝐵 +o 𝑣) ∈ On) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
201	199, 200	sylan2 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ On ∧ (𝑣 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
202	201	an12s 649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
203	184, 202	sylan 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((Lim 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
204	203	an32s 652	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣 ∈ 𝑥) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
205		onelss 6233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·o (𝐵 +o 𝑣)) ∈ On → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
206	204, 205	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣 ∈ 𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
207	206	adantlr 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
208	197, 207	syld 47	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
209	181, 208	jcad 516	. . . . . . . . . . . . . . . . . 18 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥) ∧ ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣)))))
210		oveq2 7199	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 +o 𝑣) → (𝐴 ·o 𝑧) = (𝐴 ·o (𝐵 +o 𝑣)))
211	210	sseq2d 3919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐵 +o 𝑣) → (((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧) ↔ ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
212	211	rspcev 3527	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥) ∧ ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧))
213	209, 212	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣 ∈ 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧)))
214	213	rexlimdva 3193	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) → (∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧)))
215	214	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → (∃𝑣 ∈ 𝑥 𝑤 ∈ (𝐴 ·o 𝑣) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧)))
216	175, 215	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧))
217	216	ralrimiva 3095	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) → ∀𝑤 ∈ (𝐴 ·o 𝑥)∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧))
218		iunss2 4944	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ (𝐴 ·o 𝑥)∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧) → ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
219	217, 218	syl 17	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) → ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
220	219	adantrl 716	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
221	168, 220	eqssd 3904	. . . . . . . . . 10 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧) = ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
222		oalimcl 8266	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +o 𝑥))
223	59, 222	mpanr1 703	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +o 𝑥))
224	223	ancoms 462	. . . . . . . . . . . . . 14 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → Lim (𝐵 +o 𝑥))
225	224	anim2i 620	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)))
226	225	an12s 649	. . . . . . . . . . . 12 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)))
227		ovex 7224	. . . . . . . . . . . . 13 ⊢ (𝐵 +o 𝑥) ∈ V
228		omlim 8238	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ ((𝐵 +o 𝑥) ∈ V ∧ Lim (𝐵 +o 𝑥))) → (𝐴 ·o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
229	227, 228	mpanr1 703	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)) → (𝐴 ·o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
230	226, 229	syl 17	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
231	230	adantr 484	. . . . . . . . . 10 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (𝐴 ·o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
232	21	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴 ·o 𝐵) ∈ On)
233	59	jctl 527	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑥 → (𝑥 ∈ V ∧ Lim 𝑥))
234	233	anim1ci 619	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)))
235		omlimcl 8284	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝑥))
236	234, 235	sylan 583	. . . . . . . . . . . . . 14 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝑥))
237	236	adantlrr 721	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝑥))
238		ovex 7224	. . . . . . . . . . . . 13 ⊢ (𝐴 ·o 𝑥) ∈ V
239	237, 238	jctil 523	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) ∈ V ∧ Lim (𝐴 ·o 𝑥)))
240		oalim 8237	. . . . . . . . . . . 12 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ ((𝐴 ·o 𝑥) ∈ V ∧ Lim (𝐴 ·o 𝑥))) → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
241	232, 239, 240	syl2anc 587	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
242	241	adantrr 717	. . . . . . . . . 10 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ∪ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
243	221, 231, 242	3eqtr4d 2781	. . . . . . . . 9 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))
244	243	exp43 440	. . . . . . . 8 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))))
245	244	com3l 89	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))))
246	245	imp 410	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
247	84, 246	oe0lem 8218	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
248	247	com12 32	. . . 4 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
249	5, 10, 15, 20, 30, 58, 248	tfinds3 7621	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
250	249	expdcom 418	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))))
251	250	3imp 1113	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052  Vcvv 3398   ⊆ wss 3853  ∅c0 4223  ∪ ciun 4890  Ord word 6190  Oncon0 6191  Lim wlim 6192  suc csuc 6193  (class class class)co 7191   +_o coa 8177   ·_o comu 8178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-oadd 8184  df-omul 8185

This theorem is referenced by:  omass  8286  oeeui  8308  oaabs2  8352