Theorem oeoalem 7563

Description: Lemma for oeoa 7564. (Contributed by Eric Schmidt, 26-May-2009.)

Hypotheses

Ref	Expression
oeoalem.1	⊢ 𝐴 ∈ On
oeoalem.2	⊢ ∅ ∈ 𝐴
oeoalem.3	⊢ 𝐵 ∈ On

Assertion

Ref	Expression
oeoalem	⊢ (𝐶 ∈ On → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))

Proof of Theorem oeoalem

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

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . 4 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
2	1	oveq2d 6565	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 ∅)))
3		oveq2 6557	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
4	3	oveq2d 6565	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 ∅)))
5	2, 4	eqeq12d 2625	. 2 ⊢ (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 6557	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
7	6	oveq2d 6565	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)))
8		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
9	8	oveq2d 6565	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
10	7, 9	eqeq12d 2625	. 2 ⊢ (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 6557	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
12	11	oveq2d 6565	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)))
13		oveq2 6557	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
14	13	oveq2d 6565	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 suc 𝑦)))
15	12, 14	eqeq12d 2625	. 2 ⊢ (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 6557	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
17	16	oveq2d 6565	. . 3 ⊢ (𝑥 = 𝐶 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)))
18		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐶))
19	18	oveq2d 6565	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))
20	17, 19	eqeq12d 2625	. 2 ⊢ (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		oeoalem.1	. . . . 5 ⊢ 𝐴 ∈ On
22		oeoalem.3	. . . . 5 ⊢ 𝐵 ∈ On
23		oecl 7504	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
24	21, 22, 23	mp2an 704	. . . 4 ⊢ (𝐴 ↑𝑜 𝐵) ∈ On
25		om1 7509	. . . 4 ⊢ ((𝐴 ↑𝑜 𝐵) ∈ On → ((𝐴 ↑𝑜 𝐵) ·𝑜 1𝑜) = (𝐴 ↑𝑜 𝐵))
26	24, 25	ax-mp 5	. . 3 ⊢ ((𝐴 ↑𝑜 𝐵) ·𝑜 1𝑜) = (𝐴 ↑𝑜 𝐵)
27		oe0 7489	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
28	21, 27	ax-mp 5	. . . 4 ⊢ (𝐴 ↑𝑜 ∅) = 1𝑜
29	28	oveq2i 6560	. . 3 ⊢ ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 ∅)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 1𝑜)
30		oa0 7483	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
31	22, 30	ax-mp 5	. . . 4 ⊢ (𝐵 +𝑜 ∅) = 𝐵
32	31	oveq2i 6560	. . 3 ⊢ (𝐴 ↑𝑜 (𝐵 +𝑜 ∅)) = (𝐴 ↑𝑜 𝐵)
33	26, 29, 32	3eqtr4ri 2643	. 2 ⊢ (𝐴 ↑𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 ∅))
34		oasuc 7491	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
35	34	oveq2d 6565	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 ↑𝑜 suc (𝐵 +𝑜 𝑦)))
36		oacl 7502	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
37		oesuc 7494	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴 ↑𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
38	21, 36, 37	sylancr 694	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
39	35, 38	eqtrd 2644	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
40	22, 39	mpan 702	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
41		oveq1 6556	. . . . 5 ⊢ ((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))
42	40, 41	sylan9eq 2664	. . . 4 ⊢ ((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 𝑦)) ·𝑜 𝐴))
43		oecl 7504	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
44		omass 7547	. . . . . . . . 9 ⊢ (((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	24, 21, 44	mp3an13 1407	. . . . . . . 8 ⊢ ((𝐴 ↑𝑜 𝑦) ∈ On → (((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴 ↑𝑜 𝐵) ·𝑜 ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
46	43, 45	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴 ↑𝑜 𝐵) ·𝑜 ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
47		oesuc 7494	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
48	47	oveq2d 6565	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 suc 𝑦)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
49	46, 48	eqtr4d 2647	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 suc 𝑦)))
50	21, 49	mpan 702	. . . . 5 ⊢ (𝑦 ∈ On → (((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 suc 𝑦)))
51	50	adantr 480	. . . 4 ⊢ ((𝑦 ∈ On ∧ (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦))) → (((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 suc 𝑦)))
52	42, 51	eqtrd 2644	. . 3 ⊢ ((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)))
53	52	ex 449	. 2 ⊢ (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))))
54		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
55		oalim 7499	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦))
56	22, 55	mpan 702	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦))
57	54, 56	mpan 702	. . . . . . 7 ⊢ (Lim 𝑥 → (𝐵 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦))
58	57	oveq2d 6565	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦)))
59	54	a1i 11	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 ∈ V)
60		limord 5701	. . . . . . . . . 10 ⊢ (Lim 𝑥 → Ord 𝑥)
61		ordelon 5664	. . . . . . . . . 10 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
62	60, 61	sylan 487	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
63	22, 62, 36	sylancr 694	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
64	63	ralrimiva 2949	. . . . . . 7 ⊢ (Lim 𝑥 → ∀𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦) ∈ On)
65		0ellim 5704	. . . . . . . 8 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
66		ne0i 3880	. . . . . . . 8 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
67	65, 66	syl 17	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
68		vex 3176	. . . . . . . . 9 ⊢ 𝑤 ∈ V
69		oeoalem.2	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝐴
70		oelim 7501	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
71	69, 70	mpan2 703	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
72	21, 71	mpan 702	. . . . . . . . 9 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
73	68, 72	mpan 702	. . . . . . . 8 ⊢ (Lim 𝑤 → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
74		oewordi 7558	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
75	69, 74	mpan2 703	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
76	21, 75	mp3an3 1405	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
77	76	3impia 1253	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤))
78	73, 77	onoviun 7327	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴 ↑𝑜 ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)))
79	59, 64, 67, 78	syl3anc 1318	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ↑𝑜 ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)))
80	58, 79	eqtrd 2644	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)))
81		iuneq2 4473	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
82	80, 81	sylan9eq 2664	. . . 4 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦))) → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
83		oelim 7501	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
84	69, 83	mpan2 703	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
85	21, 84	mpan 702	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
86	54, 85	mpan 702	. . . . . . 7 ⊢ (Lim 𝑥 → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
87	86	oveq2d 6565	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦)))
88	21, 62, 43	sylancr 694	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝐴 ↑𝑜 𝑦) ∈ On)
89	88	ralrimiva 2949	. . . . . . 7 ⊢ (Lim 𝑥 → ∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ∈ On)
90		omlim 7500	. . . . . . . . . 10 ⊢ (((𝐴 ↑𝑜 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧))
91	24, 90	mpan 702	. . . . . . . . 9 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧))
92	68, 91	mpan 702	. . . . . . . 8 ⊢ (Lim 𝑤 → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧))
93		omwordi 7538	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴 ↑𝑜 𝐵) ∈ On) → (𝑧 ⊆ 𝑤 → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤)))
94	24, 93	mp3an3 1405	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤)))
95	94	3impia 1253	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤))
96	92, 95	onoviun 7327	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴 ↑𝑜 𝐵) ·𝑜 ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
97	59, 89, 67, 96	syl3anc 1318	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ↑𝑜 𝐵) ·𝑜 ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
98	87, 97	eqtrd 2644	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
99	98	adantr 480	. . . 4 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦))) → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
100	82, 99	eqtr4d 2647	. . 3 ⊢ ((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)))
101	100	ex 449	. 2 ⊢ (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))))
102	5, 10, 15, 20, 33, 53, 101	tfinds 6951	1 ⊢ (𝐶 ∈ On → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440   +_𝑜 coa 7444   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

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-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-int 4411  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-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-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oeoa  7564