Theorem oeoalem 7676

Description: Lemma for oeoa 7677. (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 6658	. . . 4 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
2	1	oveq2d 6666	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 ∅)))
3		oveq2 6658	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
4	3	oveq2d 6666	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 ∅)))
5	2, 4	eqeq12d 2637	. 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 6658	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
7	6	oveq2d 6666	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)))
8		oveq2 6658	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
9	8	oveq2d 6666	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
10	7, 9	eqeq12d 2637	. 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 6658	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
12	11	oveq2d 6666	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)))
13		oveq2 6658	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
14	13	oveq2d 6666	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 suc 𝑦)))
15	12, 14	eqeq12d 2637	. 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 6658	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
17	16	oveq2d 6666	. . 3 ⊢ (𝑥 = 𝐶 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)))
18		oveq2 6658	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐶))
19	18	oveq2d 6666	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))
20	17, 19	eqeq12d 2637	. 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 7617	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
24	21, 22, 23	mp2an 708	. . . 4 ⊢ (𝐴 ↑𝑜 𝐵) ∈ On
25		om1 7622	. . . 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 7602	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
28	21, 27	ax-mp 5	. . . 4 ⊢ (𝐴 ↑𝑜 ∅) = 1𝑜
29	28	oveq2i 6661	. . 3 ⊢ ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 ∅)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 1𝑜)
30		oa0 7596	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
31	22, 30	ax-mp 5	. . . 4 ⊢ (𝐵 +𝑜 ∅) = 𝐵
32	31	oveq2i 6661	. . 3 ⊢ (𝐴 ↑𝑜 (𝐵 +𝑜 ∅)) = (𝐴 ↑𝑜 𝐵)
33	26, 29, 32	3eqtr4ri 2655	. 2 ⊢ (𝐴 ↑𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 ∅))
34		oasuc 7604	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
35	34	oveq2d 6666	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 ↑𝑜 suc (𝐵 +𝑜 𝑦)))
36		oacl 7615	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
37		oesuc 7607	. . . . . . . 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 695	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
39	35, 38	eqtrd 2656	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
40	22, 39	mpan 706	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 ↑𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
41		oveq1 6657	. . . . 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 2676	. . . 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 7617	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
44		omass 7660	. . . . . . . . 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 1415	. . . . . . . 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 7607	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
48	47	oveq2d 6666	. . . . . . 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 2659	. . . . . 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 706	. . . . 5 ⊢ (𝑦 ∈ On → (((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 suc 𝑦)))
51	50	adantr 481	. . . 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 2656	. . 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 450	. 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 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
55		oalim 7612	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦))
56	22, 55	mpan 706	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦))
57	54, 56	mpan 706	. . . . . . 7 ⊢ (Lim 𝑥 → (𝐵 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦))
58	57	oveq2d 6666	. . . . . 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 5784	. . . . . . . . . 10 ⊢ (Lim 𝑥 → Ord 𝑥)
61		ordelon 5747	. . . . . . . . . 10 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
62	60, 61	sylan 488	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
63	22, 62, 36	sylancr 695	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
64	63	ralrimiva 2966	. . . . . . 7 ⊢ (Lim 𝑥 → ∀𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦) ∈ On)
65		0ellim 5787	. . . . . . . 8 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
66		ne0i 3921	. . . . . . . 8 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
67	65, 66	syl 17	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
68		vex 3203	. . . . . . . . 9 ⊢ 𝑤 ∈ V
69		oeoalem.2	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝐴
70		oelim 7614	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
71	69, 70	mpan2 707	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
72	21, 71	mpan 706	. . . . . . . . 9 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
73	68, 72	mpan 706	. . . . . . . 8 ⊢ (Lim 𝑤 → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
74		oewordi 7671	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
75	69, 74	mpan2 707	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
76	21, 75	mp3an3 1413	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
77	76	3impia 1261	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤))
78	73, 77	onoviun 7440	. . . . . . 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 1326	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ↑𝑜 ∪ 𝑦 ∈ 𝑥 (𝐵 +𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)))
80	58, 79	eqtrd 2656	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ↑𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 +𝑜 𝑦)))
81		iuneq2 4537	. . . . 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 2676	. . . 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 7614	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
84	69, 83	mpan2 707	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
85	21, 84	mpan 706	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
86	54, 85	mpan 706	. . . . . . 7 ⊢ (Lim 𝑥 → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
87	86	oveq2d 6666	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦)))
88	21, 62, 43	sylancr 695	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝐴 ↑𝑜 𝑦) ∈ On)
89	88	ralrimiva 2966	. . . . . . 7 ⊢ (Lim 𝑥 → ∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ∈ On)
90		omlim 7613	. . . . . . . . . 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 706	. . . . . . . . 9 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧))
92	68, 91	mpan 706	. . . . . . . 8 ⊢ (Lim 𝑤 → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧))
93		omwordi 7651	. . . . . . . . . 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 1413	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤)))
95	94	3impia 1261	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴 ↑𝑜 𝐵) ·𝑜 𝑤))
96	92, 95	onoviun 7440	. . . . . . 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 1326	. . . . . 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 2656	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝑦)))
99	98	adantr 481	. . . 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 2659	. . 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 450	. 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 7059	1 ⊢ (𝐶 ∈ On → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  1_𝑜c1o 7553   +_𝑜 coa 7557   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oeoa  7677