Theorem oeoalem 8563

Description: Lemma for oeoa 8564. (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 7398	. . . 4 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
2	1	oveq2d 7406	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ↑o (𝐵 +o 𝑥)) = (𝐴 ↑o (𝐵 +o ∅)))
3		oveq2 7398	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
4	3	oveq2d 7406	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑥)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o ∅)))
5	2, 4	eqeq12d 2746	. 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 7398	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
7	6	oveq2d 7406	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o (𝐵 +o 𝑥)) = (𝐴 ↑o (𝐵 +o 𝑦)))
8		oveq2 7398	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
9	8	oveq2d 7406	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑥)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑦)))
10	7, 9	eqeq12d 2746	. 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 7398	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
12	11	oveq2d 7406	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o (𝐵 +o 𝑥)) = (𝐴 ↑o (𝐵 +o suc 𝑦)))
13		oveq2 7398	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
14	13	oveq2d 7406	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑥)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o suc 𝑦)))
15	12, 14	eqeq12d 2746	. 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 7398	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
17	16	oveq2d 7406	. . 3 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o (𝐵 +o 𝑥)) = (𝐴 ↑o (𝐵 +o 𝐶)))
18		oveq2 7398	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐶))
19	18	oveq2d 7406	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑥)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶)))
20	17, 19	eqeq12d 2746	. 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 8504	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
24	21, 22, 23	mp2an 692	. . . 4 ⊢ (𝐴 ↑o 𝐵) ∈ On
25		om1 8509	. . . 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 8489	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
28	21, 27	ax-mp 5	. . . 4 ⊢ (𝐴 ↑o ∅) = 1o
29	28	oveq2i 7401	. . 3 ⊢ ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o ∅)) = ((𝐴 ↑o 𝐵) ·o 1o)
30		oa0 8483	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
31	22, 30	ax-mp 5	. . . 4 ⊢ (𝐵 +o ∅) = 𝐵
32	31	oveq2i 7401	. . 3 ⊢ (𝐴 ↑o (𝐵 +o ∅)) = (𝐴 ↑o 𝐵)
33	26, 29, 32	3eqtr4ri 2764	. 2 ⊢ (𝐴 ↑o (𝐵 +o ∅)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o ∅))
34		oasuc 8491	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
35	34	oveq2d 7406	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 +o suc 𝑦)) = (𝐴 ↑o suc (𝐵 +o 𝑦)))
36		oacl 8502	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
37		oesuc 8494	. . . . . . . 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 587	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc (𝐵 +o 𝑦)) = ((𝐴 ↑o (𝐵 +o 𝑦)) ·o 𝐴))
39	35, 38	eqtrd 2765	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 +o suc 𝑦)) = ((𝐴 ↑o (𝐵 +o 𝑦)) ·o 𝐴))
40	22, 39	mpan 690	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 ↑o (𝐵 +o suc 𝑦)) = ((𝐴 ↑o (𝐵 +o 𝑦)) ·o 𝐴))
41		oveq1 7397	. . . . 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 2785	. . . 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 8504	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
44		omass 8547	. . . . . . . . 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 1454	. . . . . . . 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 8494	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
48	47	oveq2d 7406	. . . . . . 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 2768	. . . . . 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 690	. . . . 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 2765	. . 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 412	. 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 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
55		oalim 8499	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦))
56	22, 55	mpan 690	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦))
57	54, 56	mpan 690	. . . . . . 7 ⊢ (Lim 𝑥 → (𝐵 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦))
58	57	oveq2d 7406	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ↑o (𝐵 +o 𝑥)) = (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦)))
59		limord 6396	. . . . . . . . . 10 ⊢ (Lim 𝑥 → Ord 𝑥)
60		ordelon 6359	. . . . . . . . . 10 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
61	59, 60	sylan 580	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
62	22, 61, 36	sylancr 587	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝐵 +o 𝑦) ∈ On)
63	62	ralrimiva 3126	. . . . . . 7 ⊢ (Lim 𝑥 → ∀𝑦 ∈ 𝑥 (𝐵 +o 𝑦) ∈ On)
64		0ellim 6399	. . . . . . . 8 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
65	64	ne0d 4308	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
66		vex 3454	. . . . . . . . 9 ⊢ 𝑤 ∈ V
67		oeoalem.2	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝐴
68		oelim 8501	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
69	67, 68	mpan2 691	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
70	21, 69	mpan 690	. . . . . . . . 9 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
71	66, 70	mpan 690	. . . . . . . 8 ⊢ (Lim 𝑤 → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
72		oewordi 8558	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
73	67, 72	mpan2 691	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
74	21, 73	mp3an3 1452	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
75	74	3impia 1117	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤))
76	71, 75	onoviun 8315	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐵 +o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 +o 𝑦)))
77	54, 63, 65, 76	mp3an2i 1468	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 +o 𝑦)))
78	58, 77	eqtrd 2765	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ↑o (𝐵 +o 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 +o 𝑦)))
79		iuneq2 4978	. . . . 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)))
80	78, 79	sylan9eq 2785	. . . 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)))
81		oelim 8501	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
82	67, 81	mpan2 691	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
83	21, 82	mpan 690	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
84	54, 83	mpan 690	. . . . . . 7 ⊢ (Lim 𝑥 → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
85	84	oveq2d 7406	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑥)) = ((𝐴 ↑o 𝐵) ·o ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)))
86	21, 61, 43	sylancr 587	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝐴 ↑o 𝑦) ∈ On)
87	86	ralrimiva 3126	. . . . . . 7 ⊢ (Lim 𝑥 → ∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
88		omlim 8500	. . . . . . . . . 10 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴 ↑o 𝐵) ·o 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑o 𝐵) ·o 𝑧))
89	24, 88	mpan 690	. . . . . . . . 9 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴 ↑o 𝐵) ·o 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑o 𝐵) ·o 𝑧))
90	66, 89	mpan 690	. . . . . . . 8 ⊢ (Lim 𝑤 → ((𝐴 ↑o 𝐵) ·o 𝑤) = ∪ 𝑧 ∈ 𝑤 ((𝐴 ↑o 𝐵) ·o 𝑧))
91		omwordi 8538	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On) → (𝑧 ⊆ 𝑤 → ((𝐴 ↑o 𝐵) ·o 𝑧) ⊆ ((𝐴 ↑o 𝐵) ·o 𝑤)))
92	24, 91	mp3an3 1452	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → ((𝐴 ↑o 𝐵) ·o 𝑧) ⊆ ((𝐴 ↑o 𝐵) ·o 𝑤)))
93	92	3impia 1117	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → ((𝐴 ↑o 𝐵) ·o 𝑧) ⊆ ((𝐴 ↑o 𝐵) ·o 𝑤))
94	90, 93	onoviun 8315	. . . . . . 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)))
95	54, 87, 65, 94	mp3an2i 1468	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ↑o 𝐵) ·o ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑦)))
96	85, 95	eqtrd 2765	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑥)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝑦)))
97	96	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)))
98	80, 97	eqtr4d 2768	. . 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)))
99	98	ex 412	. 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))))
100	5, 10, 15, 20, 33, 53, 99	tfinds 7839	1 ⊢ (𝐶 ∈ On → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  ∪ ciun 4958  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337  (class class class)co 7390  1_oc1o 8430   +_o coa 8434   ·_o comu 8435   ↑_o coe 8436

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-oexp 8443

This theorem is referenced by:  oeoa  8564