Theorem oeoelem 7565

Description: Lemma for oeoe 7566. (Contributed by Eric Schmidt, 26-May-2009.)

Hypotheses

Ref	Expression
oeoelem.1	⊢ 𝐴 ∈ On
oeoelem.2	⊢ ∅ ∈ 𝐴

Assertion

Ref	Expression
oeoelem	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))

Proof of Theorem oeoelem

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

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴 ↑𝑜 𝐵) ↑𝑜 ∅))
2		oveq2 6557	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
3	2	oveq2d 6565	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 ·𝑜 ∅)))
4	1, 3	eqeq12d 2625	. . 3 ⊢ (𝑥 = ∅ → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ↑𝑜 𝐵) ↑𝑜 ∅) = (𝐴 ↑𝑜 (𝐵 ·𝑜 ∅))))
5		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦))
6		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
7	6	oveq2d 6565	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)))
8	5, 7	eqeq12d 2625	. . 3 ⊢ (𝑥 = 𝑦 → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦))))
9		oveq2 6557	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴 ↑𝑜 𝐵) ↑𝑜 suc 𝑦))
10		oveq2 6557	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
11	10	oveq2d 6565	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 ·𝑜 suc 𝑦)))
12	9, 11	eqeq12d 2625	. . 3 ⊢ (𝑥 = suc 𝑦 → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ↑𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 suc 𝑦))))
13		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶))
14		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
15	14	oveq2d 6565	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
16	13, 15	eqeq12d 2625	. . 3 ⊢ (𝑥 = 𝐶 → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶))))
17		oeoelem.1	. . . . . 6 ⊢ 𝐴 ∈ On
18		oecl 7504	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
19	17, 18	mpan 702	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑𝑜 𝐵) ∈ On)
20		oe0 7489	. . . . 5 ⊢ ((𝐴 ↑𝑜 𝐵) ∈ On → ((𝐴 ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
21	19, 20	syl 17	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
22		om0 7484	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐵 ·𝑜 ∅) = ∅)
23	22	oveq2d 6565	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑𝑜 (𝐵 ·𝑜 ∅)) = (𝐴 ↑𝑜 ∅))
24		oe0 7489	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
25	17, 24	ax-mp 5	. . . . 5 ⊢ (𝐴 ↑𝑜 ∅) = 1𝑜
26	23, 25	syl6eq 2660	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ↑𝑜 (𝐵 ·𝑜 ∅)) = 1𝑜)
27	21, 26	eqtr4d 2647	. . 3 ⊢ (𝐵 ∈ On → ((𝐴 ↑𝑜 𝐵) ↑𝑜 ∅) = (𝐴 ↑𝑜 (𝐵 ·𝑜 ∅)))
28		oveq1 6556	. . . . 5 ⊢ (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴 ↑𝑜 𝐵)) = ((𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴 ↑𝑜 𝐵)))
29		oesuc 7494	. . . . . . 7 ⊢ (((𝐴 ↑𝑜 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 suc 𝑦) = (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴 ↑𝑜 𝐵)))
30	19, 29	sylan 487	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 suc 𝑦) = (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴 ↑𝑜 𝐵)))
31		omsuc 7493	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
32	31	oveq2d 6565	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 (𝐵 ·𝑜 suc 𝑦)) = (𝐴 ↑𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
33		omcl 7503	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
34		oeoa 7564	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴 ↑𝑜 𝐵)))
35	17, 34	mp3an1 1403	. . . . . . . . 9 ⊢ (((𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴 ↑𝑜 𝐵)))
36	33, 35	sylan 487	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴 ↑𝑜 𝐵)))
37	36	anabss1 851	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴 ↑𝑜 𝐵)))
38	32, 37	eqtrd 2644	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 (𝐵 ·𝑜 suc 𝑦)) = ((𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴 ↑𝑜 𝐵)))
39	30, 38	eqeq12d 2625	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ↑𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 suc 𝑦)) ↔ (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) ·𝑜 (𝐴 ↑𝑜 𝐵)) = ((𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) ·𝑜 (𝐴 ↑𝑜 𝐵))))
40	28, 39	syl5ibr 235	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 suc 𝑦))))
41	40	expcom 450	. . 3 ⊢ (𝑦 ∈ On → (𝐵 ∈ On → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 suc 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 suc 𝑦)))))
42		iuneq2 4473	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) → ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)))
43		vex 3176	. . . . . . 7 ⊢ 𝑥 ∈ V
44		oeoelem.2	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝐴
45		oen0 7553	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐵))
46	44, 45	mpan2 703	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴 ↑𝑜 𝐵))
47		oelim 7501	. . . . . . . . . . 11 ⊢ ((((𝐴 ↑𝑜 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴 ↑𝑜 𝐵)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦))
48	18, 47	sylanl1 680	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴 ↑𝑜 𝐵)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦))
49	46, 48	sylan2 490	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦))
50	49	anabss1 851	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦))
51	17, 50	mpanl1 712	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦))
52	43, 51	mpanr1 715	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦))
53		omlim 7500	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦))
54	43, 53	mpanr1 715	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦))
55	54	oveq2d 6565	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ↑𝑜 ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦)))
56	43	a1i 11	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ∈ V)
57		limord 5701	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
58		ordelon 5664	. . . . . . . . . . . 12 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
59	57, 58	sylan 487	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
60	59, 33	sylan2 490	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ (Lim 𝑥 ∧ 𝑦 ∈ 𝑥)) → (𝐵 ·𝑜 𝑦) ∈ On)
61	60	anassrs 678	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝐵 ·𝑜 𝑦) ∈ On)
62	61	ralrimiva 2949	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦) ∈ On)
63		0ellim 5704	. . . . . . . . . 10 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
64		ne0i 3880	. . . . . . . . . 10 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
65	63, 64	syl 17	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
66	65	adantl 481	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
67		vex 3176	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
68		oelim 7501	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
69	44, 68	mpan2 703	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
70	17, 69	mpan 702	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
71	67, 70	mpan 702	. . . . . . . . 9 ⊢ (Lim 𝑤 → (𝐴 ↑𝑜 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑𝑜 𝑧))
72		oewordi 7558	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
73	44, 72	mpan2 703	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
74	17, 73	mp3an3 1405	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤)))
75	74	3impia 1253	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑𝑜 𝑧) ⊆ (𝐴 ↑𝑜 𝑤))
76	71, 75	onoviun 7327	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴 ↑𝑜 ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)))
77	56, 62, 66, 76	syl3anc 1318	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑𝑜 ∪ 𝑦 ∈ 𝑥 (𝐵 ·𝑜 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)))
78	55, 77	eqtrd 2644	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)))
79	52, 78	eqeq12d 2625	. . . . 5 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)) ↔ ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦))))
80	42, 79	syl5ibr 235	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥))))
81	80	expcom 450	. . 3 ⊢ (Lim 𝑥 → (𝐵 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑦) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝑥) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝑥)))))
82	4, 8, 12, 16, 27, 41, 81	tfinds3 6956	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶))))
83	82	impcom 445	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·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:  oeoe  7566