Theorem oeoa 7677

Description: Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.)

Assertion

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

Proof of Theorem oeoa

Step	Hyp	Ref	Expression
1		oa00 7639	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2	1	biimpar 502	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (𝐵 +𝑜 𝐶) = ∅)
3	2	oveq2d 6666	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 ∅))
4		oveq2 6658	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
5		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = (∅ ↑𝑜 ∅))
6		oe0m0 7600	. . . . . . . . . . 11 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
7	5, 6	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = 1𝑜)
8	4, 7	oveqan12d 6669	. . . . . . . . 9 ⊢ ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 ∅) ·𝑜 1𝑜))
9		0elon 5778	. . . . . . . . . . 11 ⊢ ∅ ∈ On
10		oecl 7617	. . . . . . . . . . 11 ⊢ ((∅ ∈ On ∧ ∅ ∈ On) → (∅ ↑𝑜 ∅) ∈ On)
11	9, 9, 10	mp2an 708	. . . . . . . . . 10 ⊢ (∅ ↑𝑜 ∅) ∈ On
12		om1 7622	. . . . . . . . . 10 ⊢ ((∅ ↑𝑜 ∅) ∈ On → ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅))
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅)
14	8, 13	syl6eq 2672	. . . . . . . 8 ⊢ ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
15	14	adantl 482	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
16	3, 15	eqtr4d 2659	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
17		oacl 7615	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +𝑜 𝐶) ∈ On)
18		on0eln0 5780	. . . . . . . . . 10 ⊢ ((𝐵 +𝑜 𝐶) ∈ On → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (𝐵 +𝑜 𝐶) ≠ ∅))
19	17, 18	syl 17	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (𝐵 +𝑜 𝐶) ≠ ∅))
20		oe0m1 7601	. . . . . . . . . 10 ⊢ ((𝐵 +𝑜 𝐶) ∈ On → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅))
21	17, 20	syl 17	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅))
22	1	necon3abid 2830	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) ≠ ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
23	19, 21, 22	3bitr3d 298	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
24	23	biimpar 502	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅)
25		on0eln0 5780	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
26	25	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
27		on0eln0 5780	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
28	27	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
29	26, 28	orbi12d 746	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅)))
30		neorian 2888	. . . . . . . . . 10 ⊢ ((𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅))
31	29, 30	syl6bb 276	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
32		oe0m1 7601	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
33	32	biimpa 501	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
34	33	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ·𝑜 (∅ ↑𝑜 𝐶)))
35		oecl 7617	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 𝐶) ∈ On)
36	9, 35	mpan 706	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → (∅ ↑𝑜 𝐶) ∈ On)
37		om0r 7619	. . . . . . . . . . . . . 14 ⊢ ((∅ ↑𝑜 𝐶) ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
39	34, 38	sylan9eq 2676	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
40	39	an32s 846	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
41		oe0m1 7601	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
42	41	biimpa 501	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
43	42	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
44		oecl 7617	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
45	9, 44	mpan 706	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
46		om0 7597	. . . . . . . . . . . . . 14 ⊢ ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
47	45, 46	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
48	43, 47	sylan9eqr 2678	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
49	48	anassrs 680	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
50	40, 49	jaodan 826	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
51	50	ex 450	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
52	31, 51	sylbird 250	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
53	52	imp 445	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
54	24, 53	eqtr4d 2659	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
55	16, 54	pm2.61dan 832	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
56		oveq1 6657	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 +𝑜 𝐶)))
57		oveq1 6657	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
58		oveq1 6657	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐶) = (∅ ↑𝑜 𝐶))
59	57, 58	oveq12d 6668	. . . . . 6 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
60	56, 59	eqeq12d 2637	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶))))
61	55, 60	syl5ibr 236	. . . 4 ⊢ (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶))))
62	61	impcom 446	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))
63		oveq1 6657	. . . . . . . 8 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)))
64		oveq1 6657	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ↑𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
65		oveq1 6657	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ↑𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))
66	64, 65	oveq12d 6668	. . . . . . . 8 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))
67	63, 66	eqeq12d 2637	. . . . . . 7 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))))
68	67	imbi2d 330	. . . . . 6 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐶 ∈ On → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶))) ↔ (𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))))
69		oveq1 6657	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (𝐵 +𝑜 𝐶) = (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶))
70	69	oveq2d 6666	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)))
71		oveq2 6658	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)))
72	71	oveq1d 6665	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))
73	70, 72	eqeq12d 2637	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))))
74	73	imbi2d 330	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → ((𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))) ↔ (𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))))
75		eleq1 2689	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
76		eleq2 2690	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
77	75, 76	anbi12d 747	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))))
78		eleq1 2689	. . . . . . . . . 10 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
79		eleq2 2690	. . . . . . . . . 10 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
80	78, 79	anbi12d 747	. . . . . . . . 9 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((1𝑜 ∈ On ∧ ∅ ∈ 1𝑜) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))))
81		1on 7567	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
82		0lt1o 7584	. . . . . . . . . 10 ⊢ ∅ ∈ 1𝑜
83	81, 82	pm3.2i 471	. . . . . . . . 9 ⊢ (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
84	77, 80, 83	elimhyp 4146	. . . . . . . 8 ⊢ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
85	84	simpli 474	. . . . . . 7 ⊢ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
86	84	simpri 478	. . . . . . 7 ⊢ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
87	81	elimel 4150	. . . . . . 7 ⊢ if(𝐵 ∈ On, 𝐵, 1𝑜) ∈ On
88	85, 86, 87	oeoalem 7676	. . . . . 6 ⊢ (𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))
89	68, 74, 88	dedth2h 4140	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶))))
90	89	impr 649	. . . 4 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))
91	90	an32s 846	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))
92	62, 91	oe0lem 7593	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴 ↑𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ↑𝑜 𝐵) ·𝑜 (𝐴 ↑𝑜 𝐶)))
93	92	3impb 1260	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 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  ifcif 4086  Oncon0 5723  (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:  oeoelem  7678  infxpenc  8841