Theorem oeoa 8635

Description: Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. Theorem 4.7 of [Schloeder] p. 14. (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 8597	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +o 𝐶) = ∅ ↔ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2	1	biimpar 477	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (𝐵 +o 𝐶) = ∅)
3	2	oveq2d 7447	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = (∅ ↑o ∅))
4		oveq2 7439	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
5		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (∅ ↑o 𝐶) = (∅ ↑o ∅))
6		oe0m0 8558	. . . . . . . . . . 11 ⊢ (∅ ↑o ∅) = 1o
7	5, 6	eqtrdi 2793	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → (∅ ↑o 𝐶) = 1o)
8	4, 7	oveqan12d 7450	. . . . . . . . 9 ⊢ ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ((∅ ↑o ∅) ·o 1o))
9		0elon 6438	. . . . . . . . . . 11 ⊢ ∅ ∈ On
10		oecl 8575	. . . . . . . . . . 11 ⊢ ((∅ ∈ On ∧ ∅ ∈ On) → (∅ ↑o ∅) ∈ On)
11	9, 9, 10	mp2an 692	. . . . . . . . . 10 ⊢ (∅ ↑o ∅) ∈ On
12		om1 8580	. . . . . . . . . 10 ⊢ ((∅ ↑o ∅) ∈ On → ((∅ ↑o ∅) ·o 1o) = (∅ ↑o ∅))
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ ((∅ ↑o ∅) ·o 1o) = (∅ ↑o ∅)
14	8, 13	eqtrdi 2793	. . . . . . . 8 ⊢ ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = (∅ ↑o ∅))
15	14	adantl 481	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = (∅ ↑o ∅))
16	3, 15	eqtr4d 2780	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
17		oacl 8573	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +o 𝐶) ∈ On)
18		on0eln0 6440	. . . . . . . . . 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 8559	. . . . . . . . . 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 2977	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +o 𝐶) ≠ ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
23	19, 21, 22	3bitr3d 309	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑o (𝐵 +o 𝐶)) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
24	23	biimpar 477	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = ∅)
25		on0eln0 6440	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
27		on0eln0 6440	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
28	27	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
29	26, 28	orbi12d 919	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅)))
30		neorian 3037	. . . . . . . . . 10 ⊢ ((𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅))
31	29, 30	bitrdi 287	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
32		oe0m1 8559	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
33	32	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
34	33	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = (∅ ·o (∅ ↑o 𝐶)))
35		oecl 8575	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ On ∧ 𝐶 ∈ On) → (∅ ↑o 𝐶) ∈ On)
36	9, 35	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → (∅ ↑o 𝐶) ∈ On)
37		om0r 8577	. . . . . . . . . . . . . 14 ⊢ ((∅ ↑o 𝐶) ∈ On → (∅ ·o (∅ ↑o 𝐶)) = ∅)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ ·o (∅ ↑o 𝐶)) = ∅)
39	34, 38	sylan9eq 2797	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐶 ∈ On) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
40	39	an32s 652	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
41		oe0m1 8559	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑o 𝐶) = ∅))
42	41	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑o 𝐶) = ∅)
43	42	oveq2d 7447	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ((∅ ↑o 𝐵) ·o ∅))
44		oecl 8575	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
45	9, 44	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
46		om0 8555	. . . . . . . . . . . . . 14 ⊢ ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
47	45, 46	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
48	43, 47	sylan9eqr 2799	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
49	48	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
50	40, 49	jaodan 960	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
51	50	ex 412	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅))
52	31, 51	sylbird 260	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅))
53	52	imp 406	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
54	24, 53	eqtr4d 2780	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
55	16, 54	pm2.61dan 813	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ↑o (𝐵 +o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
56		oveq1 7438	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑o (𝐵 +o 𝐶)) = (∅ ↑o (𝐵 +o 𝐶)))
57		oveq1 7438	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
58		oveq1 7438	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐶) = (∅ ↑o 𝐶))
59	57, 58	oveq12d 7449	. . . . . 6 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
60	56, 59	eqeq12d 2753	. . . . 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	imbitrrid 246	. . . 4 ⊢ (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶))))
62	61	impcom 407	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶)))
63		oveq1 7438	. . . . . . . 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 7438	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ↑o 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵))
65		oveq1 7438	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ↑o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐶))
66	64, 65	oveq12d 7449	. . . . . . . 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 2753	. . . . . . 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 340	. . . . . 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 7438	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, 1o) → (𝐵 +o 𝐶) = (if(𝐵 ∈ On, 𝐵, 1o) +o 𝐶))
70	69	oveq2d 7447	. . . . . . . 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 7439	. . . . . . . . 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 7446	. . . . . . . 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 2753	. . . . . . 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 340	. . . . . 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 2829	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
76		eleq2 2830	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
77	75, 76	anbi12d 632	. . . . . . . . 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 2829	. . . . . . . . . 10 ⊢ (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (1o ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
79		eleq2 2830	. . . . . . . . . 10 ⊢ (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 1o ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
80	78, 79	anbi12d 632	. . . . . . . . 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 8518	. . . . . . . . . 10 ⊢ 1o ∈ On
82		0lt1o 8542	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
83	81, 82	pm3.2i 470	. . . . . . . . 9 ⊢ (1o ∈ On ∧ ∅ ∈ 1o)
84	77, 80, 83	elimhyp 4591	. . . . . . . 8 ⊢ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o))
85	84	simpli 483	. . . . . . 7 ⊢ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On
86	84	simpri 485	. . . . . . 7 ⊢ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)
87	81	elimel 4595	. . . . . . 7 ⊢ if(𝐵 ∈ On, 𝐵, 1o) ∈ On
88	85, 86, 87	oeoalem 8634	. . . . . 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 4585	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶))))
90	89	impr 454	. . . 4 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶)))
91	90	an32s 652	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶)))
92	62, 91	oe0lem 8551	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶)))
93	92	3impb 1115	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 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∅c0 4333  ifcif 4525  Oncon0 6384  (class class class)co 7431  1_oc1o 8499   +_o coa 8503   ·_o comu 8504   ↑_o coe 8505

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512

This theorem is referenced by:  oeoelem  8636  infxpenc  10058  omabs2  43345