Theorem oeoe 8595

Description: Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.)

Assertion

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

Proof of Theorem oeoe

Step	Hyp	Ref	Expression
1		oveq2 7412	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8515	. . . . . . . . . . . 12 ⊢ (∅ ↑o ∅) = 1o
3	1, 2	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = 1o)
4	3	oveq1d 7419	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = (1o ↑o 𝐶))
5		oe1m 8541	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (1o ↑o 𝐶) = 1o)
6	4, 5	sylan9eqr 2795	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
7	6	adantll 713	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
8		oveq2 7412	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o ∅))
9		0elon 6415	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
10		oecl 8532	. . . . . . . . . . . 12 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
11	9, 10	mpan 689	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
12		oe0 8517	. . . . . . . . . . 11 ⊢ ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
14	8, 13	sylan9eqr 2795	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
15	14	adantlr 714	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
16	7, 15	jaodan 957	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
17		om00 8571	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·o 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
18	17	biimpar 479	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·o 𝐶) = ∅)
19	18	oveq2d 7420	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = (∅ ↑o ∅))
20	19, 2	eqtrdi 2789	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = 1o)
21	16, 20	eqtr4d 2776	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
22		on0eln0 6417	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
23		on0eln0 6417	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
24	22, 23	bi2anan9 638	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25		neanior 3036	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
26	24, 25	bitrdi 287	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27		oe0m1 8516	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
28	27	biimpa 478	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
29	28	oveq1d 7419	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o 𝐶))
30		oe0m1 8516	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑o 𝐶) = ∅))
31	30	biimpa 478	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑o 𝐶) = ∅)
32	29, 31	sylan9eq 2793	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
33	32	an4s 659	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
34		om00el 8572	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35		omcl 8531	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·o 𝐶) ∈ On)
36		oe0m1 8516	. . . . . . . . . . . . 13 ⊢ ((𝐵 ·o 𝐶) ∈ On → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ↑o (𝐵 ·o 𝐶)) = ∅))
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ↑o (𝐵 ·o 𝐶)) = ∅))
38	34, 37	bitr3d 281	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (∅ ↑o (𝐵 ·o 𝐶)) = ∅))
39	38	biimpa 478	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑o (𝐵 ·o 𝐶)) = ∅)
40	33, 39	eqtr4d 2776	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
41	40	ex 414	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶))))
42	26, 41	sylbird 260	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∨ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶))))
43	42	imp 408	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
44	21, 43	pm2.61dan 812	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
45		oveq1 7411	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
46	45	oveq1d 7419	. . . . . 6 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o 𝐶))
47		oveq1 7411	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑o (𝐵 ·o 𝐶)) = (∅ ↑o (𝐵 ·o 𝐶)))
48	46, 47	eqeq12d 2749	. . . . 5 ⊢ (𝐴 = ∅ → (((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)) ↔ ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶))))
49	44, 48	imbitrrid 245	. . . 4 ⊢ (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
50	49	impcom 409	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
51		oveq1 7411	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ↑o 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵))
52	51	oveq1d 7419	. . . . . . . 8 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶))
53		oveq1 7411	. . . . . . . 8 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ↑o (𝐵 ·o 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶)))
54	52, 53	eqeq12d 2749	. . . . . . 7 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)) ↔ ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶))))
55	54	imbi2d 341	. . . . . 6 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))) ↔ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶)))))
56		eleq1 2822	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
57		eleq2 2823	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
58	56, 57	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))))
59		eleq1 2822	. . . . . . . . . 10 ⊢ (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (1o ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
60		eleq2 2823	. . . . . . . . . 10 ⊢ (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 1o ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
61	59, 60	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))))
62		1on 8473	. . . . . . . . . 10 ⊢ 1o ∈ On
63		0lt1o 8499	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
64	62, 63	pm3.2i 472	. . . . . . . . 9 ⊢ (1o ∈ On ∧ ∅ ∈ 1o)
65	58, 61, 64	elimhyp 4592	. . . . . . . 8 ⊢ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o))
66	65	simpli 485	. . . . . . 7 ⊢ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On
67	65	simpri 487	. . . . . . 7 ⊢ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)
68	66, 67	oeoelem 8594	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶)))
69	55, 68	dedth 4585	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
70	69	imp 408	. . . 4 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
71	70	an32s 651	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
72	50, 71	oe0lem 8508	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
73	72	3impb 1116	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∅c0 4321  ifcif 4527  Oncon0 6361  (class class class)co 7404  1_oc1o 8454   ·_o comu 8459   ↑_o coe 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-oexp 8467

This theorem is referenced by:  infxpenc  10009