Theorem oeoe 8509

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 7349	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8430	. . . . . . . . . . . 12 ⊢ (∅ ↑o ∅) = 1o
3	1, 2	eqtrdi 2781	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = 1o)
4	3	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = (1o ↑o 𝐶))
5		oe1m 8455	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (1o ↑o 𝐶) = 1o)
6	4, 5	sylan9eqr 2787	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
7	6	adantll 714	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
8		oveq2 7349	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o ∅))
9		0elon 6357	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
10		oecl 8447	. . . . . . . . . . . 12 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
11	9, 10	mpan 690	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
12		oe0 8432	. . . . . . . . . . 11 ⊢ ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
14	8, 13	sylan9eqr 2787	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
15	14	adantlr 715	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
16	7, 15	jaodan 959	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
17		om00 8485	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·o 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
18	17	biimpar 477	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·o 𝐶) = ∅)
19	18	oveq2d 7357	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = (∅ ↑o ∅))
20	19, 2	eqtrdi 2781	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = 1o)
21	16, 20	eqtr4d 2768	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
22		on0eln0 6359	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
23		on0eln0 6359	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
24	22, 23	bi2anan9 638	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25		neanior 3019	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
26	24, 25	bitrdi 287	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27		oe0m1 8431	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
28	27	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
29	28	oveq1d 7356	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o 𝐶))
30		oe0m1 8431	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑o 𝐶) = ∅))
31	30	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑o 𝐶) = ∅)
32	29, 31	sylan9eq 2785	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
33	32	an4s 660	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
34		om00el 8486	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35		omcl 8446	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·o 𝐶) ∈ On)
36		oe0m1 8431	. . . . . . . . . . . . 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 476	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑o (𝐵 ·o 𝐶)) = ∅)
40	33, 39	eqtr4d 2768	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
41	40	ex 412	. . . . . . . 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 406	. . . . . 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 7348	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
46	45	oveq1d 7356	. . . . . 6 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o 𝐶))
47		oveq1 7348	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑o (𝐵 ·o 𝐶)) = (∅ ↑o (𝐵 ·o 𝐶)))
48	46, 47	eqeq12d 2746	. . . . 5 ⊢ (𝐴 = ∅ → (((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)) ↔ ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶))))
49	44, 48	imbitrrid 246	. . . 4 ⊢ (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
50	49	impcom 407	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
51		oveq1 7348	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ↑o 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵))
52	51	oveq1d 7356	. . . . . . . 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 7348	. . . . . . . 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 2746	. . . . . . 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 340	. . . . . 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 2817	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
57		eleq2 2818	. . . . . . . . . 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 2817	. . . . . . . . . 10 ⊢ (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (1o ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
60		eleq2 2818	. . . . . . . . . 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 8392	. . . . . . . . . 10 ⊢ 1o ∈ On
63		0lt1o 8414	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
64	62, 63	pm3.2i 470	. . . . . . . . 9 ⊢ (1o ∈ On ∧ ∅ ∈ 1o)
65	58, 61, 64	elimhyp 4539	. . . . . . . 8 ⊢ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o))
66	65	simpli 483	. . . . . . 7 ⊢ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On
67	65	simpri 485	. . . . . . 7 ⊢ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)
68	66, 67	oeoelem 8508	. . . . . 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 4532	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
70	69	imp 406	. . . 4 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
71	70	an32s 652	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
72	50, 71	oe0lem 8423	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
73	72	3impb 1114	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110   ≠ wne 2926  ∅c0 4281  ifcif 4473  Oncon0 6302  (class class class)co 7341  1_oc1o 8373   ·_o comu 8378   ↑_o coe 8379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  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 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-omul 8385  df-oexp 8386

This theorem is referenced by:  infxpenc  9901