Theorem oeoe 8526

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 7366	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8446	. . . . . . . . . . . 12 ⊢ (∅ ↑o ∅) = 1o
3	1, 2	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = 1o)
4	3	oveq1d 7373	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = (1o ↑o 𝐶))
5		oe1m 8471	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (1o ↑o 𝐶) = 1o)
6	4, 5	sylan9eqr 2794	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
7	6	adantll 715	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
8		oveq2 7366	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o ∅))
9		0elon 6370	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
10		oecl 8463	. . . . . . . . . . . 12 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
11	9, 10	mpan 691	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
12		oe0 8448	. . . . . . . . . . 11 ⊢ ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
14	8, 13	sylan9eqr 2794	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
15	14	adantlr 716	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
16	7, 15	jaodan 960	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
17		om00 8501	. . . . . . . . . 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 7374	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = (∅ ↑o ∅))
20	19, 2	eqtrdi 2788	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = 1o)
21	16, 20	eqtr4d 2775	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
22		on0eln0 6372	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
23		on0eln0 6372	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
24	22, 23	bi2anan9 639	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25		neanior 3026	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
26	24, 25	bitrdi 287	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27		oe0m1 8447	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
28	27	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
29	28	oveq1d 7373	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o 𝐶))
30		oe0m1 8447	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑o 𝐶) = ∅))
31	30	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑o 𝐶) = ∅)
32	29, 31	sylan9eq 2792	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
33	32	an4s 661	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
34		om00el 8502	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35		omcl 8462	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·o 𝐶) ∈ On)
36		oe0m1 8447	. . . . . . . . . . . . 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 2775	. . . . . . . . 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 813	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
45		oveq1 7365	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
46	45	oveq1d 7373	. . . . . 6 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o 𝐶))
47		oveq1 7365	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑o (𝐵 ·o 𝐶)) = (∅ ↑o (𝐵 ·o 𝐶)))
48	46, 47	eqeq12d 2753	. . . . 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 7365	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ↑o 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵))
52	51	oveq1d 7373	. . . . . . . 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 7365	. . . . . . . 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 2753	. . . . . . 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 2825	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
57		eleq2 2826	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
58	56, 57	anbi12d 633	. . . . . . . . 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 2825	. . . . . . . . . 10 ⊢ (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (1o ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
60		eleq2 2826	. . . . . . . . . 10 ⊢ (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 1o ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
61	59, 60	anbi12d 633	. . . . . . . . 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 8408	. . . . . . . . . 10 ⊢ 1o ∈ On
63		0lt1o 8430	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
64	62, 63	pm3.2i 470	. . . . . . . . 9 ⊢ (1o ∈ On ∧ ∅ ∈ 1o)
65	58, 61, 64	elimhyp 4533	. . . . . . . 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 8525	. . . . . 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 4526	. . . . 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 653	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
72	50, 71	oe0lem 8439	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))
73	72	3impb 1115	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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  ifcif 4467  Oncon0 6315  (class class class)co 7358  1_oc1o 8389   ·_o comu 8394   ↑_o coe 8395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-omul 8401  df-oexp 8402

This theorem is referenced by:  infxpenc  9929