Theorem oeoe 5274

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 e. On /\ B e. On /\ C e. On) -> ((A ^o B) ^o C) = (A ^o (B .o C)))

Proof of Theorem oeoe

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . . 7 |- (A = (/) -> (A ^o B) = ((/) ^o B))
2	1	opreq1d 4897	. . . . . 6 |- (A = (/) -> ((A ^o B) ^o C) = (((/) ^o B) ^o C))
3		opreq1 4889	. . . . . 6 |- (A = (/) -> (A ^o (B .o C)) = ((/) ^o (B .o C)))
4	2, 3	eqeq12d 1899	. . . . 5 |- (A = (/) -> (((A ^o B) ^o C) = (A ^o (B .o C)) <-> (((/) ^o B) ^o C) = ((/) ^o (B .o C))))
5		opreq2 4890	. . . . . . . . . . . 12 |- (B = (/) -> ((/) ^o B) = ((/) ^o (/)))
6		oe0m0 5204	. . . . . . . . . . . 12 |- ((/) ^o (/)) = 1o
7	5, 6	syl6eq 1944	. . . . . . . . . . 11 |- (B = (/) -> ((/) ^o B) = 1o)
8	7	opreq1d 4897	. . . . . . . . . 10 |- (B = (/) -> (((/) ^o B) ^o C) = (1o ^o C))
9		oe1m 5226	. . . . . . . . . 10 |- (C e. On -> (1o ^o C) = 1o)
10	8, 9	sylan9eqr 1951	. . . . . . . . 9 |- ((C e. On /\ B = (/)) -> (((/) ^o B) ^o C) = 1o)
11	10	adantll 428	. . . . . . . 8 |- (((B e. On /\ C e. On) /\ B = (/)) -> (((/) ^o B) ^o C) = 1o)
12		opreq2 4890	. . . . . . . . . 10 |- (C = (/) -> (((/) ^o B) ^o C) = (((/) ^o B) ^o (/)))
13		0elon 3716	. . . . . . . . . . . 12 |- (/) e. On
14		oecl 5218	. . . . . . . . . . . 12 |- (((/) e. On /\ B e. On) -> ((/) ^o B) e. On)
15	13, 14	mpan 759	. . . . . . . . . . 11 |- (B e. On -> ((/) ^o B) e. On)
16		oe0 5206	. . . . . . . . . . 11 |- (((/) ^o B) e. On -> (((/) ^o B) ^o (/)) = 1o)
17	15, 16	syl 12	. . . . . . . . . 10 |- (B e. On -> (((/) ^o B) ^o (/)) = 1o)
18	12, 17	sylan9eqr 1951	. . . . . . . . 9 |- ((B e. On /\ C = (/)) -> (((/) ^o B) ^o C) = 1o)
19	18	adantlr 429	. . . . . . . 8 |- (((B e. On /\ C e. On) /\ C = (/)) -> (((/) ^o B) ^o C) = 1o)
20	11, 19	jaodan 471	. . . . . . 7 |- (((B e. On /\ C e. On) /\ (B = (/) \/ C = (/))) -> (((/) ^o B) ^o C) = 1o)
21		om00 5254	. . . . . . . . . 10 |- ((B e. On /\ C e. On) -> ((B .o C) = (/) <-> (B = (/) \/ C = (/))))
22	21	biimpar 461	. . . . . . . . 9 |- (((B e. On /\ C e. On) /\ (B = (/) \/ C = (/))) -> (B .o C) = (/))
23	22	opreq2d 4898	. . . . . . . 8 |- (((B e. On /\ C e. On) /\ (B = (/) \/ C = (/))) -> ((/) ^o (B .o C)) = ((/) ^o (/)))
24	23, 6	syl6eq 1944	. . . . . . 7 |- (((B e. On /\ C e. On) /\ (B = (/) \/ C = (/))) -> ((/) ^o (B .o C)) = 1o)
25	20, 24	eqtr4d 1928	. . . . . 6 |- (((B e. On /\ C e. On) /\ (B = (/) \/ C = (/))) -> (((/) ^o B) ^o C) = ((/) ^o (B .o C)))
26		on0eln0 3718	. . . . . . . . . 10 |- (B e. On -> ((/) e. B <-> B =/= (/)))
27		on0eln0 3718	. . . . . . . . . 10 |- (C e. On -> ((/) e. C <-> C =/= (/)))
28	26, 27	bi2anan9 694	. . . . . . . . 9 |- ((B e. On /\ C e. On) -> (((/) e. B /\ (/) e. C) <-> (B =/= (/) /\ C =/= (/))))
29		neanior 2097	. . . . . . . . 9 |- ((B =/= (/) /\ C =/= (/)) <-> -. (B = (/) \/ C = (/)))
30	28, 29	syl6bb 595	. . . . . . . 8 |- ((B e. On /\ C e. On) -> (((/) e. B /\ (/) e. C) <-> -. (B = (/) \/ C = (/))))
31		oe0m1 5205	. . . . . . . . . . . . . 14 |- (B e. On -> ((/) e. B <-> ((/) ^o B) = (/)))
32	31	biimpa 460	. . . . . . . . . . . . 13 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
33	32	opreq1d 4897	. . . . . . . . . . . 12 |- ((B e. On /\ (/) e. B) -> (((/) ^o B) ^o C) = ((/) ^o C))
34		oe0m1 5205	. . . . . . . . . . . . 13 |- (C e. On -> ((/) e. C <-> ((/) ^o C) = (/)))
35	34	biimpa 460	. . . . . . . . . . . 12 |- ((C e. On /\ (/) e. C) -> ((/) ^o C) = (/))
36	33, 35	sylan9eq 1948	. . . . . . . . . . 11 |- (((B e. On /\ (/) e. B) /\ (C e. On /\ (/) e. C)) -> (((/) ^o B) ^o C) = (/))
37	36	an4s 566	. . . . . . . . . 10 |- (((B e. On /\ C e. On) /\ ((/) e. B /\ (/) e. C)) -> (((/) ^o B) ^o C) = (/))
38		om00el 5255	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> ((/) e. (B .o C) <-> ((/) e. B /\ (/) e. C)))
39		omcl 5216	. . . . . . . . . . . . 13 |- ((B e. On /\ C e. On) -> (B .o C) e. On)
40		oe0m1 5205	. . . . . . . . . . . . 13 |- ((B .o C) e. On -> ((/) e. (B .o C) <-> ((/) ^o (B .o C)) = (/)))
41	39, 40	syl 12	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> ((/) e. (B .o C) <-> ((/) ^o (B .o C)) = (/)))
42	38, 41	bitr3d 589	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (((/) e. B /\ (/) e. C) <-> ((/) ^o (B .o C)) = (/)))
43	42	biimpa 460	. . . . . . . . . 10 |- (((B e. On /\ C e. On) /\ ((/) e. B /\ (/) e. C)) -> ((/) ^o (B .o C)) = (/))
44	37, 43	eqtr4d 1928	. . . . . . . . 9 |- (((B e. On /\ C e. On) /\ ((/) e. B /\ (/) e. C)) -> (((/) ^o B) ^o C) = ((/) ^o (B .o C)))
45	44	ex 402	. . . . . . . 8 |- ((B e. On /\ C e. On) -> (((/) e. B /\ (/) e. C) -> (((/) ^o B) ^o C) = ((/) ^o (B .o C))))
46	30, 45	sylbird 222	. . . . . . 7 |- ((B e. On /\ C e. On) -> (-. (B = (/) \/ C = (/)) -> (((/) ^o B) ^o C) = ((/) ^o (B .o C))))
47	46	imp 377	. . . . . 6 |- (((B e. On /\ C e. On) /\ -. (B = (/) \/ C = (/))) -> (((/) ^o B) ^o C) = ((/) ^o (B .o C)))
48	25, 47	pm2.61dan 535	. . . . 5 |- ((B e. On /\ C e. On) -> (((/) ^o B) ^o C) = ((/) ^o (B .o C)))
49	4, 48	syl5bir 227	. . . 4 |- (A = (/) -> ((B e. On /\ C e. On) -> ((A ^o B) ^o C) = (A ^o (B .o C))))
50	49	impcom 378	. . 3 |- (((B e. On /\ C e. On) /\ A = (/)) -> ((A ^o B) ^o C) = (A ^o (B .o C)))
51		opreq1 4889	. . . . . . . . 9 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> (A ^o B) = (if((A e. On /\ (/) e. A), A, 1o) ^o B))
52	51	opreq1d 4897	. . . . . . . 8 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> ((A ^o B) ^o C) = ((if((A e. On /\ (/) e. A), A, 1o) ^o B) ^o C))
53		opreq1 4889	. . . . . . . 8 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> (A ^o (B .o C)) = (if((A e. On /\ (/) e. A), A, 1o) ^o (B .o C)))
54	52, 53	eqeq12d 1899	. . . . . . 7 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> (((A ^o B) ^o C) = (A ^o (B .o C)) <-> ((if((A e. On /\ (/) e. A), A, 1o) ^o B) ^o C) = (if((A e. On /\ (/) e. A), A, 1o) ^o (B .o C))))
55	54	imbi2d 674	. . . . . 6 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> (((B e. On /\ C e. On) -> ((A ^o B) ^o C) = (A ^o (B .o C))) <-> ((B e. On /\ C e. On) -> ((if((A e. On /\ (/) e. A), A, 1o) ^o B) ^o C) = (if((A e. On /\ (/) e. A), A, 1o) ^o (B .o C)))))
56		eleq1 1957	. . . . . . . . . 10 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> (A e. On <-> if((A e. On /\ (/) e. A), A, 1o) e. On))
57		eleq2 1958	. . . . . . . . . 10 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> ((/) e. A <-> (/) e. if((A e. On /\ (/) e. A), A, 1o)))
58	56, 57	anbi12d 690	. . . . . . . . 9 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> ((A e. On /\ (/) e. A) <-> (if((A e. On /\ (/) e. A), A, 1o) e. On /\ (/) e. if((A e. On /\ (/) e. A), A, 1o))))
59		eleq1 1957	. . . . . . . . . 10 |- (1o = if((A e. On /\ (/) e. A), A, 1o) -> (1o e. On <-> if((A e. On /\ (/) e. A), A, 1o) e. On))
60		eleq2 1958	. . . . . . . . . 10 |- (1o = if((A e. On /\ (/) e. A), A, 1o) -> ((/) e. 1o <-> (/) e. if((A e. On /\ (/) e. A), A, 1o)))
61	59, 60	anbi12d 690	. . . . . . . . 9 |- (1o = if((A e. On /\ (/) e. A), A, 1o) -> ((1o e. On /\ (/) e. 1o) <-> (if((A e. On /\ (/) e. A), A, 1o) e. On /\ (/) e. if((A e. On /\ (/) e. A), A, 1o))))
62		1on 5182	. . . . . . . . . 10 |- 1o e. On
63		0lt1o 5192	. . . . . . . . . 10 |- (/) e. 1o
64	62, 63	pm3.2i 307	. . . . . . . . 9 |- (1o e. On /\ (/) e. 1o)
65	58, 61, 64	elimhyp 3021	. . . . . . . 8 |- (if((A e. On /\ (/) e. A), A, 1o) e. On /\ (/) e. if((A e. On /\ (/) e. A), A, 1o))
66	65	simpli 347	. . . . . . 7 |- if((A e. On /\ (/) e. A), A, 1o) e. On
67	65	simpri 351	. . . . . . 7 |- (/) e. if((A e. On /\ (/) e. A), A, 1o)
68	66, 67	oeoelem 5273	. . . . . 6 |- ((B e. On /\ C e. On) -> ((if((A e. On /\ (/) e. A), A, 1o) ^o B) ^o C) = (if((A e. On /\ (/) e. A), A, 1o) ^o (B .o C)))
69	55, 68	dedth 3011	. . . . 5 |- ((A e. On /\ (/) e. A) -> ((B e. On /\ C e. On) -> ((A ^o B) ^o C) = (A ^o (B .o C))))
70	69	imp 377	. . . 4 |- (((A e. On /\ (/) e. A) /\ (B e. On /\ C e. On)) -> ((A ^o B) ^o C) = (A ^o (B .o C)))
71	70	an1rs 547	. . 3 |- (((A e. On /\ (B e. On /\ C e. On)) /\ (/) e. A) -> ((A ^o B) ^o C) = (A ^o (B .o C)))
72	50, 71	oe0lem 5197	. 2 |- ((A e. On /\ (B e. On /\ C e. On)) -> ((A ^o B) ^o C) = (A ^o (B .o C)))
73	72	3impb 1063	1 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o B) ^o C) = (A ^o (B .o C)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  (/)c0 2875  ifcif 2982  Oncon0 3657  (class class class)co 4884  1oc1o 5172   .o comu 5175   ^o coe 5176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-oexp 5181

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