Theorem oeoa 5272

Description: Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.)

Assertion

Ref	Expression
oeoa	|- ((A e. On /\ B e. On /\ C e. On) -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C)))

Proof of Theorem oeoa

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . 6 |- (A = (/) -> (A ^o (B +o C)) = ((/) ^o (B +o C)))
2		opreq1 4889	. . . . . . 7 |- (A = (/) -> (A ^o B) = ((/) ^o B))
3		opreq1 4889	. . . . . . 7 |- (A = (/) -> (A ^o C) = ((/) ^o C))
4	2, 3	opreq12d 4900	. . . . . 6 |- (A = (/) -> ((A ^o B) .o (A ^o C)) = (((/) ^o B) .o ((/) ^o C)))
5	1, 4	eqeq12d 1899	. . . . 5 |- (A = (/) -> ((A ^o (B +o C)) = ((A ^o B) .o (A ^o C)) <-> ((/) ^o (B +o C)) = (((/) ^o B) .o ((/) ^o C))))
6		oa00 5241	. . . . . . . . 9 |- ((B e. On /\ C e. On) -> ((B +o C) = (/) <-> (B = (/) /\ C = (/))))
7	6	biimpar 461	. . . . . . . 8 |- (((B e. On /\ C e. On) /\ (B = (/) /\ C = (/))) -> (B +o C) = (/))
8	7	opreq2d 4898	. . . . . . 7 |- (((B e. On /\ C e. On) /\ (B = (/) /\ C = (/))) -> ((/) ^o (B +o C)) = ((/) ^o (/)))
9		opreq2 4890	. . . . . . . . . 10 |- (B = (/) -> ((/) ^o B) = ((/) ^o (/)))
10		opreq2 4890	. . . . . . . . . . 11 |- (C = (/) -> ((/) ^o C) = ((/) ^o (/)))
11		oe0m0 5204	. . . . . . . . . . 11 |- ((/) ^o (/)) = 1o
12	10, 11	syl6eq 1944	. . . . . . . . . 10 |- (C = (/) -> ((/) ^o C) = 1o)
13	9, 12	opreqan12d 4902	. . . . . . . . 9 |- ((B = (/) /\ C = (/)) -> (((/) ^o B) .o ((/) ^o C)) = (((/) ^o (/)) .o 1o))
14		0elon 3716	. . . . . . . . . . 11 |- (/) e. On
15		oecl 5218	. . . . . . . . . . 11 |- (((/) e. On /\ (/) e. On) -> ((/) ^o (/)) e. On)
16	14, 14, 15	mp2an 761	. . . . . . . . . 10 |- ((/) ^o (/)) e. On
17		om1 5223	. . . . . . . . . 10 |- (((/) ^o (/)) e. On -> (((/) ^o (/)) .o 1o) = ((/) ^o (/)))
18	16, 17	ax-mp 7	. . . . . . . . 9 |- (((/) ^o (/)) .o 1o) = ((/) ^o (/))
19	13, 18	syl6eq 1944	. . . . . . . 8 |- ((B = (/) /\ C = (/)) -> (((/) ^o B) .o ((/) ^o C)) = ((/) ^o (/)))
20	19	adantl 424	. . . . . . 7 |- (((B e. On /\ C e. On) /\ (B = (/) /\ C = (/))) -> (((/) ^o B) .o ((/) ^o C)) = ((/) ^o (/)))
21	8, 20	eqtr4d 1928	. . . . . 6 |- (((B e. On /\ C e. On) /\ (B = (/) /\ C = (/))) -> ((/) ^o (B +o C)) = (((/) ^o B) .o ((/) ^o C)))
22		oacl 5215	. . . . . . . . . 10 |- ((B e. On /\ C e. On) -> (B +o C) e. On)
23		on0eln0 3718	. . . . . . . . . 10 |- ((B +o C) e. On -> ((/) e. (B +o C) <-> (B +o C) =/= (/)))
24	22, 23	syl 12	. . . . . . . . 9 |- ((B e. On /\ C e. On) -> ((/) e. (B +o C) <-> (B +o C) =/= (/)))
25		oe0m1 5205	. . . . . . . . . 10 |- ((B +o C) e. On -> ((/) e. (B +o C) <-> ((/) ^o (B +o C)) = (/)))
26	22, 25	syl 12	. . . . . . . . 9 |- ((B e. On /\ C e. On) -> ((/) e. (B +o C) <-> ((/) ^o (B +o C)) = (/)))
27	6	necon3abid 2033	. . . . . . . . 9 |- ((B e. On /\ C e. On) -> ((B +o C) =/= (/) <-> -. (B = (/) /\ C = (/))))
28	24, 26, 27	3bitr3d 607	. . . . . . . 8 |- ((B e. On /\ C e. On) -> (((/) ^o (B +o C)) = (/) <-> -. (B = (/) /\ C = (/))))
29	28	biimpar 461	. . . . . . 7 |- (((B e. On /\ C e. On) /\ -. (B = (/) /\ C = (/))) -> ((/) ^o (B +o C)) = (/))
30		on0eln0 3718	. . . . . . . . . . . 12 |- (B e. On -> ((/) e. B <-> B =/= (/)))
31	30	adantr 425	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> ((/) e. B <-> B =/= (/)))
32		on0eln0 3718	. . . . . . . . . . . 12 |- (C e. On -> ((/) e. C <-> C =/= (/)))
33	32	adantl 424	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> ((/) e. C <-> C =/= (/)))
34	31, 33	orbi12d 689	. . . . . . . . . 10 |- ((B e. On /\ C e. On) -> (((/) e. B \/ (/) e. C) <-> (B =/= (/) \/ C =/= (/))))
35		neorian 2098	. . . . . . . . . 10 |- ((B =/= (/) \/ C =/= (/)) <-> -. (B = (/) /\ C = (/)))
36	34, 35	syl6bb 595	. . . . . . . . 9 |- ((B e. On /\ C e. On) -> (((/) e. B \/ (/) e. C) <-> -. (B = (/) /\ C = (/))))
37		oe0m1 5205	. . . . . . . . . . . . . . 15 |- (B e. On -> ((/) e. B <-> ((/) ^o B) = (/)))
38	37	biimpa 460	. . . . . . . . . . . . . 14 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
39	38	opreq1d 4897	. . . . . . . . . . . . 13 |- ((B e. On /\ (/) e. B) -> (((/) ^o B) .o ((/) ^o C)) = ((/) .o ((/) ^o C)))
40		oecl 5218	. . . . . . . . . . . . . . 15 |- (((/) e. On /\ C e. On) -> ((/) ^o C) e. On)
41	14, 40	mpan 759	. . . . . . . . . . . . . 14 |- (C e. On -> ((/) ^o C) e. On)
42		om0r 5221	. . . . . . . . . . . . . 14 |- (((/) ^o C) e. On -> ((/) .o ((/) ^o C)) = (/))
43	41, 42	syl 12	. . . . . . . . . . . . 13 |- (C e. On -> ((/) .o ((/) ^o C)) = (/))
44	39, 43	sylan9eq 1948	. . . . . . . . . . . 12 |- (((B e. On /\ (/) e. B) /\ C e. On) -> (((/) ^o B) .o ((/) ^o C)) = (/))
45	44	an1rs 547	. . . . . . . . . . 11 |- (((B e. On /\ C e. On) /\ (/) e. B) -> (((/) ^o B) .o ((/) ^o C)) = (/))
46		oe0m1 5205	. . . . . . . . . . . . . . 15 |- (C e. On -> ((/) e. C <-> ((/) ^o C) = (/)))
47	46	biimpa 460	. . . . . . . . . . . . . 14 |- ((C e. On /\ (/) e. C) -> ((/) ^o C) = (/))
48	47	opreq2d 4898	. . . . . . . . . . . . 13 |- ((C e. On /\ (/) e. C) -> (((/) ^o B) .o ((/) ^o C)) = (((/) ^o B) .o (/)))
49		oecl 5218	. . . . . . . . . . . . . . 15 |- (((/) e. On /\ B e. On) -> ((/) ^o B) e. On)
50	14, 49	mpan 759	. . . . . . . . . . . . . 14 |- (B e. On -> ((/) ^o B) e. On)
51		om0 5201	. . . . . . . . . . . . . 14 |- (((/) ^o B) e. On -> (((/) ^o B) .o (/)) = (/))
52	50, 51	syl 12	. . . . . . . . . . . . 13 |- (B e. On -> (((/) ^o B) .o (/)) = (/))
53	48, 52	sylan9eqr 1951	. . . . . . . . . . . 12 |- ((B e. On /\ (C e. On /\ (/) e. C)) -> (((/) ^o B) .o ((/) ^o C)) = (/))
54	53	anassrs 489	. . . . . . . . . . 11 |- (((B e. On /\ C e. On) /\ (/) e. C) -> (((/) ^o B) .o ((/) ^o C)) = (/))
55	45, 54	jaodan 471	. . . . . . . . . 10 |- (((B e. On /\ C e. On) /\ ((/) e. B \/ (/) e. C)) -> (((/) ^o B) .o ((/) ^o C)) = (/))
56	55	ex 402	. . . . . . . . 9 |- ((B e. On /\ C e. On) -> (((/) e. B \/ (/) e. C) -> (((/) ^o B) .o ((/) ^o C)) = (/)))
57	36, 56	sylbird 222	. . . . . . . 8 |- ((B e. On /\ C e. On) -> (-. (B = (/) /\ C = (/)) -> (((/) ^o B) .o ((/) ^o C)) = (/)))
58	57	imp 377	. . . . . . 7 |- (((B e. On /\ C e. On) /\ -. (B = (/) /\ C = (/))) -> (((/) ^o B) .o ((/) ^o C)) = (/))
59	29, 58	eqtr4d 1928	. . . . . 6 |- (((B e. On /\ C e. On) /\ -. (B = (/) /\ C = (/))) -> ((/) ^o (B +o C)) = (((/) ^o B) .o ((/) ^o C)))
60	21, 59	pm2.61dan 535	. . . . 5 |- ((B e. On /\ C e. On) -> ((/) ^o (B +o C)) = (((/) ^o B) .o ((/) ^o C)))
61	5, 60	syl5bir 227	. . . 4 |- (A = (/) -> ((B e. On /\ C e. On) -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C))))
62	61	impcom 378	. . 3 |- (((B e. On /\ C e. On) /\ A = (/)) -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C)))
63		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)))
64		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))
65		opreq1 4889	. . . . . . . . 9 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> (A ^o C) = (if((A e. On /\ (/) e. A), A, 1o) ^o C))
66	64, 65	opreq12d 4900	. . . . . . . 8 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> ((A ^o B) .o (A ^o C)) = ((if((A e. On /\ (/) e. A), A, 1o) ^o B) .o (if((A e. On /\ (/) e. A), A, 1o) ^o C)))
67	63, 66	eqeq12d 1899	. . . . . . 7 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> ((A ^o (B +o C)) = ((A ^o B) .o (A ^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 (if((A e. On /\ (/) e. A), A, 1o) ^o C))))
68	67	imbi2d 674	. . . . . 6 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> ((C e. On -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C))) <-> (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 (if((A e. On /\ (/) e. A), A, 1o) ^o C)))))
69		opreq1 4889	. . . . . . . . 9 |- (B = if(B e. On, B, 1o) -> (B +o C) = (if(B e. On, B, 1o) +o C))
70	69	opreq2d 4898	. . . . . . . 8 |- (B = if(B e. On, B, 1o) -> (if((A e. On /\ (/) e. A), A, 1o) ^o (B +o C)) = (if((A e. On /\ (/) e. A), A, 1o) ^o (if(B e. On, B, 1o) +o C)))
71		opreq2 4890	. . . . . . . . 9 |- (B = if(B e. On, B, 1o) -> (if((A e. On /\ (/) e. A), A, 1o) ^o B) = (if((A e. On /\ (/) e. A), A, 1o) ^o if(B e. On, B, 1o)))
72	71	opreq1d 4897	. . . . . . . 8 |- (B = if(B e. On, B, 1o) -> ((if((A e. On /\ (/) e. A), A, 1o) ^o B) .o (if((A e. On /\ (/) e. A), A, 1o) ^o C)) = ((if((A e. On /\ (/) e. A), A, 1o) ^o if(B e. On, B, 1o)) .o (if((A e. On /\ (/) e. A), A, 1o) ^o C)))
73	70, 72	eqeq12d 1899	. . . . . . 7 |- (B = if(B e. On, B, 1o) -> ((if((A e. On /\ (/) e. A), A, 1o) ^o (B +o C)) = ((if((A e. On /\ (/) e. A), A, 1o) ^o B) .o (if((A e. On /\ (/) e. A), A, 1o) ^o C)) <-> (if((A e. On /\ (/) e. A), A, 1o) ^o (if(B e. On, B, 1o) +o C)) = ((if((A e. On /\ (/) e. A), A, 1o) ^o if(B e. On, B, 1o)) .o (if((A e. On /\ (/) e. A), A, 1o) ^o C))))
74	73	imbi2d 674	. . . . . 6 |- (B = if(B e. On, B, 1o) -> ((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 (if((A e. On /\ (/) e. A), A, 1o) ^o C))) <-> (C e. On -> (if((A e. On /\ (/) e. A), A, 1o) ^o (if(B e. On, B, 1o) +o C)) = ((if((A e. On /\ (/) e. A), A, 1o) ^o if(B e. On, B, 1o)) .o (if((A e. On /\ (/) e. A), A, 1o) ^o C)))))
75		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))
76		eleq2 1958	. . . . . . . . . 10 |- (A = if((A e. On /\ (/) e. A), A, 1o) -> ((/) e. A <-> (/) e. if((A e. On /\ (/) e. A), A, 1o)))
77	75, 76	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))))
78		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))
79		eleq2 1958	. . . . . . . . . 10 |- (1o = if((A e. On /\ (/) e. A), A, 1o) -> ((/) e. 1o <-> (/) e. if((A e. On /\ (/) e. A), A, 1o)))
80	78, 79	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))))
81		1on 5182	. . . . . . . . . 10 |- 1o e. On
82		0lt1o 5192	. . . . . . . . . 10 |- (/) e. 1o
83	81, 82	pm3.2i 307	. . . . . . . . 9 |- (1o e. On /\ (/) e. 1o)
84	77, 80, 83	elimhyp 3021	. . . . . . . 8 |- (if((A e. On /\ (/) e. A), A, 1o) e. On /\ (/) e. if((A e. On /\ (/) e. A), A, 1o))
85	84	simpli 347	. . . . . . 7 |- if((A e. On /\ (/) e. A), A, 1o) e. On
86	84	simpri 351	. . . . . . 7 |- (/) e. if((A e. On /\ (/) e. A), A, 1o)
87	81	elimel 3025	. . . . . . 7 |- if(B e. On, B, 1o) e. On
88	85, 86, 87	oeoalem 5271	. . . . . 6 |- (C e. On -> (if((A e. On /\ (/) e. A), A, 1o) ^o (if(B e. On, B, 1o) +o C)) = ((if((A e. On /\ (/) e. A), A, 1o) ^o if(B e. On, B, 1o)) .o (if((A e. On /\ (/) e. A), A, 1o) ^o C)))
89	68, 74, 88	dedth2h 3015	. . . . 5 |- (((A e. On /\ (/) e. A) /\ B e. On) -> (C e. On -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C))))
90	89	impr 422	. . . 4 |- (((A e. On /\ (/) e. A) /\ (B e. On /\ C e. On)) -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C)))
91	90	an1rs 547	. . 3 |- (((A e. On /\ (B e. On /\ C e. On)) /\ (/) e. A) -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C)))
92	62, 91	oe0lem 5197	. 2 |- ((A e. On /\ (B e. On /\ C e. On)) -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C)))
93	92	3impb 1063	1 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o (B +o C)) = ((A ^o B) .o (A ^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 coa 5174   .o comu 5175   ^o coe 5176

This theorem is referenced by:  oeoelem 5273

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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