Theorem oeoe 7679

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		oveq2 6658	. . . . . . . . . . . 12 |- ( B = (/) -> ( (/) ^o B ) = ( (/) ^o (/) ) )
2		oe0m0 7600	. . . . . . . . . . . 12 |- ( (/) ^o (/) ) = 1o
3	1, 2	syl6eq 2672	. . . . . . . . . . 11 |- ( B = (/) -> ( (/) ^o B ) = 1o )
4	3	oveq1d 6665	. . . . . . . . . 10 |- ( B = (/) -> ( ( (/) ^o B ) ^o C ) = ( 1o ^o C ) )
5		oe1m 7625	. . . . . . . . . 10 |- ( C e. On -> ( 1o ^o C ) = 1o )
6	4, 5	sylan9eqr 2678	. . . . . . . . 9 |- ( ( C e. On /\ B = (/) ) -> ( ( (/) ^o B ) ^o C ) = 1o )
7	6	adantll 750	. . . . . . . 8 |- ( ( ( B e. On /\ C e. On ) /\ B = (/) ) -> ( ( (/) ^o B ) ^o C ) = 1o )
8		oveq2 6658	. . . . . . . . . 10 |- ( C = (/) -> ( ( (/) ^o B ) ^o C ) = ( ( (/) ^o B ) ^o (/) ) )
9		0elon 5778	. . . . . . . . . . . 12 |- (/) e. On
10		oecl 7617	. . . . . . . . . . . 12 |- ( ( (/) e. On /\ B e. On ) -> ( (/) ^o B ) e. On )
11	9, 10	mpan 706	. . . . . . . . . . 11 |- ( B e. On -> ( (/) ^o B ) e. On )
12		oe0 7602	. . . . . . . . . . 11 |- ( ( (/) ^o B ) e. On -> ( ( (/) ^o B ) ^o (/) ) = 1o )
13	11, 12	syl 17	. . . . . . . . . 10 |- ( B e. On -> ( ( (/) ^o B ) ^o (/) ) = 1o )
14	8, 13	sylan9eqr 2678	. . . . . . . . 9 |- ( ( B e. On /\ C = (/) ) -> ( ( (/) ^o B ) ^o C ) = 1o )
15	14	adantlr 751	. . . . . . . 8 |- ( ( ( B e. On /\ C e. On ) /\ C = (/) ) -> ( ( (/) ^o B ) ^o C ) = 1o )
16	7, 15	jaodan 826	. . . . . . 7 |- ( ( ( B e. On /\ C e. On ) /\ ( B = (/) \/ C = (/) ) ) -> ( ( (/) ^o B ) ^o C ) = 1o )
17		om00 7655	. . . . . . . . . 10 |- ( ( B e. On /\ C e. On ) -> ( ( B .o C ) = (/) <-> ( B = (/) \/ C = (/) ) ) )
18	17	biimpar 502	. . . . . . . . 9 |- ( ( ( B e. On /\ C e. On ) /\ ( B = (/) \/ C = (/) ) ) -> ( B .o C ) = (/) )
19	18	oveq2d 6666	. . . . . . . 8 |- ( ( ( B e. On /\ C e. On ) /\ ( B = (/) \/ C = (/) ) ) -> ( (/) ^o ( B .o C ) ) = ( (/) ^o (/) ) )
20	19, 2	syl6eq 2672	. . . . . . 7 |- ( ( ( B e. On /\ C e. On ) /\ ( B = (/) \/ C = (/) ) ) -> ( (/) ^o ( B .o C ) ) = 1o )
21	16, 20	eqtr4d 2659	. . . . . 6 |- ( ( ( B e. On /\ C e. On ) /\ ( B = (/) \/ C = (/) ) ) -> ( ( (/) ^o B ) ^o C ) = ( (/) ^o ( B .o C ) ) )
22		on0eln0 5780	. . . . . . . . . 10 |- ( B e. On -> ( (/) e. B <-> B =/= (/) ) )
23		on0eln0 5780	. . . . . . . . . 10 |- ( C e. On -> ( (/) e. C <-> C =/= (/) ) )
24	22, 23	bi2anan9 917	. . . . . . . . 9 |- ( ( B e. On /\ C e. On ) -> ( ( (/) e. B /\ (/) e. C ) <-> ( B =/= (/) /\ C =/= (/) ) ) )
25		neanior 2886	. . . . . . . . 9 |- ( ( B =/= (/) /\ C =/= (/) ) <-> -. ( B = (/) \/ C = (/) ) )
26	24, 25	syl6bb 276	. . . . . . . 8 |- ( ( B e. On /\ C e. On ) -> ( ( (/) e. B /\ (/) e. C ) <-> -. ( B = (/) \/ C = (/) ) ) )
27		oe0m1 7601	. . . . . . . . . . . . . 14 |- ( B e. On -> ( (/) e. B <-> ( (/) ^o B ) = (/) ) )
28	27	biimpa 501	. . . . . . . . . . . . 13 |- ( ( B e. On /\ (/) e. B ) -> ( (/) ^o B ) = (/) )
29	28	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( B e. On /\ (/) e. B ) -> ( ( (/) ^o B ) ^o C ) = ( (/) ^o C ) )
30		oe0m1 7601	. . . . . . . . . . . . 13 |- ( C e. On -> ( (/) e. C <-> ( (/) ^o C ) = (/) ) )
31	30	biimpa 501	. . . . . . . . . . . 12 |- ( ( C e. On /\ (/) e. C ) -> ( (/) ^o C ) = (/) )
32	29, 31	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ( B e. On /\ (/) e. B ) /\ ( C e. On /\ (/) e. C ) ) -> ( ( (/) ^o B ) ^o C ) = (/) )
33	32	an4s 869	. . . . . . . . . 10 |- ( ( ( B e. On /\ C e. On ) /\ ( (/) e. B /\ (/) e. C ) ) -> ( ( (/) ^o B ) ^o C ) = (/) )
34		om00el 7656	. . . . . . . . . . . 12 |- ( ( B e. On /\ C e. On ) -> ( (/) e. ( B .o C ) <-> ( (/) e. B /\ (/) e. C ) ) )
35		omcl 7616	. . . . . . . . . . . . 13 |- ( ( B e. On /\ C e. On ) -> ( B .o C ) e. On )
36		oe0m1 7601	. . . . . . . . . . . . 13 |- ( ( B .o C ) e. On -> ( (/) e. ( B .o C ) <-> ( (/) ^o ( B .o C ) ) = (/) ) )
37	35, 36	syl 17	. . . . . . . . . . . 12 |- ( ( B e. On /\ C e. On ) -> ( (/) e. ( B .o C ) <-> ( (/) ^o ( B .o C ) ) = (/) ) )
38	34, 37	bitr3d 270	. . . . . . . . . . 11 |- ( ( B e. On /\ C e. On ) -> ( ( (/) e. B /\ (/) e. C ) <-> ( (/) ^o ( B .o C ) ) = (/) ) )
39	38	biimpa 501	. . . . . . . . . 10 |- ( ( ( B e. On /\ C e. On ) /\ ( (/) e. B /\ (/) e. C ) ) -> ( (/) ^o ( B .o C ) ) = (/) )
40	33, 39	eqtr4d 2659	. . . . . . . . 9 |- ( ( ( B e. On /\ C e. On ) /\ ( (/) e. B /\ (/) e. C ) ) -> ( ( (/) ^o B ) ^o C ) = ( (/) ^o ( B .o C ) ) )
41	40	ex 450	. . . . . . . 8 |- ( ( B e. On /\ C e. On ) -> ( ( (/) e. B /\ (/) e. C ) -> ( ( (/) ^o B ) ^o C ) = ( (/) ^o ( B .o C ) ) ) )
42	26, 41	sylbird 250	. . . . . . 7 |- ( ( B e. On /\ C e. On ) -> ( -. ( B = (/) \/ C = (/) ) -> ( ( (/) ^o B ) ^o C ) = ( (/) ^o ( B .o C ) ) ) )
43	42	imp 445	. . . . . 6 |- ( ( ( B e. On /\ C e. On ) /\ -. ( B = (/) \/ C = (/) ) ) -> ( ( (/) ^o B ) ^o C ) = ( (/) ^o ( B .o C ) ) )
44	21, 43	pm2.61dan 832	. . . . 5 |- ( ( B e. On /\ C e. On ) -> ( ( (/) ^o B ) ^o C ) = ( (/) ^o ( B .o C ) ) )
45		oveq1 6657	. . . . . . 7 |- ( A = (/) -> ( A ^o B ) = ( (/) ^o B ) )
46	45	oveq1d 6665	. . . . . 6 |- ( A = (/) -> ( ( A ^o B ) ^o C ) = ( ( (/) ^o B ) ^o C ) )
47		oveq1 6657	. . . . . 6 |- ( A = (/) -> ( A ^o ( B .o C ) ) = ( (/) ^o ( B .o C ) ) )
48	46, 47	eqeq12d 2637	. . . . 5 |- ( A = (/) -> ( ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) <-> ( ( (/) ^o B ) ^o C ) = ( (/) ^o ( B .o C ) ) ) )
49	44, 48	syl5ibr 236	. . . 4 |- ( A = (/) -> ( ( B e. On /\ C e. On ) -> ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) ) )
50	49	impcom 446	. . 3 |- ( ( ( B e. On /\ C e. On ) /\ A = (/) ) -> ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) )
51		oveq1 6657	. . . . . . . . 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	oveq1d 6665	. . . . . . . 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		oveq1 6657	. . . . . . . 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 2637	. . . . . . 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 330	. . . . . 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 2689	. . . . . . . . . 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 2690	. . . . . . . . . 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 747	. . . . . . . . 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 2689	. . . . . . . . . 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 2690	. . . . . . . . . 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 747	. . . . . . . . 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 7567	. . . . . . . . . 10 |- 1o e. On
63		0lt1o 7584	. . . . . . . . . 10 |- (/) e. 1o
64	62, 63	pm3.2i 471	. . . . . . . . 9 |- ( 1o e. On /\ (/) e. 1o )
65	58, 61, 64	elimhyp 4146	. . . . . . . 8 |- ( if ( ( A e. On /\ (/) e. A ) , A , 1o ) e. On /\ (/) e. if ( ( A e. On /\ (/) e. A ) , A , 1o ) )
66	65	simpli 474	. . . . . . 7 |- if ( ( A e. On /\ (/) e. A ) , A , 1o ) e. On
67	65	simpri 478	. . . . . . 7 |- (/) e. if ( ( A e. On /\ (/) e. A ) , A , 1o )
68	66, 67	oeoelem 7678	. . . . . 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 4139	. . . . 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 445	. . . 4 |- ( ( ( A e. On /\ (/) e. A ) /\ ( B e. On /\ C e. On ) ) -> ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) )
71	70	an32s 846	. . 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 7593	. 2 |- ( ( A e. On /\ ( B e. On /\ C e. On ) ) -> ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) )
73	72	3impb 1260	1 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   ifcif 4086   Oncon0 5723  (class class class)co 6650   1oc1o 7553   .o comu 7558   ^o coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  infxpenc  8841