Theorem oewordi 7242

Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)

Assertion

Ref	Expression
oewordi	|- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) )

Proof of Theorem oewordi

Step	Hyp	Ref	Expression
1		eloni 4878	. . . . . 6 |- ( C e. On -> Ord C )
2		ordgt0ge1 7149	. . . . . 6 |- ( Ord C -> ( (/) e. C <-> 1o C_ C ) )
3	1, 2	syl 16	. . . . 5 |- ( C e. On -> ( (/) e. C <-> 1o C_ C ) )
4		1on 7139	. . . . . 6 |- 1o e. On
5		onsseleq 4909	. . . . . 6 |- ( ( 1o e. On /\ C e. On ) -> ( 1o C_ C <-> ( 1o e. C \/ 1o = C ) ) )
6	4, 5	mpan 670	. . . . 5 |- ( C e. On -> ( 1o C_ C <-> ( 1o e. C \/ 1o = C ) ) )
7	3, 6	bitrd 253	. . . 4 |- ( C e. On -> ( (/) e. C <-> ( 1o e. C \/ 1o = C ) ) )
8	7	3ad2ant3 1020	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( (/) e. C <-> ( 1o e. C \/ 1o = C ) ) )
9		ondif2 7154	. . . . . . 7 |- ( C e. ( On \ 2o ) <-> ( C e. On /\ 1o e. C ) )
10		oeword 7241	. . . . . . . . 9 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A C_ B <-> ( C ^o A ) C_ ( C ^o B ) ) )
11	10	biimpd 207	. . . . . . . 8 |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) )
12	11	3expia 1199	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( C e. ( On \ 2o ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
13	9, 12	syl5bir 218	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( C e. On /\ 1o e. C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
14	13	expd 436	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( C e. On -> ( 1o e. C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) ) )
15	14	3impia 1194	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( 1o e. C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
16		oe1m 7196	. . . . . . . . . 10 |- ( A e. On -> ( 1o ^o A ) = 1o )
17	16	adantr 465	. . . . . . . . 9 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o A ) = 1o )
18		oe1m 7196	. . . . . . . . . 10 |- ( B e. On -> ( 1o ^o B ) = 1o )
19	18	adantl 466	. . . . . . . . 9 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o B ) = 1o )
20	17, 19	eqtr4d 2487	. . . . . . . 8 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o A ) = ( 1o ^o B ) )
21		eqimss 3541	. . . . . . . 8 |- ( ( 1o ^o A ) = ( 1o ^o B ) -> ( 1o ^o A ) C_ ( 1o ^o B ) )
22	20, 21	syl 16	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( 1o ^o A ) C_ ( 1o ^o B ) )
23		oveq1 6288	. . . . . . . 8 |- ( 1o = C -> ( 1o ^o A ) = ( C ^o A ) )
24		oveq1 6288	. . . . . . . 8 |- ( 1o = C -> ( 1o ^o B ) = ( C ^o B ) )
25	23, 24	sseq12d 3518	. . . . . . 7 |- ( 1o = C -> ( ( 1o ^o A ) C_ ( 1o ^o B ) <-> ( C ^o A ) C_ ( C ^o B ) ) )
26	22, 25	syl5ibcom 220	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( 1o = C -> ( C ^o A ) C_ ( C ^o B ) ) )
27	26	3adant3 1017	. . . . 5 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( 1o = C -> ( C ^o A ) C_ ( C ^o B ) ) )
28	27	a1dd 46	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( 1o = C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
29	15, 28	jaod 380	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( 1o e. C \/ 1o = C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
30	8, 29	sylbid 215	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( (/) e. C -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) )
31	30	imp 429	1 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) -> ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   \ cdif 3458   C_ wss 3461   (/)c0 3770   Ord word 4867   Oncon0 4868  (class class class)co 6281   1oc1o 7125   2oc2o 7126   ^o coe 7131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138

This theorem is referenced by:  oelim2  7246  oeoalem  7247  oeoelem  7249  oaabs2  7296  cantnflt  8094  cantnfltOLD  8124  cnfcom  8147  cnfcomOLD  8155