Theorem oewordi 5266

Description: Weak ordering property of ordinal exponentiation.

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 3667	. . . . . 6 |- (C e. On -> Ord C)
2		ordgt0ge1 5189	. . . . . 6 |- (Ord C -> ((/) e. C <-> 1o C_ C))
3	1, 2	syl 12	. . . . 5 |- (C e. On -> ((/) e. C <-> 1o C_ C))
4		1on 5182	. . . . . 6 |- 1o e. On
5		onsseleq 3704	. . . . . 6 |- ((1o e. On /\ C e. On) -> (1o C_ C <-> (1o e. C \/ 1o = C)))
6	4, 5	mpan 759	. . . . 5 |- (C e. On -> (1o C_ C <-> (1o e. C \/ 1o = C)))
7	3, 6	bitrd 587	. . . 4 |- (C e. On -> ((/) e. C <-> (1o e. C \/ 1o = C)))
8	7	3ad2ant3 899	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> ((/) e. C <-> (1o e. C \/ 1o = C)))
9		oeword 5265	. . . . . 6 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A C_ B <-> (C ^o A) C_ (C ^o B)))
10	9	biimpd 170	. . . . 5 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A C_ B -> (C ^o A) C_ (C ^o B)))
11	10	ex 402	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (1o e. C -> (A C_ B -> (C ^o A) C_ (C ^o B))))
12		opreq1 4889	. . . . . . . 8 |- (1o = C -> (1o ^o A) = (C ^o A))
13		opreq1 4889	. . . . . . . 8 |- (1o = C -> (1o ^o B) = (C ^o B))
14	12, 13	sseq12d 2646	. . . . . . 7 |- (1o = C -> ((1o ^o A) C_ (1o ^o B) <-> (C ^o A) C_ (C ^o B)))
15		oe1m 5226	. . . . . . . . . 10 |- (A e. On -> (1o ^o A) = 1o)
16	15	adantr 425	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (1o ^o A) = 1o)
17		oe1m 5226	. . . . . . . . . 10 |- (B e. On -> (1o ^o B) = 1o)
18	17	adantl 424	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (1o ^o B) = 1o)
19	16, 18	eqtr4d 1928	. . . . . . . 8 |- ((A e. On /\ B e. On) -> (1o ^o A) = (1o ^o B))
20		eqimss 2665	. . . . . . . 8 |- ((1o ^o A) = (1o ^o B) -> (1o ^o A) C_ (1o ^o B))
21	19, 20	syl 12	. . . . . . 7 |- ((A e. On /\ B e. On) -> (1o ^o A) C_ (1o ^o B))
22	14, 21	syl5cbi 226	. . . . . 6 |- ((A e. On /\ B e. On) -> (1o = C -> (C ^o A) C_ (C ^o B)))
23	22	3adant3 896	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) -> (1o = C -> (C ^o A) C_ (C ^o B)))
24	23	a1dd 53	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (1o = C -> (A C_ B -> (C ^o A) C_ (C ^o B))))
25	11, 24	jaod 469	. . 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))))
26	8, 25	sylbid 220	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> ((/) e. C -> (A C_ B -> (C ^o A) C_ (C ^o B))))
27	26	imp 377	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 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)c0 2875  Ord word 3656  Oncon0 3657  (class class class)co 4884  1oc1o 5172   ^o coe 5176

This theorem is referenced by:  oelim2 5270  oeoalem 5271  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-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-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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