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Theorem omword 7231
Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omword  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( C  .o  A
)  C_  ( C  .o  B ) ) )

Proof of Theorem omword
StepHypRef Expression
1 omord2 7228 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) )
2 3anrot 978 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On  /\  B  e.  On )  <->  ( A  e.  On  /\  B  e.  On  /\  C  e.  On ) )
3 omcan 7230 . . . . 5  |-  ( ( ( C  e.  On  /\  A  e.  On  /\  B  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  =  ( C  .o  B
)  <->  A  =  B
) )
42, 3sylanbr 473 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  =  ( C  .o  B
)  <->  A  =  B
) )
54bicomd 201 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  =  B  <->  ( C  .o  A )  =  ( C  .o  B ) ) )
61, 5orbi12d 709 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( A  e.  B  \/  A  =  B )  <->  ( ( C  .o  A )  e.  ( C  .o  B
)  \/  ( C  .o  A )  =  ( C  .o  B
) ) ) )
7 onsseleq 4925 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
873adant3 1016 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
98adantr 465 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( A  e.  B  \/  A  =  B
) ) )
10 omcl 7198 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  .o  A
)  e.  On )
11 omcl 7198 . . . . . . 7  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  .o  B
)  e.  On )
1210, 11anim12dan 835 . . . . . 6  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  .o  A )  e.  On  /\  ( C  .o  B )  e.  On ) )
1312ancoms 453 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  On  /\  ( C  .o  B )  e.  On ) )
14133impa 1191 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  .o  A
)  e.  On  /\  ( C  .o  B
)  e.  On ) )
15 onsseleq 4925 . . . 4  |-  ( ( ( C  .o  A
)  e.  On  /\  ( C  .o  B
)  e.  On )  ->  ( ( C  .o  A )  C_  ( C  .o  B
)  <->  ( ( C  .o  A )  e.  ( C  .o  B
)  \/  ( C  .o  A )  =  ( C  .o  B
) ) ) )
1614, 15syl 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  .o  A
)  C_  ( C  .o  B )  <->  ( ( C  .o  A )  e.  ( C  .o  B
)  \/  ( C  .o  A )  =  ( C  .o  B
) ) ) )
1716adantr 465 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  C_  ( C  .o  B
)  <->  ( ( C  .o  A )  e.  ( C  .o  B
)  \/  ( C  .o  A )  =  ( C  .o  B
) ) ) )
186, 9, 173bitr4d 285 1  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   (/)c0 3790   Oncon0 4884  (class class class)co 6295    .o comu 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-oadd 7146  df-omul 7147
This theorem is referenced by:  omwordi  7232  omeulem2  7244  oeeui  7263
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