MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omword Structured version   Unicode version

Theorem omword 6813
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 6810 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) )
2 3anrot 941 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On  /\  B  e.  On )  <->  ( A  e.  On  /\  B  e.  On  /\  C  e.  On ) )
3 omcan 6812 . . . . 5  |-  ( ( ( C  e.  On  /\  A  e.  On  /\  B  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  =  ( C  .o  B
)  <->  A  =  B
) )
42, 3sylanbr 460 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  =  ( C  .o  B
)  <->  A  =  B
) )
54bicomd 193 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  =  B  <->  ( C  .o  A )  =  ( C  .o  B ) ) )
61, 5orbi12d 691 . 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 4622 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
873adant3 977 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
98adantr 452 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( A  e.  B  \/  A  =  B
) ) )
10 omcl 6780 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  .o  A
)  e.  On )
11 omcl 6780 . . . . . . 7  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  .o  B
)  e.  On )
1210, 11anim12dan 811 . . . . . 6  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  .o  A )  e.  On  /\  ( C  .o  B )  e.  On ) )
1312ancoms 440 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  On  /\  ( C  .o  B )  e.  On ) )
14133impa 1148 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  .o  A
)  e.  On  /\  ( C  .o  B
)  e.  On ) )
15 onsseleq 4622 . . . 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 452 . 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 277 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 set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   (/)c0 3628   Oncon0 4581  (class class class)co 6081    .o comu 6722
This theorem is referenced by:  omwordi  6814  omeulem2  6826  oeeui  6845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-oadd 6728  df-omul 6729
  Copyright terms: Public domain W3C validator