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

Theorem omwordi 6814
Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omwordi  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) )

Proof of Theorem omwordi
StepHypRef Expression
1 omword 6813 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
21biimpd 199 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B ) ) )
32ex 424 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) ) )
4 eloni 4591 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
5 ord0eln0 4635 . . . . . . 7  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
65necon2bbid 2662 . . . . . 6  |-  ( Ord 
C  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
74, 6syl 16 . . . . 5  |-  ( C  e.  On  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
873ad2ant3 980 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
9 ssid 3367 . . . . . . 7  |-  (/)  C_  (/)
10 om0r 6783 . . . . . . . . 9  |-  ( A  e.  On  ->  ( (/) 
.o  A )  =  (/) )
1110adantr 452 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  =  (/) )
12 om0r 6783 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
1312adantl 453 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  B
)  =  (/) )
1411, 13sseq12d 3377 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  .o  A
)  C_  ( (/)  .o  B
)  <->  (/)  C_  (/) ) )
159, 14mpbiri 225 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) )
16 oveq1 6088 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  A )  =  ( (/)  .o  A
) )
17 oveq1 6088 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
1816, 17sseq12d 3377 . . . . . 6  |-  ( C  =  (/)  ->  ( ( C  .o  A ) 
C_  ( C  .o  B )  <->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) ) )
1915, 18syl5ibrcom 214 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
20193adant3 977 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A )  C_  ( C  .o  B
) ) )
218, 20sylbird 227 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  (/)  e.  C  -> 
( C  .o  A
)  C_  ( C  .o  B ) ) )
2221a1dd 44 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  (/)  e.  C  -> 
( A  C_  B  ->  ( C  .o  A
)  C_  ( C  .o  B ) ) ) )
233, 22pm2.61d 152 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   (/)c0 3628   Ord word 4580   Oncon0 4581  (class class class)co 6081    .o comu 6722
This theorem is referenced by:  omword1  6816  omass  6823  omeulem1  6825  oewordri  6835  oeoalem  6839  oeeui  6845  oaabs2  6888  omxpenlem  7209  cantnflt  7627  cantnflem1d  7644
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-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-oadd 6728  df-omul 6729
  Copyright terms: Public domain W3C validator