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Theorem omwordi 7115
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 7114 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
21biimpd 207 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B ) ) )
32ex 434 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) ) )
4 eloni 4832 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
5 ord0eln0 4876 . . . . . . 7  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
65necon2bbid 2705 . . . . . 6  |-  ( Ord 
C  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
74, 6syl 16 . . . . 5  |-  ( C  e.  On  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
873ad2ant3 1011 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
9 ssid 3478 . . . . . . 7  |-  (/)  C_  (/)
10 om0r 7084 . . . . . . . . 9  |-  ( A  e.  On  ->  ( (/) 
.o  A )  =  (/) )
1110adantr 465 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  =  (/) )
12 om0r 7084 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
1312adantl 466 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  B
)  =  (/) )
1411, 13sseq12d 3488 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  .o  A
)  C_  ( (/)  .o  B
)  <->  (/)  C_  (/) ) )
159, 14mpbiri 233 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) )
16 oveq1 6202 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  A )  =  ( (/)  .o  A
) )
17 oveq1 6202 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
1816, 17sseq12d 3488 . . . . . 6  |-  ( C  =  (/)  ->  ( ( C  .o  A ) 
C_  ( C  .o  B )  <->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) ) )
1915, 18syl5ibrcom 222 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
20193adant3 1008 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A )  C_  ( C  .o  B
) ) )
218, 20sylbird 235 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  (/)  e.  C  -> 
( C  .o  A
)  C_  ( C  .o  B ) ) )
2221a1dd 46 . 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 158 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3431   (/)c0 3740   Ord word 4821   Oncon0 4822  (class class class)co 6195    .o comu 7023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-oadd 7029  df-omul 7030
This theorem is referenced by:  omword1  7117  omass  7124  omeulem1  7126  oewordri  7136  oeoalem  7140  oeeui  7146  oaabs2  7189  omxpenlem  7517  cantnflt  7986  cantnflem1d  8002  cantnfltOLD  8016  cantnflem1dOLD  8025
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