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Theorem omwordi 6816
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 6815 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
21biimpd 200 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B ) ) )
32ex 425 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) ) )
4 eloni 4593 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
5 ord0eln0 4637 . . . . . . 7  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
65necon2bbid 2664 . . . . . 6  |-  ( Ord 
C  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
74, 6syl 16 . . . . 5  |-  ( C  e.  On  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
873ad2ant3 981 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
9 ssid 3369 . . . . . . 7  |-  (/)  C_  (/)
10 om0r 6785 . . . . . . . . 9  |-  ( A  e.  On  ->  ( (/) 
.o  A )  =  (/) )
1110adantr 453 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  =  (/) )
12 om0r 6785 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
1312adantl 454 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  B
)  =  (/) )
1411, 13sseq12d 3379 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  .o  A
)  C_  ( (/)  .o  B
)  <->  (/)  C_  (/) ) )
159, 14mpbiri 226 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) )
16 oveq1 6090 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  A )  =  ( (/)  .o  A
) )
17 oveq1 6090 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
1816, 17sseq12d 3379 . . . . . 6  |-  ( C  =  (/)  ->  ( ( C  .o  A ) 
C_  ( C  .o  B )  <->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) ) )
1915, 18syl5ibrcom 215 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
20193adant3 978 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A )  C_  ( C  .o  B
) ) )
218, 20sylbird 228 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  (/)  e.  C  -> 
( C  .o  A
)  C_  ( C  .o  B ) ) )
2221a1dd 45 . 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 153 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   (/)c0 3630   Ord word 4582   Oncon0 4583  (class class class)co 6083    .o comu 6724
This theorem is referenced by:  omword1  6818  omass  6825  omeulem1  6827  oewordri  6837  oeoalem  6841  oeeui  6847  oaabs2  6890  omxpenlem  7211  cantnflt  7629  cantnflem1d  7646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-oadd 6730  df-omul 6731
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