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Theorem omwordi 7212
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 7211 . . . 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 432 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) ) )
4 eloni 4877 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
5 ord0eln0 4921 . . . . . . 7  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
65necon2bbid 2710 . . . . . 6  |-  ( Ord 
C  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
74, 6syl 16 . . . . 5  |-  ( C  e.  On  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
873ad2ant3 1017 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
9 ssid 3508 . . . . . . 7  |-  (/)  C_  (/)
10 om0r 7181 . . . . . . . . 9  |-  ( A  e.  On  ->  ( (/) 
.o  A )  =  (/) )
1110adantr 463 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  =  (/) )
12 om0r 7181 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
1312adantl 464 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  B
)  =  (/) )
1411, 13sseq12d 3518 . . . . . . 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 6277 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  A )  =  ( (/)  .o  A
) )
17 oveq1 6277 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
1816, 17sseq12d 3518 . . . . . 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 1014 . . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   (/)c0 3783   Ord word 4866   Oncon0 4867  (class class class)co 6270    .o comu 7120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-oadd 7126  df-omul 7127
This theorem is referenced by:  omword1  7214  omass  7221  omeulem1  7223  oewordri  7233  oeoalem  7237  oeeui  7243  oaabs2  7286  omxpenlem  7611  cantnflt  8082  cantnflem1d  8098  cantnfltOLD  8112  cantnflem1dOLD  8121
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