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Theorem omwordi 6585
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 6584 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
<->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
21biimpd 198 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B ) ) )
32ex 423 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B
) ) ) )
4 eloni 4418 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
5 ord0eln0 4462 . . . . . . 7  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
65necon2bbid 2517 . . . . . 6  |-  ( Ord 
C  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
74, 6syl 15 . . . . 5  |-  ( C  e.  On  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
873ad2ant3 978 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  <->  -.  (/)  e.  C
) )
9 ssid 3210 . . . . . . 7  |-  (/)  C_  (/)
10 om0r 6554 . . . . . . . . 9  |-  ( A  e.  On  ->  ( (/) 
.o  A )  =  (/) )
1110adantr 451 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  =  (/) )
12 om0r 6554 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
1312adantl 452 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  B
)  =  (/) )
1411, 13sseq12d 3220 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  .o  A
)  C_  ( (/)  .o  B
)  <->  (/)  C_  (/) ) )
159, 14mpbiri 224 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) )
16 oveq1 5881 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  A )  =  ( (/)  .o  A
) )
17 oveq1 5881 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
1816, 17sseq12d 3220 . . . . . 6  |-  ( C  =  (/)  ->  ( ( C  .o  A ) 
C_  ( C  .o  B )  <->  ( (/)  .o  A
)  C_  ( (/)  .o  B
) ) )
1915, 18syl5ibrcom 213 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A
)  C_  ( C  .o  B ) ) )
20193adant3 975 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  =  (/)  ->  ( C  .o  A )  C_  ( C  .o  B
) ) )
218, 20sylbird 226 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  (/)  e.  C  -> 
( C  .o  A
)  C_  ( C  .o  B ) ) )
2221a1dd 42 . 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 150 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   (/)c0 3468   Ord word 4407   Oncon0 4408  (class class class)co 5874    .o comu 6493
This theorem is referenced by:  omword1  6587  omass  6594  omeulem1  6596  oewordri  6606  oeoalem  6610  oeeui  6616  oaabs2  6659  omxpenlem  6979  cantnflt  7389  cantnflem1d  7406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-omul 6500
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