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Theorem omord 6561
Description: Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
Assertion
Ref Expression
omord  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  e.  B  /\  (/)  e.  C )  <-> 
( C  .o  A
)  e.  ( C  .o  B ) ) )

Proof of Theorem omord
StepHypRef Expression
1 omord2 6560 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) )
21ex 425 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) ) )
32pm5.32rd 623 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  e.  B  /\  (/)  e.  C )  <-> 
( ( C  .o  A )  e.  ( C  .o  B )  /\  (/)  e.  C ) ) )
4 simpl 445 . . 3  |-  ( ( ( C  .o  A
)  e.  ( C  .o  B )  /\  (/) 
e.  C )  -> 
( C  .o  A
)  e.  ( C  .o  B ) )
5 ne0i 3462 . . . . . . . 8  |-  ( ( C  .o  A )  e.  ( C  .o  B )  ->  ( C  .o  B )  =/=  (/) )
6 om0r 6533 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
7 oveq1 5826 . . . . . . . . . . 11  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
87eqeq1d 2292 . . . . . . . . . 10  |-  ( C  =  (/)  ->  ( ( C  .o  B )  =  (/)  <->  ( (/)  .o  B
)  =  (/) ) )
96, 8syl5ibrcom 215 . . . . . . . . 9  |-  ( B  e.  On  ->  ( C  =  (/)  ->  ( C  .o  B )  =  (/) ) )
109necon3d 2485 . . . . . . . 8  |-  ( B  e.  On  ->  (
( C  .o  B
)  =/=  (/)  ->  C  =/=  (/) ) )
115, 10syl5 30 . . . . . . 7  |-  ( B  e.  On  ->  (
( C  .o  A
)  e.  ( C  .o  B )  ->  C  =/=  (/) ) )
1211adantr 453 . . . . . 6  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  ( C  .o  B )  ->  C  =/=  (/) ) )
13 on0eln0 4446 . . . . . . 7  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  C  =/=  (/) ) )
1413adantl 454 . . . . . 6  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
1512, 14sylibrd 227 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  ( C  .o  B )  ->  (/)  e.  C ) )
16153adant1 975 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  .o  A
)  e.  ( C  .o  B )  ->  (/) 
e.  C ) )
1716ancld 538 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  .o  A
)  e.  ( C  .o  B )  -> 
( ( C  .o  A )  e.  ( C  .o  B )  /\  (/)  e.  C ) ) )
184, 17impbid2 197 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( ( C  .o  A )  e.  ( C  .o  B )  /\  (/)  e.  C )  <-> 
( C  .o  A
)  e.  ( C  .o  B ) ) )
193, 18bitrd 246 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  e.  B  /\  (/)  e.  C )  <-> 
( C  .o  A
)  e.  ( C  .o  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   (/)c0 3456   Oncon0 4391  (class class class)co 5819    .o comu 6472
This theorem is referenced by:  omlimcl  6571  oneo  6574
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-oadd 6478  df-omul 6479
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