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Theorem omord 6802
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 6801 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) )
21ex 424 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) ) )
32pm5.32rd 622 . 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 444 . . 3  |-  ( ( ( C  .o  A
)  e.  ( C  .o  B )  /\  (/) 
e.  C )  -> 
( C  .o  A
)  e.  ( C  .o  B ) )
5 ne0i 3626 . . . . . . . 8  |-  ( ( C  .o  A )  e.  ( C  .o  B )  ->  ( C  .o  B )  =/=  (/) )
6 om0r 6774 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
7 oveq1 6079 . . . . . . . . . . 11  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
87eqeq1d 2443 . . . . . . . . . 10  |-  ( C  =  (/)  ->  ( ( C  .o  B )  =  (/)  <->  ( (/)  .o  B
)  =  (/) ) )
96, 8syl5ibrcom 214 . . . . . . . . 9  |-  ( B  e.  On  ->  ( C  =  (/)  ->  ( C  .o  B )  =  (/) ) )
109necon3d 2636 . . . . . . . 8  |-  ( B  e.  On  ->  (
( C  .o  B
)  =/=  (/)  ->  C  =/=  (/) ) )
115, 10syl5 30 . . . . . . 7  |-  ( B  e.  On  ->  (
( C  .o  A
)  e.  ( C  .o  B )  ->  C  =/=  (/) ) )
1211adantr 452 . . . . . 6  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  ( C  .o  B )  ->  C  =/=  (/) ) )
13 on0eln0 4628 . . . . . . 7  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  C  =/=  (/) ) )
1413adantl 453 . . . . . 6  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
1512, 14sylibrd 226 . . . . 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 537 . . 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 196 . 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 245 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 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   Oncon0 4573  (class class class)co 6072    .o comu 6713
This theorem is referenced by:  omlimcl  6812  oneo  6815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-oadd 6719  df-omul 6720
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