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Theorem omord 7019
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 7018 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) )
21ex 434 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) ) )
32pm5.32rd 640 . 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 457 . . 3  |-  ( ( ( C  .o  A
)  e.  ( C  .o  B )  /\  (/) 
e.  C )  -> 
( C  .o  A
)  e.  ( C  .o  B ) )
5 ne0i 3655 . . . . . . . 8  |-  ( ( C  .o  A )  e.  ( C  .o  B )  ->  ( C  .o  B )  =/=  (/) )
6 om0r 6991 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
7 oveq1 6110 . . . . . . . . . . 11  |-  ( C  =  (/)  ->  ( C  .o  B )  =  ( (/)  .o  B
) )
87eqeq1d 2451 . . . . . . . . . 10  |-  ( C  =  (/)  ->  ( ( C  .o  B )  =  (/)  <->  ( (/)  .o  B
)  =  (/) ) )
96, 8syl5ibrcom 222 . . . . . . . . 9  |-  ( B  e.  On  ->  ( C  =  (/)  ->  ( C  .o  B )  =  (/) ) )
109necon3d 2658 . . . . . . . 8  |-  ( B  e.  On  ->  (
( C  .o  B
)  =/=  (/)  ->  C  =/=  (/) ) )
115, 10syl5 32 . . . . . . 7  |-  ( B  e.  On  ->  (
( C  .o  A
)  e.  ( C  .o  B )  ->  C  =/=  (/) ) )
1211adantr 465 . . . . . 6  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  ( C  .o  B )  ->  C  =/=  (/) ) )
13 on0eln0 4786 . . . . . . 7  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  C  =/=  (/) ) )
1413adantl 466 . . . . . 6  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  C  <->  C  =/=  (/) ) )
1512, 14sylibrd 234 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  ( C  .o  B )  ->  (/)  e.  C ) )
16153adant1 1006 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  .o  A
)  e.  ( C  .o  B )  ->  (/) 
e.  C ) )
1716ancld 553 . . 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 204 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   (/)c0 3649   Oncon0 4731  (class class class)co 6103    .o comu 6930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-oadd 6936  df-omul 6937
This theorem is referenced by:  omlimcl  7029  oneo  7032
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