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Theorem omord2 6802
Description: Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)
Assertion
Ref Expression
omord2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) )

Proof of Theorem omord2
StepHypRef Expression
1 omordi 6801 . . 3  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  ->  ( C  .o  A )  e.  ( C  .o  B ) ) )
213adantl1 1113 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  ->  ( C  .o  A )  e.  ( C  .o  B ) ) )
3 oveq2 6081 . . . . . 6  |-  ( A  =  B  ->  ( C  .o  A )  =  ( C  .o  B
) )
43a1i 11 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  =  B  ->  ( C  .o  A )  =  ( C  .o  B ) ) )
5 omordi 6801 . . . . . 6  |-  ( ( ( A  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( B  e.  A  ->  ( C  .o  B )  e.  ( C  .o  A ) ) )
653adantl2 1114 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( B  e.  A  ->  ( C  .o  B )  e.  ( C  .o  A ) ) )
74, 6orim12d 812 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( A  =  B  \/  B  e.  A )  ->  (
( C  .o  A
)  =  ( C  .o  B )  \/  ( C  .o  B
)  e.  ( C  .o  A ) ) ) )
87con3d 127 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( -.  (
( C  .o  A
)  =  ( C  .o  B )  \/  ( C  .o  B
)  e.  ( C  .o  A ) )  ->  -.  ( A  =  B  \/  B  e.  A ) ) )
9 omcl 6772 . . . . . . . 8  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  .o  A
)  e.  On )
10 omcl 6772 . . . . . . . 8  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  .o  B
)  e.  On )
11 eloni 4583 . . . . . . . . 9  |-  ( ( C  .o  A )  e.  On  ->  Ord  ( C  .o  A
) )
12 eloni 4583 . . . . . . . . 9  |-  ( ( C  .o  B )  e.  On  ->  Ord  ( C  .o  B
) )
13 ordtri2 4608 . . . . . . . . 9  |-  ( ( Ord  ( C  .o  A )  /\  Ord  ( C  .o  B
) )  ->  (
( C  .o  A
)  e.  ( C  .o  B )  <->  -.  (
( C  .o  A
)  =  ( C  .o  B )  \/  ( C  .o  B
)  e.  ( C  .o  A ) ) ) )
1411, 12, 13syl2an 464 . . . . . . . 8  |-  ( ( ( C  .o  A
)  e.  On  /\  ( C  .o  B
)  e.  On )  ->  ( ( C  .o  A )  e.  ( C  .o  B
)  <->  -.  ( ( C  .o  A )  =  ( C  .o  B
)  \/  ( C  .o  B )  e.  ( C  .o  A
) ) ) )
159, 10, 14syl2an 464 . . . . . . 7  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) )  -> 
( ( C  .o  A )  e.  ( C  .o  B )  <->  -.  ( ( C  .o  A )  =  ( C  .o  B )  \/  ( C  .o  B )  e.  ( C  .o  A ) ) ) )
1615anandis 804 . . . . . 6  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  ->  ( ( C  .o  A )  e.  ( C  .o  B
)  <->  -.  ( ( C  .o  A )  =  ( C  .o  B
)  \/  ( C  .o  B )  e.  ( C  .o  A
) ) ) )
1716ancoms 440 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On )  ->  ( ( C  .o  A )  e.  ( C  .o  B
)  <->  -.  ( ( C  .o  A )  =  ( C  .o  B
)  \/  ( C  .o  B )  e.  ( C  .o  A
) ) ) )
18173impa 1148 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  .o  A
)  e.  ( C  .o  B )  <->  -.  (
( C  .o  A
)  =  ( C  .o  B )  \/  ( C  .o  B
)  e.  ( C  .o  A ) ) ) )
1918adantr 452 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  e.  ( C  .o  B
)  <->  -.  ( ( C  .o  A )  =  ( C  .o  B
)  \/  ( C  .o  B )  e.  ( C  .o  A
) ) ) )
20 eloni 4583 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
21 eloni 4583 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
22 ordtri2 4608 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2320, 21, 22syl2an 464 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A
) ) )
24233adant3 977 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
2524adantr 452 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
268, 19, 253imtr4d 260 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  e.  ( C  .o  B
)  ->  A  e.  B ) )
272, 26impbid 184 1  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  ( C  .o  A )  e.  ( C  .o  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   (/)c0 3620   Ord word 4572   Oncon0 4573  (class class class)co 6073    .o comu 6714
This theorem is referenced by:  omord  6803  omword  6805  oeeui  6837  omabs  6882  omxpenlem  7201  cantnflt  7619  cnfcom  7649
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 4693
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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-oadd 6720  df-omul 6721
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