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Theorem omord2 6581
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 6580 . . 3  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  ->  ( C  .o  A )  e.  ( C  .o  B ) ) )
213adantl1 1111 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  ->  ( C  .o  A )  e.  ( C  .o  B ) ) )
3 oveq2 5882 . . . . . 6  |-  ( A  =  B  ->  ( C  .o  A )  =  ( C  .o  B
) )
43a1i 10 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  =  B  ->  ( C  .o  A )  =  ( C  .o  B ) ) )
5 omordi 6580 . . . . . 6  |-  ( ( ( A  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( B  e.  A  ->  ( C  .o  B )  e.  ( C  .o  A ) ) )
653adantl2 1112 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( B  e.  A  ->  ( C  .o  B )  e.  ( C  .o  A ) ) )
74, 6orim12d 811 . . . 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 125 . . 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 6551 . . . . . . . 8  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  .o  A
)  e.  On )
10 omcl 6551 . . . . . . . 8  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  .o  B
)  e.  On )
11 eloni 4418 . . . . . . . . 9  |-  ( ( C  .o  A )  e.  On  ->  Ord  ( C  .o  A
) )
12 eloni 4418 . . . . . . . . 9  |-  ( ( C  .o  B )  e.  On  ->  Ord  ( C  .o  B
) )
13 ordtri2 4443 . . . . . . . . 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 463 . . . . . . . 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 463 . . . . . . 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 803 . . . . . 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 439 . . . . 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 1146 . . . 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 451 . . 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 4418 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
21 eloni 4418 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
22 ordtri2 4443 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2320, 21, 22syl2an 463 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A
) ) )
24233adant3 975 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
2524adantr 451 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
268, 19, 253imtr4d 259 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( ( C  .o  A )  e.  ( C  .o  B
)  ->  A  e.  B ) )
272, 26impbid 183 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   (/)c0 3468   Ord word 4407   Oncon0 4408  (class class class)co 5874    .o comu 6493
This theorem is referenced by:  omord  6582  omword  6584  oeeui  6616  omabs  6661  omxpenlem  6979  cantnflt  7389  cnfcom  7419
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-recs 6404  df-rdg 6439  df-oadd 6499  df-omul 6500
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