MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omcan Unicode version

Theorem omcan 6771
Description: Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)
Assertion
Ref Expression
omcan  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C
)  <->  B  =  C
) )

Proof of Theorem omcan
StepHypRef Expression
1 omordi 6768 . . . . . . . . 9  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) )
21ex 424 . . . . . . . 8  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( (/)  e.  A  ->  ( B  e.  C  ->  ( A  .o  B
)  e.  ( A  .o  C ) ) ) )
32ancoms 440 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( (/)  e.  A  ->  ( B  e.  C  ->  ( A  .o  B
)  e.  ( A  .o  C ) ) ) )
433adant2 976 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  A  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) ) )
54imp 419 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) )
6 omordi 6768 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) )
76ex 424 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  A  ->  ( C  e.  B  ->  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
87ancoms 440 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  ->  ( C  e.  B  ->  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
983adant3 977 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  A  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) ) )
109imp 419 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) )
115, 10orim12d 812 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( B  e.  C  \/  C  e.  B )  ->  (
( A  .o  B
)  e.  ( A  .o  C )  \/  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
1211con3d 127 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( -.  (
( A  .o  B
)  e.  ( A  .o  C )  \/  ( A  .o  C
)  e.  ( A  .o  B ) )  ->  -.  ( B  e.  C  \/  C  e.  B ) ) )
13 omcl 6739 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
14 eloni 4551 . . . . . . 7  |-  ( ( A  .o  B )  e.  On  ->  Ord  ( A  .o  B
) )
1513, 14syl 16 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  .o  B ) )
16 omcl 6739 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  .o  C
)  e.  On )
17 eloni 4551 . . . . . . 7  |-  ( ( A  .o  C )  e.  On  ->  Ord  ( A  .o  C
) )
1816, 17syl 16 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  On )  ->  Ord  ( A  .o  C ) )
19 ordtri3 4577 . . . . . 6  |-  ( ( Ord  ( A  .o  B )  /\  Ord  ( A  .o  C
) )  ->  (
( A  .o  B
)  =  ( A  .o  C )  <->  -.  (
( A  .o  B
)  e.  ( A  .o  C )  \/  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
2015, 18, 19syl2an 464 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  ( A  e.  On  /\  C  e.  On ) )  -> 
( ( A  .o  B )  =  ( A  .o  C )  <->  -.  ( ( A  .o  B )  e.  ( A  .o  C )  \/  ( A  .o  C )  e.  ( A  .o  B ) ) ) )
21203impdi 1239 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  .o  B
)  =  ( A  .o  C )  <->  -.  (
( A  .o  B
)  e.  ( A  .o  C )  \/  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
2221adantr 452 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C
)  <->  -.  ( ( A  .o  B )  e.  ( A  .o  C
)  \/  ( A  .o  C )  e.  ( A  .o  B
) ) ) )
23 eloni 4551 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
24 eloni 4551 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
25 ordtri3 4577 . . . . . 6  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2623, 24, 25syl2an 464 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
27263adant1 975 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
2827adantr 452 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2912, 22, 283imtr4d 260 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C
)  ->  B  =  C ) )
30 oveq2 6048 . 2  |-  ( B  =  C  ->  ( A  .o  B )  =  ( A  .o  C
) )
3129, 30impbid1 195 1  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C
)  <->  B  =  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   (/)c0 3588   Ord word 4540   Oncon0 4541  (class class class)co 6040    .o comu 6681
This theorem is referenced by:  omword  6772  fin1a2lem4  8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-oadd 6687  df-omul 6688
  Copyright terms: Public domain W3C validator