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

Theorem omcan 7110
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 7107 . . . . . . . . 9  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) )
21ex 434 . . . . . . . 8  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( (/)  e.  A  ->  ( B  e.  C  ->  ( A  .o  B
)  e.  ( A  .o  C ) ) ) )
32ancoms 453 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( (/)  e.  A  ->  ( B  e.  C  ->  ( A  .o  B
)  e.  ( A  .o  C ) ) ) )
433adant2 1007 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  A  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) ) )
54imp 429 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) )
6 omordi 7107 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) )
76ex 434 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  A  ->  ( C  e.  B  ->  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
87ancoms 453 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  ->  ( C  e.  B  ->  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
983adant3 1008 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  A  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) ) )
109imp 429 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) )
115, 10orim12d 834 . . . 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 133 . . 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 7078 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
14 eloni 4829 . . . . . . 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 7078 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  .o  C
)  e.  On )
17 eloni 4829 . . . . . . 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 4855 . . . . . 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 477 . . . . 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 1274 . . . 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 465 . . 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 4829 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
24 eloni 4829 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
25 ordtri3 4855 . . . . . 6  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2623, 24, 25syl2an 477 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
27263adant1 1006 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
2827adantr 465 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2912, 22, 283imtr4d 268 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C
)  ->  B  =  C ) )
30 oveq2 6200 . 2  |-  ( B  =  C  ->  ( A  .o  B )  =  ( A  .o  C
) )
3129, 30impbid1 203 1  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C
)  <->  B  =  C
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   (/)c0 3737   Ord word 4818   Oncon0 4819  (class class class)co 6192    .o comu 7020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-oadd 7026  df-omul 7027
This theorem is referenced by:  omword  7111  fin1a2lem4  8675
  Copyright terms: Public domain W3C validator