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Theorem omcan 6709
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 6706 . . . . . . . . 9  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) )
21ex 423 . . . . . . . 8  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( (/)  e.  A  ->  ( B  e.  C  ->  ( A  .o  B
)  e.  ( A  .o  C ) ) ) )
32ancoms 439 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( (/)  e.  A  ->  ( B  e.  C  ->  ( A  .o  B
)  e.  ( A  .o  C ) ) ) )
433adant2 975 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  A  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) ) )
54imp 418 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( B  e.  C  ->  ( A  .o  B )  e.  ( A  .o  C ) ) )
6 omordi 6706 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  A  e.  On )  /\  (/)  e.  A )  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) )
76ex 423 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  A  ->  ( C  e.  B  ->  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
87ancoms 439 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  ->  ( C  e.  B  ->  ( A  .o  C
)  e.  ( A  .o  B ) ) ) )
983adant3 976 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  A  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) ) )
109imp 418 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( C  e.  B  ->  ( A  .o  C )  e.  ( A  .o  B ) ) )
115, 10orim12d 811 . . . 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 125 . . 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 6677 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
14 eloni 4505 . . . . . . 7  |-  ( ( A  .o  B )  e.  On  ->  Ord  ( A  .o  B
) )
1513, 14syl 15 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  .o  B ) )
16 omcl 6677 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  .o  C
)  e.  On )
17 eloni 4505 . . . . . . 7  |-  ( ( A  .o  C )  e.  On  ->  Ord  ( A  .o  C
) )
1816, 17syl 15 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  On )  ->  Ord  ( A  .o  C ) )
19 ordtri3 4531 . . . . . 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 463 . . . . 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 1238 . . . 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 451 . . 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 4505 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
24 eloni 4505 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
25 ordtri3 4531 . . . . . 6  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2623, 24, 25syl2an 463 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
27263adant1 974 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
2827adantr 451 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2912, 22, 283imtr4d 259 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C
)  ->  B  =  C ) )
30 oveq2 5989 . 2  |-  ( B  =  C  ->  ( A  .o  B )  =  ( A  .o  C
) )
3129, 30impbid1 194 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 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   (/)c0 3543   Ord word 4494   Oncon0 4495  (class class class)co 5981    .o comu 6619
This theorem is referenced by:  omword  6710  fin1a2lem4  8176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-recs 6530  df-rdg 6565  df-oadd 6625  df-omul 6626
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