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Theorem omcan 6569
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 6566 . . . . . . . . 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 974 . . . . . 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 6566 . . . . . . . . 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 975 . . . . . 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 6537 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
14 eloni 4404 . . . . . . 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 6537 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  .o  C
)  e.  On )
17 eloni 4404 . . . . . . 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 4430 . . . . . 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 1237 . . . 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 4404 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
24 eloni 4404 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
25 ordtri3 4430 . . . . . 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 973 . . . 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 5868 . 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 934    = wceq 1625    e. wcel 1686   (/)c0 3457   Ord word 4393   Oncon0 4394  (class class class)co 5860    .o comu 6479
This theorem is referenced by:  omword  6570  fin1a2lem4  8031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-oadd 6485  df-omul 6486
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