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Theorem omopthi 6650
Description: An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11279. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypotheses
Ref Expression
omopth.1  |-  A  e. 
om
omopth.2  |-  B  e. 
om
omopth.3  |-  C  e. 
om
omopth.4  |-  D  e. 
om
Assertion
Ref Expression
omopthi  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem omopthi
StepHypRef Expression
1 omopth.1 . . . . . . . . . . . . 13  |-  A  e. 
om
2 omopth.2 . . . . . . . . . . . . 13  |-  B  e. 
om
31, 2nnacli 6607 . . . . . . . . . . . 12  |-  ( A  +o  B )  e. 
om
43nnoni 4662 . . . . . . . . . . 11  |-  ( A  +o  B )  e.  On
54onordi 4496 . . . . . . . . . 10  |-  Ord  ( A  +o  B )
6 omopth.3 . . . . . . . . . . . . 13  |-  C  e. 
om
7 omopth.4 . . . . . . . . . . . . 13  |-  D  e. 
om
86, 7nnacli 6607 . . . . . . . . . . . 12  |-  ( C  +o  D )  e. 
om
98nnoni 4662 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
109onordi 4496 . . . . . . . . . 10  |-  Ord  ( C  +o  D )
11 ordtri3 4427 . . . . . . . . . 10  |-  ( ( Ord  ( A  +o  B )  /\  Ord  ( C  +o  D
) )  ->  (
( A  +o  B
)  =  ( C  +o  D )  <->  -.  (
( A  +o  B
)  e.  ( C  +o  D )  \/  ( C  +o  D
)  e.  ( A  +o  B ) ) ) )
125, 10, 11mp2an 656 . . . . . . . . 9  |-  ( ( A  +o  B )  =  ( C  +o  D )  <->  -.  (
( A  +o  B
)  e.  ( C  +o  D )  \/  ( C  +o  D
)  e.  ( A  +o  B ) ) )
1312con2bii 324 . . . . . . . 8  |-  ( ( ( A  +o  B
)  e.  ( C  +o  D )  \/  ( C  +o  D
)  e.  ( A  +o  B ) )  <->  -.  ( A  +o  B
)  =  ( C  +o  D ) )
141, 2, 8, 7omopthlem2 6649 . . . . . . . . . 10  |-  ( ( A  +o  B )  e.  ( C  +o  D )  ->  -.  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  =  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B ) )
15 eqcom 2286 . . . . . . . . . 10  |-  ( ( ( ( C  +o  D )  .o  ( C  +o  D ) )  +o  D )  =  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  <->  ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D ) )  +o  D ) )
1614, 15sylnib 297 . . . . . . . . 9  |-  ( ( A  +o  B )  e.  ( C  +o  D )  ->  -.  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  =  ( ( ( C  +o  D
)  .o  ( C  +o  D ) )  +o  D ) )
176, 7, 3, 2omopthlem2 6649 . . . . . . . . 9  |-  ( ( C  +o  D )  e.  ( A  +o  B )  ->  -.  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  =  ( ( ( C  +o  D
)  .o  ( C  +o  D ) )  +o  D ) )
1816, 17jaoi 370 . . . . . . . 8  |-  ( ( ( A  +o  B
)  e.  ( C  +o  D )  \/  ( C  +o  D
)  e.  ( A  +o  B ) )  ->  -.  ( (
( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
) )
1913, 18sylbir 206 . . . . . . 7  |-  ( -.  ( A  +o  B
)  =  ( C  +o  D )  ->  -.  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  =  ( ( ( C  +o  D
)  .o  ( C  +o  D ) )  +o  D ) )
2019con4i 124 . . . . . 6  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( A  +o  B )  =  ( C  +o  D ) )
21 id 21 . . . . . . . . 9  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( (
( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
) )
2220, 20oveq12d 5837 . . . . . . . . . 10  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( ( A  +o  B )  .o  ( A  +o  B
) )  =  ( ( C  +o  D
)  .o  ( C  +o  D ) ) )
2322oveq1d 5834 . . . . . . . . 9  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( (
( A  +o  B
)  .o  ( A  +o  B ) )  +o  D )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
) )
2421, 23eqtr4d 2319 . . . . . . . 8  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( (
( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  D
) )
253, 3nnmcli 6608 . . . . . . . . 9  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
26 nnacan 6621 . . . . . . . . 9  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  e.  om  /\  B  e.  om  /\  D  e. 
om )  ->  (
( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  =  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  D )  <->  B  =  D ) )
2725, 2, 7, 26mp3an 1282 . . . . . . . 8  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  D
)  <->  B  =  D
)
2824, 27sylib 190 . . . . . . 7  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  B  =  D )
2928oveq2d 5835 . . . . . 6  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( C  +o  B )  =  ( C  +o  D ) )
3020, 29eqtr4d 2319 . . . . 5  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( A  +o  B )  =  ( C  +o  B ) )
31 nnacom 6610 . . . . . 6  |-  ( ( B  e.  om  /\  A  e.  om )  ->  ( B  +o  A
)  =  ( A  +o  B ) )
322, 1, 31mp2an 656 . . . . 5  |-  ( B  +o  A )  =  ( A  +o  B
)
33 nnacom 6610 . . . . . 6  |-  ( ( B  e.  om  /\  C  e.  om )  ->  ( B  +o  C
)  =  ( C  +o  B ) )
342, 6, 33mp2an 656 . . . . 5  |-  ( B  +o  C )  =  ( C  +o  B
)
3530, 32, 343eqtr4g 2341 . . . 4  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( B  +o  A )  =  ( B  +o  C ) )
36 nnacan 6621 . . . . 5  |-  ( ( B  e.  om  /\  A  e.  om  /\  C  e.  om )  ->  (
( B  +o  A
)  =  ( B  +o  C )  <->  A  =  C ) )
372, 1, 6, 36mp3an 1282 . . . 4  |-  ( ( B  +o  A )  =  ( B  +o  C )  <->  A  =  C )
3835, 37sylib 190 . . 3  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  A  =  C )
3938, 28jca 520 . 2  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  ( A  =  C  /\  B  =  D ) )
40 oveq12 5828 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +o  B
)  =  ( C  +o  D ) )
4140, 40oveq12d 5837 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  +o  B )  .o  ( A  +o  B ) )  =  ( ( C  +o  D )  .o  ( C  +o  D
) ) )
42 simpr 449 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
4341, 42oveq12d 5837 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  =  ( ( ( C  +o  D
)  .o  ( C  +o  D ) )  +o  D ) )
4439, 43impbii 182 1  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1628    e. wcel 1688   Ord word 4390   omcom 4655  (class class class)co 5819    +o coa 6471    .o comu 6472
This theorem is referenced by:  omopth  6651
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479
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