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Theorem omopthi 6859
Description: An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11518. (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 6816 . . . . . . . . . . . 12  |-  ( A  +o  B )  e. 
om
43nnoni 4811 . . . . . . . . . . 11  |-  ( A  +o  B )  e.  On
54onordi 4645 . . . . . . . . . 10  |-  Ord  ( A  +o  B )
6 omopth.3 . . . . . . . . . . . . 13  |-  C  e. 
om
7 omopth.4 . . . . . . . . . . . . 13  |-  D  e. 
om
86, 7nnacli 6816 . . . . . . . . . . . 12  |-  ( C  +o  D )  e. 
om
98nnoni 4811 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
109onordi 4645 . . . . . . . . . 10  |-  Ord  ( C  +o  D )
11 ordtri3 4577 . . . . . . . . . 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 654 . . . . . . . . 9  |-  ( ( A  +o  B )  =  ( C  +o  D )  <->  -.  (
( A  +o  B
)  e.  ( C  +o  D )  \/  ( C  +o  D
)  e.  ( A  +o  B ) ) )
1312con2bii 323 . . . . . . . 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 6858 . . . . . . . . . 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 2406 . . . . . . . . . 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 296 . . . . . . . . 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 6858 . . . . . . . . 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 369 . . . . . . . 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 205 . . . . . . 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 20 . . . . . . . . 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 6058 . . . . . . . . . 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 6055 . . . . . . . . 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 2439 . . . . . . . 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 6817 . . . . . . . . 9  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
26 nnacan 6830 . . . . . . . . 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 1279 . . . . . . . 8  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  D
)  <->  B  =  D
)
2824, 27sylib 189 . . . . . . 7  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  B  =  D )
2928oveq2d 6056 . . . . . 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 2439 . . . . 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 6819 . . . . . 6  |-  ( ( B  e.  om  /\  A  e.  om )  ->  ( B  +o  A
)  =  ( A  +o  B ) )
322, 1, 31mp2an 654 . . . . 5  |-  ( B  +o  A )  =  ( A  +o  B
)
33 nnacom 6819 . . . . . 6  |-  ( ( B  e.  om  /\  C  e.  om )  ->  ( B  +o  C
)  =  ( C  +o  B ) )
342, 6, 33mp2an 654 . . . . 5  |-  ( B  +o  C )  =  ( C  +o  B
)
3530, 32, 343eqtr4g 2461 . . . 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 6830 . . . . 5  |-  ( ( B  e.  om  /\  A  e.  om  /\  C  e.  om )  ->  (
( B  +o  A
)  =  ( B  +o  C )  <->  A  =  C ) )
372, 1, 6, 36mp3an 1279 . . . 4  |-  ( ( B  +o  A )  =  ( B  +o  C )  <->  A  =  C )
3835, 37sylib 189 . . 3  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  A  =  C )
3938, 28jca 519 . 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 6049 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +o  B
)  =  ( C  +o  D ) )
4140, 40oveq12d 6058 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  +o  B )  .o  ( A  +o  B ) )  =  ( ( C  +o  D )  .o  ( C  +o  D
) ) )
42 simpr 448 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
4341, 42oveq12d 6058 . 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 181 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 3    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   Ord word 4540   omcom 4804  (class class class)co 6040    +o coa 6680    .o comu 6681
This theorem is referenced by:  omopth  6860
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-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-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688
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