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Theorem omopthi 7242
Description: An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 12271. (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 7199 . . . . . . . . . . . 12  |-  ( A  +o  B )  e. 
om
43nnoni 6624 . . . . . . . . . . 11  |-  ( A  +o  B )  e.  On
54onordi 4909 . . . . . . . . . 10  |-  Ord  ( A  +o  B )
6 omopth.3 . . . . . . . . . . . . 13  |-  C  e. 
om
7 omopth.4 . . . . . . . . . . . . 13  |-  D  e. 
om
86, 7nnacli 7199 . . . . . . . . . . . 12  |-  ( C  +o  D )  e. 
om
98nnoni 6624 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
109onordi 4909 . . . . . . . . . 10  |-  Ord  ( C  +o  D )
11 ordtri3 4841 . . . . . . . . . 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 670 . . . . . . . . 9  |-  ( ( A  +o  B )  =  ( C  +o  D )  <->  -.  (
( A  +o  B
)  e.  ( C  +o  D )  \/  ( C  +o  D
)  e.  ( A  +o  B ) ) )
1312con2bii 330 . . . . . . . 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 7241 . . . . . . . . . 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 2401 . . . . . . . . . 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 302 . . . . . . . . 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 7241 . . . . . . . . 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 377 . . . . . . . 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 213 . . . . . . 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 130 . . . . . 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 22 . . . . . . . . 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 6232 . . . . . . . . . 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 6229 . . . . . . . . 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 2436 . . . . . . . 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 7200 . . . . . . . . 9  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
26 nnacan 7213 . . . . . . . . 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 1322 . . . . . . . 8  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  D
)  <->  B  =  D
)
2824, 27sylib 196 . . . . . . 7  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  B  =  D )
2928oveq2d 6230 . . . . . 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 2436 . . . . 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 7202 . . . . . 6  |-  ( ( B  e.  om  /\  A  e.  om )  ->  ( B  +o  A
)  =  ( A  +o  B ) )
322, 1, 31mp2an 670 . . . . 5  |-  ( B  +o  A )  =  ( A  +o  B
)
33 nnacom 7202 . . . . . 6  |-  ( ( B  e.  om  /\  C  e.  om )  ->  ( B  +o  C
)  =  ( C  +o  B ) )
342, 6, 33mp2an 670 . . . . 5  |-  ( B  +o  C )  =  ( C  +o  B
)
3530, 32, 343eqtr4g 2458 . . . 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 7213 . . . . 5  |-  ( ( B  e.  om  /\  A  e.  om  /\  C  e.  om )  ->  (
( B  +o  A
)  =  ( B  +o  C )  <->  A  =  C ) )
372, 1, 6, 36mp3an 1322 . . . 4  |-  ( ( B  +o  A )  =  ( B  +o  C )  <->  A  =  C )
3835, 37sylib 196 . . 3  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( ( ( C  +o  D )  .o  ( C  +o  D
) )  +o  D
)  ->  A  =  C )
3938, 28jca 530 . 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 6223 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +o  B
)  =  ( C  +o  D ) )
4140, 40oveq12d 6232 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  +o  B )  .o  ( A  +o  B ) )  =  ( ( C  +o  D )  .o  ( C  +o  D
) ) )
42 simpr 459 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
4341, 42oveq12d 6232 . 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 188 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 setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1836   Ord word 4804  (class class class)co 6214   omcom 6617    +o coa 7063    .o comu 7064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-1st 6717  df-2nd 6718  df-recs 6978  df-rdg 7012  df-1o 7066  df-2o 7067  df-oadd 7070  df-omul 7071
This theorem is referenced by:  omopth  7243
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