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Theorem omopthi 6892
Description: An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11555. (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 6849 . . . . . . . . . . . 12  |-  ( A  +o  B )  e. 
om
43nnoni 4844 . . . . . . . . . . 11  |-  ( A  +o  B )  e.  On
54onordi 4678 . . . . . . . . . 10  |-  Ord  ( A  +o  B )
6 omopth.3 . . . . . . . . . . . . 13  |-  C  e. 
om
7 omopth.4 . . . . . . . . . . . . 13  |-  D  e. 
om
86, 7nnacli 6849 . . . . . . . . . . . 12  |-  ( C  +o  D )  e. 
om
98nnoni 4844 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
109onordi 4678 . . . . . . . . . 10  |-  Ord  ( C  +o  D )
11 ordtri3 4609 . . . . . . . . . 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 6891 . . . . . . . . . 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 2437 . . . . . . . . . 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 6891 . . . . . . . . 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 6091 . . . . . . . . . 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 6088 . . . . . . . . 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 2470 . . . . . . . 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 6850 . . . . . . . . 9  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
26 nnacan 6863 . . . . . . . . 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 6089 . . . . . 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 2470 . . . . 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 6852 . . . . . 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 6852 . . . . . 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 2492 . . . 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 6863 . . . . 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 6082 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +o  B
)  =  ( C  +o  D ) )
4140, 40oveq12d 6091 . . 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 6091 . 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 1652    e. wcel 1725   Ord word 4572   omcom 4837  (class class class)co 6073    +o coa 6713    .o comu 6714
This theorem is referenced by:  omopth  6893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721
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