MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omopthi Unicode version

Theorem omopthi 6609
Description: An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11237. (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 6566 . . . . . . . . . . . 12  |-  ( A  +o  B )  e. 
om
43nnoni 4621 . . . . . . . . . . 11  |-  ( A  +o  B )  e.  On
54onordi 4455 . . . . . . . . . 10  |-  Ord  ( A  +o  B )
6 omopth.3 . . . . . . . . . . . . 13  |-  C  e. 
om
7 omopth.4 . . . . . . . . . . . . 13  |-  D  e. 
om
86, 7nnacli 6566 . . . . . . . . . . . 12  |-  ( C  +o  D )  e. 
om
98nnoni 4621 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
109onordi 4455 . . . . . . . . . 10  |-  Ord  ( C  +o  D )
11 ordtri3 4386 . . . . . . . . . 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 6608 . . . . . . . . . 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 2258 . . . . . . . . . 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 6608 . . . . . . . . 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 5796 . . . . . . . . . 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 5793 . . . . . . . . 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 2291 . . . . . . . 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 6567 . . . . . . . . 9  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
26 nnacan 6580 . . . . . . . . 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 5794 . . . . . 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 2291 . . . . 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 6569 . . . . . 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 6569 . . . . . 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 2313 . . . 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 6580 . . . . 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 5787 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +o  B
)  =  ( C  +o  D ) )
4140, 40oveq12d 5796 . . 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 5796 . 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 1619    e. wcel 1621   Ord word 4349   omcom 4614  (class class class)co 5778    +o coa 6430    .o comu 6431
This theorem is referenced by:  omopth  6610
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-omul 6438
  Copyright terms: Public domain W3C validator