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Theorem dfplpq2 6338
Description: Alternative definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
Assertion
Ref Expression
dfplpq2  +pQ  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >. }
Distinct variable group:   ,,,,,,

Proof of Theorem dfplpq2
StepHypRef Expression
1 df-mpt2 5460 . 2  N.  X.  N. ,  N.  X.  N.  |->  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  N.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >. }
2 df-plpq 6328 . 2  +pQ  N.  X.  N. ,  N. 
X.  N.  |->  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
3 1st2nd2 5743 . . . . . . . . . 10  N.  X.  N.  <. 1st `  ,  2nd `  >.
43eqeq1d 2045 . . . . . . . . 9  N.  X.  N.  <. ,  >. 
<. 1st `  ,  2nd `  >.  <. ,  >.
5 1st2nd2 5743 . . . . . . . . . 10  N.  X.  N.  <. 1st `  ,  2nd `  >.
65eqeq1d 2045 . . . . . . . . 9  N.  X.  N.  <. ,  >. 
<. 1st `  ,  2nd `  >.  <. ,  >.
74, 6bi2anan9 538 . . . . . . . 8  N. 
X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.
87anbi1d 438 . . . . . . 7  N. 
X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  +N  .N  ,  .N  >.
9 xp1st 5734 . . . . . . . . . . . . . 14  N.  X.  N.  1st ` 
N.
109ad2antlr 458 . . . . . . . . . . . . 13  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  1st ` 
N.
117biimpa 280 . . . . . . . . . . . . . . 15  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.
1211simprd 107 . . . . . . . . . . . . . 14  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <. 1st `  ,  2nd `  >.  <. ,  >.
13 vex 2554 . . . . . . . . . . . . . . . . 17 
_V
14 vex 2554 . . . . . . . . . . . . . . . . 17 
_V
1513, 14opth2 3968 . . . . . . . . . . . . . . . 16  <. 1st `  ,  2nd `  >.  <. ,  >.  1st `  2nd `
1615simplbi 259 . . . . . . . . . . . . . . 15  <. 1st `  ,  2nd `  >.  <. ,  >.  1st `
1716eleq1d 2103 . . . . . . . . . . . . . 14  <. 1st `  ,  2nd `  >.  <. ,  >.  1st `  N. 
N.
1812, 17syl 14 . . . . . . . . . . . . 13  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  1st `  N. 
N.
1910, 18mpbid 135 . . . . . . . . . . . 12  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  N.
20 xp2nd 5735 . . . . . . . . . . . . . 14  N.  X.  N.  2nd ` 
N.
2120ad2antrr 457 . . . . . . . . . . . . 13  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  2nd ` 
N.
2211simpld 105 . . . . . . . . . . . . . 14  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <. 1st `  ,  2nd `  >.  <. ,  >.
23 vex 2554 . . . . . . . . . . . . . . . . 17 
_V
24 vex 2554 . . . . . . . . . . . . . . . . 17 
_V
2523, 24opth2 3968 . . . . . . . . . . . . . . . 16  <. 1st `  ,  2nd `  >.  <. ,  >.  1st `  2nd `
2625simprbi 260 . . . . . . . . . . . . . . 15  <. 1st `  ,  2nd `  >.  <. ,  >.  2nd `
2726eleq1d 2103 . . . . . . . . . . . . . 14  <. 1st `  ,  2nd `  >.  <. ,  >.  2nd `  N. 
N.
2822, 27syl 14 . . . . . . . . . . . . 13  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  2nd `  N. 
N.
2921, 28mpbid 135 . . . . . . . . . . . 12  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  N.
30 mulcompig 6315 . . . . . . . . . . . 12  N.  N.  .N  .N
3119, 29, 30syl2anc 391 . . . . . . . . . . 11  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  .N  .N
3231oveq2d 5471 . . . . . . . . . 10  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  .N  +N  .N  .N  +N  .N
3332opeq1d 3546 . . . . . . . . 9  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N 
>.  <.  .N  +N  .N  ,  .N 
>.
3433eqeq2d 2048 . . . . . . . 8  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N 
>.  <.  .N  +N  .N  ,  .N 
>.
3534pm5.32da 425 . . . . . . 7  N. 
X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >. 
<. ,  >.  <. ,  >.  <.  .N  +N  .N  ,  .N  >.
368, 35bitr3d 179 . . . . . 6  N. 
X.  N.  N.  X.  N.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  +N  .N  ,  .N  >. 
<. ,  >.  <. ,  >.  <.  .N  +N  .N  ,  .N  >.
37364exbidv 1747 . . . . 5  N. 
X.  N.  N.  X.  N. 
<. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  +N  .N  ,  .N  >.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >.
38 xp1st 5734 . . . . . . 7  N.  X.  N.  1st ` 
N.
3938, 20jca 290 . . . . . 6  N.  X.  N.  1st `  N.  2nd ` 
N.
40 xp2nd 5735 . . . . . . 7  N.  X.  N.  2nd ` 
N.
419, 40jca 290 . . . . . 6  N.  X.  N.  1st `  N.  2nd ` 
N.
42 simpll 481 . . . . . . . . . . 11  1st `  2nd `  1st `  2nd `  1st `
43 simprr 484 . . . . . . . . . . 11  1st `  2nd `  1st `  2nd `  2nd `
4442, 43oveq12d 5473 . . . . . . . . . 10  1st `  2nd `  1st `  2nd `  .N  1st `  .N  2nd `
45 simprl 483 . . . . . . . . . . 11  1st `  2nd `  1st `  2nd `  1st `
46 simplr 482 . . . . . . . . . . 11  1st `  2nd `  1st `  2nd `  2nd `
4745, 46oveq12d 5473 . . . . . . . . . 10  1st `  2nd `  1st `  2nd `  .N  1st `  .N  2nd `
4844, 47oveq12d 5473 . . . . . . . . 9  1st `  2nd `  1st `  2nd `  .N  +N  .N  1st `  .N  2nd `  +N  1st `  .N  2nd `
4946, 43oveq12d 5473 . . . . . . . . 9  1st `  2nd `  1st `  2nd `  .N  2nd `  .N  2nd `
5048, 49opeq12d 3548 . . . . . . . 8  1st `  2nd `  1st `  2nd `  <.  .N  +N  .N  ,  .N 
>.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
5150eqeq2d 2048 . . . . . . 7  1st `  2nd `  1st `  2nd `  <.  .N  +N  .N  ,  .N  >.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
5251copsex4g 3975 . . . . . 6  1st `  N.  2nd ` 
N.  1st `  N.  2nd ` 
N.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  +N  .N  ,  .N  >.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
5339, 41, 52syl2an 273 . . . . 5  N. 
X.  N.  N.  X.  N. 
<. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  +N  .N  ,  .N  >.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
5437, 53bitr3d 179 . . . 4  N. 
X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
5554pm5.32i 427 . . 3  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >.  N.  X.  N.  N.  X.  N.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
5655oprabbii 5502 . 2  { <. <. ,  >. ,  >.  |  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >. }  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  N.  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >. }
571, 2, 563eqtr4i 2067 1  +pQ  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  +N  .N  ,  .N  >. }
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   <.cop 3370    X. cxp 4286   ` cfv 4845  (class class class)co 5455   {coprab 5456    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   N.cnpi 6256    +N cpli 6257    .N cmi 6258    +pQ cplpq 6260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-ni 6288  df-mi 6290  df-plpq 6328
This theorem is referenced by:  addpipqqs  6354
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