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Theorem dfmpq2 6339
Description: Alternative definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
Assertion
Ref Expression
dfmpq2  .pQ  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  ,  .N  >. }
Distinct variable group:   ,,,,,,

Proof of Theorem dfmpq2
StepHypRef Expression
1 df-mpt2 5460 . 2  N.  X.  N. ,  N.  X.  N.  |->  <. 1st `  .N  1st `  ,  2nd `  .N  2nd ` 
>.  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  N.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >. }
2 df-mpq 6329 . 2  .pQ  N.  X.  N. ,  N. 
X.  N.  |->  <. 1st `  .N  1st `  ,  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  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  ,  .N  >.
98bicomd 129 . . . . . 6  N. 
X.  N.  N.  X.  N.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  ,  .N  >. 
<. ,  >.  <. ,  >.  <.  .N  ,  .N  >.
1094exbidv 1747 . . . . 5  N. 
X.  N.  N.  X.  N. 
<. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  ,  .N  >.  <. ,  >.  <. , 
>.  <.  .N  ,  .N  >.
11 xp1st 5734 . . . . . . 7  N.  X.  N.  1st ` 
N.
12 xp2nd 5735 . . . . . . 7  N.  X.  N.  2nd ` 
N.
1311, 12jca 290 . . . . . 6  N.  X.  N.  1st `  N.  2nd ` 
N.
14 xp1st 5734 . . . . . . 7  N.  X.  N.  1st ` 
N.
15 xp2nd 5735 . . . . . . 7  N.  X.  N.  2nd ` 
N.
1614, 15jca 290 . . . . . 6  N.  X.  N.  1st `  N.  2nd ` 
N.
17 simpll 481 . . . . . . . . . 10  1st `  2nd `  1st `  2nd `  1st `
18 simprl 483 . . . . . . . . . 10  1st `  2nd `  1st `  2nd `  1st `
1917, 18oveq12d 5473 . . . . . . . . 9  1st `  2nd `  1st `  2nd `  .N  1st `  .N  1st `
20 simplr 482 . . . . . . . . . 10  1st `  2nd `  1st `  2nd `  2nd `
21 simprr 484 . . . . . . . . . 10  1st `  2nd `  1st `  2nd `  2nd `
2220, 21oveq12d 5473 . . . . . . . . 9  1st `  2nd `  1st `  2nd `  .N  2nd `  .N  2nd `
2319, 22opeq12d 3548 . . . . . . . 8  1st `  2nd `  1st `  2nd `  <.  .N  ,  .N  >.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
2423eqeq2d 2048 . . . . . . 7  1st `  2nd `  1st `  2nd `  <.  .N  ,  .N  >.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd ` 
>.
2524copsex4g 3975 . . . . . 6  1st `  N.  2nd ` 
N.  1st `  N.  2nd ` 
N.  <. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  ,  .N  >.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
2613, 16, 25syl2an 273 . . . . 5  N. 
X.  N.  N.  X.  N. 
<. 1st `  ,  2nd `  >.  <. ,  >.  <. 1st `  ,  2nd `  >.  <. ,  >.  <.  .N  ,  .N  >.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
2710, 26bitr3d 179 . . . 4  N. 
X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  ,  .N  >.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
2827pm5.32i 427 . . 3  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  ,  .N  >.  N.  X.  N.  N.  X.  N.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd ` 
>.
2928oprabbii 5502 . 2  { <. <. ,  >. ,  >.  |  N.  X.  N.  N.  X.  N.  <. ,  >.  <. , 
>.  <.  .N  ,  .N  >. }  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  N.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >. }
301, 2, 293eqtr4i 2067 1  .pQ  { <. <. , 
>. ,  >.  |  N.  X.  N.  N.  X.  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 cmi 6258    .pQ cmpq 6261
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-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-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-mpq 6329
This theorem is referenced by:  mulpipqqs  6357
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