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Theorem mulpipq2 6355
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
Assertion
Ref Expression
mulpipq2  N. 
X.  N.  N.  X.  N.  .pQ 
<. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.

Proof of Theorem mulpipq2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 5734 . . . 4  N.  X.  N.  1st ` 
N.
2 xp1st 5734 . . . 4  N.  X.  N.  1st ` 
N.
3 mulclpi 6312 . . . 4  1st `  N.  1st ` 
N.  1st `  .N  1st `  N.
41, 2, 3syl2an 273 . . 3  N. 
X.  N.  N.  X.  N.  1st `  .N  1st `  N.
5 xp2nd 5735 . . . 4  N.  X.  N.  2nd ` 
N.
6 xp2nd 5735 . . . 4  N.  X.  N.  2nd ` 
N.
7 mulclpi 6312 . . . 4  2nd `  N.  2nd ` 
N.  2nd `  .N  2nd `  N.
85, 6, 7syl2an 273 . . 3  N. 
X.  N.  N.  X.  N.  2nd `  .N  2nd `  N.
9 opexg 3955 . . 3  1st `  .N  1st `  N.  2nd `  .N  2nd `  N.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd ` 
>.  _V
104, 8, 9syl2anc 391 . 2  N. 
X.  N.  N.  X.  N.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.  _V
11 fveq2 5121 . . . . 5  1st `  1st `
1211oveq1d 5470 . . . 4  1st `  .N  1st `  1st `  .N  1st `
13 fveq2 5121 . . . . 5  2nd `  2nd `
1413oveq1d 5470 . . . 4  2nd `  .N  2nd `  2nd `  .N  2nd `
1512, 14opeq12d 3548 . . 3  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
16 fveq2 5121 . . . . 5  1st `  1st `
1716oveq2d 5471 . . . 4  1st `  .N  1st `  1st `  .N  1st `
18 fveq2 5121 . . . . 5  2nd `  2nd `
1918oveq2d 5471 . . . 4  2nd `  .N  2nd `  2nd `  .N  2nd `
2017, 19opeq12d 3548 . . 3  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
21 df-mpq 6329 . . 3  .pQ  N.  X.  N. ,  N. 
X.  N.  |->  <. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
2215, 20, 21ovmpt2g 5577 . 2  N. 
X.  N.  N.  X.  N.  <. 1st `  .N  1st `  ,  2nd `  .N  2nd ` 
>.  _V  .pQ  <. 1st `  .N  1st `  ,  2nd `  .N  2nd ` 
>.
2310, 22mpd3an3 1232 1  N. 
X.  N.  N.  X.  N.  .pQ 
<. 1st `  .N  1st `  ,  2nd `  .N  2nd `  >.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   _Vcvv 2551   <.cop 3370    X. cxp 4286   ` cfv 4845  (class class class)co 5455   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-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-dc 742  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-mpq 6329
This theorem is referenced by:  mulpipq  6356
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