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Theorem mulpipq 7292
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)
Assertion
Ref Expression
mulpipq  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( A  .N  C ) ,  ( B  .N  D ) >. )

Proof of Theorem mulpipq
StepHypRef Expression
1 opelxpi 4618 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 opelxpi 4618 . . 3  |-  ( ( C  e.  N.  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( N.  X.  N. ) )
3 mulpipq2 7291 . . 3  |-  ( (
<. A ,  B >.  e.  ( N.  X.  N. )  /\  <. C ,  D >.  e.  ( N.  X.  N. ) )  ->  ( <. A ,  B >.  .pQ 
<. C ,  D >. )  =  <. ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. ) ) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
) >. )
41, 2, 3syl2an 287 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. )
) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. ) ) >. )
5 op1stg 6098 . . . 4  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 1st `  <. A ,  B >. )  =  A )
6 op1stg 6098 . . . 4  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 1st `  <. C ,  D >. )  =  C )
75, 6oveqan12d 5843 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. ) )  =  ( A  .N  C ) )
8 op2ndg 6099 . . . 4  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
9 op2ndg 6099 . . . 4  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 2nd `  <. C ,  D >. )  =  D )
108, 9oveqan12d 5843 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. ) )  =  ( B  .N  D ) )
117, 10opeq12d 3749 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  <. ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. ) ) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
) >.  =  <. ( A  .N  C ) ,  ( B  .N  D
) >. )
124, 11eqtrd 2190 1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( A  .N  C ) ,  ( B  .N  D ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   <.cop 3563    X. cxp 4584   ` cfv 5170  (class class class)co 5824   1stc1st 6086   2ndc2nd 6087   N.cnpi 7192    .N cmi 7194    .pQ cmpq 7197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-oadd 6367  df-omul 6368  df-ni 7224  df-mi 7226  df-mpq 7265
This theorem is referenced by: (None)
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