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Theorem mulpipq 7385
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 4670 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 opelxpi 4670 . . 3  |-  ( ( C  e.  N.  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( N.  X.  N. ) )
3 mulpipq2 7384 . . 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 289 . 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 6165 . . . 4  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 1st `  <. A ,  B >. )  =  A )
6 op1stg 6165 . . . 4  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 1st `  <. C ,  D >. )  =  C )
75, 6oveqan12d 5907 . . 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 6166 . . . 4  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
9 op2ndg 6166 . . . 4  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 2nd `  <. C ,  D >. )  =  D )
108, 9oveqan12d 5907 . . 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 3798 . 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 2220 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 104    = wceq 1363    e. wcel 2158   <.cop 3607    X. cxp 4636   ` cfv 5228  (class class class)co 5888   1stc1st 6153   2ndc2nd 6154   N.cnpi 7285    .N cmi 7287    .pQ cmpq 7290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-ni 7317  df-mi 7319  df-mpq 7358
This theorem is referenced by: (None)
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