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Theorem mulpipqqs 6471
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
Assertion
Ref Expression
mulpipqqs  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( [ <. A ,  B >. ]  ~Q  .Q  [ <. C ,  D >. ]  ~Q  )  =  [ <. ( A  .N  C ) ,  ( B  .N  D
) >. ]  ~Q  )

Proof of Theorem mulpipqqs
Dummy variables  x  y  z  w  v  u  t  s  f  g  h  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpi 6426 . . . 4  |-  ( ( A  e.  N.  /\  C  e.  N. )  ->  ( A  .N  C
)  e.  N. )
2 mulclpi 6426 . . . 4  |-  ( ( B  e.  N.  /\  D  e.  N. )  ->  ( B  .N  D
)  e.  N. )
3 opelxpi 4376 . . . 4  |-  ( ( ( A  .N  C
)  e.  N.  /\  ( B  .N  D
)  e.  N. )  -> 
<. ( A  .N  C
) ,  ( B  .N  D ) >.  e.  ( N.  X.  N. ) )
41, 2, 3syl2an 273 . . 3  |-  ( ( ( A  e.  N.  /\  C  e.  N. )  /\  ( B  e.  N.  /\  D  e.  N. )
)  ->  <. ( A  .N  C ) ,  ( B  .N  D
) >.  e.  ( N. 
X.  N. ) )
54an4s 522 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  <. ( A  .N  C ) ,  ( B  .N  D
) >.  e.  ( N. 
X.  N. ) )
6 mulclpi 6426 . . . 4  |-  ( ( a  e.  N.  /\  g  e.  N. )  ->  ( a  .N  g
)  e.  N. )
7 mulclpi 6426 . . . 4  |-  ( ( b  e.  N.  /\  h  e.  N. )  ->  ( b  .N  h
)  e.  N. )
8 opelxpi 4376 . . . 4  |-  ( ( ( a  .N  g
)  e.  N.  /\  ( b  .N  h
)  e.  N. )  -> 
<. ( a  .N  g
) ,  ( b  .N  h ) >.  e.  ( N.  X.  N. ) )
96, 7, 8syl2an 273 . . 3  |-  ( ( ( a  e.  N.  /\  g  e.  N. )  /\  ( b  e.  N.  /\  h  e.  N. )
)  ->  <. ( a  .N  g ) ,  ( b  .N  h
) >.  e.  ( N. 
X.  N. ) )
109an4s 522 . 2  |-  ( ( ( a  e.  N.  /\  b  e.  N. )  /\  ( g  e.  N.  /\  h  e.  N. )
)  ->  <. ( a  .N  g ) ,  ( b  .N  h
) >.  e.  ( N. 
X.  N. ) )
11 mulclpi 6426 . . . 4  |-  ( ( c  e.  N.  /\  t  e.  N. )  ->  ( c  .N  t
)  e.  N. )
12 mulclpi 6426 . . . 4  |-  ( ( d  e.  N.  /\  s  e.  N. )  ->  ( d  .N  s
)  e.  N. )
13 opelxpi 4376 . . . 4  |-  ( ( ( c  .N  t
)  e.  N.  /\  ( d  .N  s
)  e.  N. )  -> 
<. ( c  .N  t
) ,  ( d  .N  s ) >.  e.  ( N.  X.  N. ) )
1411, 12, 13syl2an 273 . . 3  |-  ( ( ( c  e.  N.  /\  t  e.  N. )  /\  ( d  e.  N.  /\  s  e.  N. )
)  ->  <. ( c  .N  t ) ,  ( d  .N  s
) >.  e.  ( N. 
X.  N. ) )
1514an4s 522 . 2  |-  ( ( ( c  e.  N.  /\  d  e.  N. )  /\  ( t  e.  N.  /\  s  e.  N. )
)  ->  <. ( c  .N  t ) ,  ( d  .N  s
) >.  e.  ( N. 
X.  N. ) )
16 enqex 6458 . 2  |-  ~Q  e.  _V
17 enqer 6456 . 2  |-  ~Q  Er  ( N.  X.  N. )
18 df-enq 6445 . 2  |-  ~Q  =  { <. x ,  y
>.  |  ( (
x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) ) ) }
19 simpll 481 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  z  =  a )
20 simprr 484 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  u  =  d )
2119, 20oveq12d 5530 . . 3  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( z  .N  u )  =  ( a  .N  d ) )
22 simplr 482 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  w  =  b )
23 simprl 483 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  v  =  c )
2422, 23oveq12d 5530 . . 3  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( w  .N  v )  =  ( b  .N  c ) )
2521, 24eqeq12d 2054 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
26 simpll 481 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  z  =  g )
27 simprr 484 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  u  =  s )
2826, 27oveq12d 5530 . . 3  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( z  .N  u )  =  ( g  .N  s ) )
29 simplr 482 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  w  =  h )
30 simprl 483 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  v  =  t )
3129, 30oveq12d 5530 . . 3  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( w  .N  v )  =  ( h  .N  t ) )
3228, 31eqeq12d 2054 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
33 dfmpq2 6453 . 2  |-  .pQ  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )
) }
34 simpll 481 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  w  =  a )
35 simprl 483 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  u  =  g )
3634, 35oveq12d 5530 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( w  .N  u
)  =  ( a  .N  g ) )
37 simplr 482 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
v  =  b )
38 simprr 484 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
f  =  h )
3937, 38oveq12d 5530 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( v  .N  f
)  =  ( b  .N  h ) )
4036, 39opeq12d 3557 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( w  .N  u
) ,  ( v  .N  f ) >.  =  <. ( a  .N  g ) ,  ( b  .N  h )
>. )
41 simpll 481 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  w  =  c )
42 simprl 483 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  u  =  t )
4341, 42oveq12d 5530 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( w  .N  u )  =  ( c  .N  t ) )
44 simplr 482 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  v  =  d )
45 simprr 484 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  f  =  s )
4644, 45oveq12d 5530 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( v  .N  f )  =  ( d  .N  s ) )
4743, 46opeq12d 3557 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( w  .N  u ) ,  ( v  .N  f )
>.  =  <. ( c  .N  t ) ,  ( d  .N  s
) >. )
48 simpll 481 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  w  =  A )
49 simprl 483 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  u  =  C )
5048, 49oveq12d 5530 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( w  .N  u
)  =  ( A  .N  C ) )
51 simplr 482 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
v  =  B )
52 simprr 484 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
f  =  D )
5351, 52oveq12d 5530 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( v  .N  f
)  =  ( B  .N  D ) )
5450, 53opeq12d 3557 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  .N  u
) ,  ( v  .N  f ) >.  =  <. ( A  .N  C ) ,  ( B  .N  D )
>. )
55 df-mqqs 6448 . 2  |-  .Q  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
Q.  /\  y  e.  Q. )  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  ~Q  /\  y  =  [ <. c ,  d >. ]  ~Q  )  /\  z  =  [
( <. a ,  b
>.  .pQ  <. c ,  d
>. ) ]  ~Q  )
) }
56 df-nqqs 6446 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
57 mulcmpblnq 6466 . 2  |-  ( ( ( ( a  e. 
N.  /\  b  e.  N. )  /\  (
c  e.  N.  /\  d  e.  N. )
)  /\  ( (
g  e.  N.  /\  h  e.  N. )  /\  ( t  e.  N.  /\  s  e.  N. )
) )  ->  (
( ( a  .N  d )  =  ( b  .N  c )  /\  ( g  .N  s )  =  ( h  .N  t ) )  ->  <. ( a  .N  g ) ,  ( b  .N  h
) >.  ~Q  <. ( c  .N  t ) ,  ( d  .N  s
) >. ) )
585, 10, 15, 16, 17, 18, 25, 32, 33, 40, 47, 54, 55, 56, 57oviec 6212 1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( [ <. A ,  B >. ]  ~Q  .Q  [ <. C ,  D >. ]  ~Q  )  =  [ <. ( A  .N  C ) ,  ( B  .N  D
) >. ]  ~Q  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   <.cop 3378    X. cxp 4343  (class class class)co 5512   [cec 6104   N.cnpi 6370    .N cmi 6372    .pQ cmpq 6375    ~Q ceq 6377   Q.cnq 6378    .Q cmq 6381
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448
This theorem is referenced by:  mulclnq  6474  mulcomnqg  6481  mulassnqg  6482  distrnqg  6485  mulidnq  6487  recexnq  6488  ltmnqg  6499  nqnq0m  6553
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