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Theorem mulpipqqs 6932
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 6887 . . . 4  |-  ( ( A  e.  N.  /\  C  e.  N. )  ->  ( A  .N  C
)  e.  N. )
2 mulclpi 6887 . . . 4  |-  ( ( B  e.  N.  /\  D  e.  N. )  ->  ( B  .N  D
)  e.  N. )
3 opelxpi 4469 . . . 4  |-  ( ( ( A  .N  C
)  e.  N.  /\  ( B  .N  D
)  e.  N. )  -> 
<. ( A  .N  C
) ,  ( B  .N  D ) >.  e.  ( N.  X.  N. ) )
41, 2, 3syl2an 283 . . 3  |-  ( ( ( A  e.  N.  /\  C  e.  N. )  /\  ( B  e.  N.  /\  D  e.  N. )
)  ->  <. ( A  .N  C ) ,  ( B  .N  D
) >.  e.  ( N. 
X.  N. ) )
54an4s 555 . 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 6887 . . . 4  |-  ( ( a  e.  N.  /\  g  e.  N. )  ->  ( a  .N  g
)  e.  N. )
7 mulclpi 6887 . . . 4  |-  ( ( b  e.  N.  /\  h  e.  N. )  ->  ( b  .N  h
)  e.  N. )
8 opelxpi 4469 . . . 4  |-  ( ( ( a  .N  g
)  e.  N.  /\  ( b  .N  h
)  e.  N. )  -> 
<. ( a  .N  g
) ,  ( b  .N  h ) >.  e.  ( N.  X.  N. ) )
96, 7, 8syl2an 283 . . 3  |-  ( ( ( a  e.  N.  /\  g  e.  N. )  /\  ( b  e.  N.  /\  h  e.  N. )
)  ->  <. ( a  .N  g ) ,  ( b  .N  h
) >.  e.  ( N. 
X.  N. ) )
109an4s 555 . 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 6887 . . . 4  |-  ( ( c  e.  N.  /\  t  e.  N. )  ->  ( c  .N  t
)  e.  N. )
12 mulclpi 6887 . . . 4  |-  ( ( d  e.  N.  /\  s  e.  N. )  ->  ( d  .N  s
)  e.  N. )
13 opelxpi 4469 . . . 4  |-  ( ( ( c  .N  t
)  e.  N.  /\  ( d  .N  s
)  e.  N. )  -> 
<. ( c  .N  t
) ,  ( d  .N  s ) >.  e.  ( N.  X.  N. ) )
1411, 12, 13syl2an 283 . . 3  |-  ( ( ( c  e.  N.  /\  t  e.  N. )  /\  ( d  e.  N.  /\  s  e.  N. )
)  ->  <. ( c  .N  t ) ,  ( d  .N  s
) >.  e.  ( N. 
X.  N. ) )
1514an4s 555 . 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 6919 . 2  |-  ~Q  e.  _V
17 enqer 6917 . 2  |-  ~Q  Er  ( N.  X.  N. )
18 df-enq 6906 . 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 496 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  z  =  a )
20 simprr 499 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  u  =  d )
2119, 20oveq12d 5670 . . 3  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( z  .N  u )  =  ( a  .N  d ) )
22 simplr 497 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  w  =  b )
23 simprl 498 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  v  =  c )
2422, 23oveq12d 5670 . . 3  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( w  .N  v )  =  ( b  .N  c ) )
2521, 24eqeq12d 2102 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
26 simpll 496 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  z  =  g )
27 simprr 499 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  u  =  s )
2826, 27oveq12d 5670 . . 3  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( z  .N  u )  =  ( g  .N  s ) )
29 simplr 497 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  w  =  h )
30 simprl 498 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  v  =  t )
3129, 30oveq12d 5670 . . 3  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( w  .N  v )  =  ( h  .N  t ) )
3228, 31eqeq12d 2102 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
33 dfmpq2 6914 . 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 496 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  w  =  a )
35 simprl 498 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  u  =  g )
3634, 35oveq12d 5670 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( w  .N  u
)  =  ( a  .N  g ) )
37 simplr 497 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
v  =  b )
38 simprr 499 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
f  =  h )
3937, 38oveq12d 5670 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( v  .N  f
)  =  ( b  .N  h ) )
4036, 39opeq12d 3630 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( w  .N  u
) ,  ( v  .N  f ) >.  =  <. ( a  .N  g ) ,  ( b  .N  h )
>. )
41 simpll 496 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  w  =  c )
42 simprl 498 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  u  =  t )
4341, 42oveq12d 5670 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( w  .N  u )  =  ( c  .N  t ) )
44 simplr 497 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  v  =  d )
45 simprr 499 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  f  =  s )
4644, 45oveq12d 5670 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( v  .N  f )  =  ( d  .N  s ) )
4743, 46opeq12d 3630 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( w  .N  u ) ,  ( v  .N  f )
>.  =  <. ( c  .N  t ) ,  ( d  .N  s
) >. )
48 simpll 496 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  w  =  A )
49 simprl 498 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  u  =  C )
5048, 49oveq12d 5670 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( w  .N  u
)  =  ( A  .N  C ) )
51 simplr 497 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
v  =  B )
52 simprr 499 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
f  =  D )
5351, 52oveq12d 5670 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( v  .N  f
)  =  ( B  .N  D ) )
5450, 53opeq12d 3630 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  .N  u
) ,  ( v  .N  f ) >.  =  <. ( A  .N  C ) ,  ( B  .N  D )
>. )
55 df-mqqs 6909 . 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 6907 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
57 mulcmpblnq 6927 . 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 6398 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 102    = wceq 1289    e. wcel 1438   <.cop 3449    X. cxp 4436  (class class class)co 5652   [cec 6290   N.cnpi 6831    .N cmi 6833    .pQ cmpq 6836    ~Q ceq 6838   Q.cnq 6839    .Q cmq 6842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-mi 6865  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-mqqs 6909
This theorem is referenced by:  mulclnq  6935  mulcomnqg  6942  mulassnqg  6943  distrnqg  6946  mulidnq  6948  recexnq  6949  ltmnqg  6960  nqnq0m  7014
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