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Theorem mulpipqqs 6529
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 6484 . . . 4  |-  ( ( A  e.  N.  /\  C  e.  N. )  ->  ( A  .N  C
)  e.  N. )
2 mulclpi 6484 . . . 4  |-  ( ( B  e.  N.  /\  D  e.  N. )  ->  ( B  .N  D
)  e.  N. )
3 opelxpi 4404 . . . 4  |-  ( ( ( A  .N  C
)  e.  N.  /\  ( B  .N  D
)  e.  N. )  -> 
<. ( A  .N  C
) ,  ( B  .N  D ) >.  e.  ( N.  X.  N. ) )
41, 2, 3syl2an 277 . . 3  |-  ( ( ( A  e.  N.  /\  C  e.  N. )  /\  ( B  e.  N.  /\  D  e.  N. )
)  ->  <. ( A  .N  C ) ,  ( B  .N  D
) >.  e.  ( N. 
X.  N. ) )
54an4s 530 . 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 6484 . . . 4  |-  ( ( a  e.  N.  /\  g  e.  N. )  ->  ( a  .N  g
)  e.  N. )
7 mulclpi 6484 . . . 4  |-  ( ( b  e.  N.  /\  h  e.  N. )  ->  ( b  .N  h
)  e.  N. )
8 opelxpi 4404 . . . 4  |-  ( ( ( a  .N  g
)  e.  N.  /\  ( b  .N  h
)  e.  N. )  -> 
<. ( a  .N  g
) ,  ( b  .N  h ) >.  e.  ( N.  X.  N. ) )
96, 7, 8syl2an 277 . . 3  |-  ( ( ( a  e.  N.  /\  g  e.  N. )  /\  ( b  e.  N.  /\  h  e.  N. )
)  ->  <. ( a  .N  g ) ,  ( b  .N  h
) >.  e.  ( N. 
X.  N. ) )
109an4s 530 . 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 6484 . . . 4  |-  ( ( c  e.  N.  /\  t  e.  N. )  ->  ( c  .N  t
)  e.  N. )
12 mulclpi 6484 . . . 4  |-  ( ( d  e.  N.  /\  s  e.  N. )  ->  ( d  .N  s
)  e.  N. )
13 opelxpi 4404 . . . 4  |-  ( ( ( c  .N  t
)  e.  N.  /\  ( d  .N  s
)  e.  N. )  -> 
<. ( c  .N  t
) ,  ( d  .N  s ) >.  e.  ( N.  X.  N. ) )
1411, 12, 13syl2an 277 . . 3  |-  ( ( ( c  e.  N.  /\  t  e.  N. )  /\  ( d  e.  N.  /\  s  e.  N. )
)  ->  <. ( c  .N  t ) ,  ( d  .N  s
) >.  e.  ( N. 
X.  N. ) )
1514an4s 530 . 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 6516 . 2  |-  ~Q  e.  _V
17 enqer 6514 . 2  |-  ~Q  Er  ( N.  X.  N. )
18 df-enq 6503 . 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 489 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  z  =  a )
20 simprr 492 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  u  =  d )
2119, 20oveq12d 5558 . . 3  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( z  .N  u )  =  ( a  .N  d ) )
22 simplr 490 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  w  =  b )
23 simprl 491 . . . 4  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  v  =  c )
2422, 23oveq12d 5558 . . 3  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( w  .N  v )  =  ( b  .N  c ) )
2521, 24eqeq12d 2070 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
26 simpll 489 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  z  =  g )
27 simprr 492 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  u  =  s )
2826, 27oveq12d 5558 . . 3  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( z  .N  u )  =  ( g  .N  s ) )
29 simplr 490 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  w  =  h )
30 simprl 491 . . . 4  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  v  =  t )
3129, 30oveq12d 5558 . . 3  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( w  .N  v )  =  ( h  .N  t ) )
3228, 31eqeq12d 2070 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
33 dfmpq2 6511 . 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 489 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  w  =  a )
35 simprl 491 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  u  =  g )
3634, 35oveq12d 5558 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( w  .N  u
)  =  ( a  .N  g ) )
37 simplr 490 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
v  =  b )
38 simprr 492 . . . 4  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
f  =  h )
3937, 38oveq12d 5558 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( v  .N  f
)  =  ( b  .N  h ) )
4036, 39opeq12d 3585 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( w  .N  u
) ,  ( v  .N  f ) >.  =  <. ( a  .N  g ) ,  ( b  .N  h )
>. )
41 simpll 489 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  w  =  c )
42 simprl 491 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  u  =  t )
4341, 42oveq12d 5558 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( w  .N  u )  =  ( c  .N  t ) )
44 simplr 490 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  v  =  d )
45 simprr 492 . . . 4  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  f  =  s )
4644, 45oveq12d 5558 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( v  .N  f )  =  ( d  .N  s ) )
4743, 46opeq12d 3585 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( w  .N  u ) ,  ( v  .N  f )
>.  =  <. ( c  .N  t ) ,  ( d  .N  s
) >. )
48 simpll 489 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  w  =  A )
49 simprl 491 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  u  =  C )
5048, 49oveq12d 5558 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( w  .N  u
)  =  ( A  .N  C ) )
51 simplr 490 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
v  =  B )
52 simprr 492 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
f  =  D )
5351, 52oveq12d 5558 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( v  .N  f
)  =  ( B  .N  D ) )
5450, 53opeq12d 3585 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  .N  u
) ,  ( v  .N  f ) >.  =  <. ( A  .N  C ) ,  ( B  .N  D )
>. )
55 df-mqqs 6506 . 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 6504 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
57 mulcmpblnq 6524 . 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 6243 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 101    = wceq 1259    e. wcel 1409   <.cop 3406    X. cxp 4371  (class class class)co 5540   [cec 6135   N.cnpi 6428    .N cmi 6430    .pQ cmpq 6433    ~Q ceq 6435   Q.cnq 6436    .Q cmq 6439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-mqqs 6506
This theorem is referenced by:  mulclnq  6532  mulcomnqg  6539  mulassnqg  6540  distrnqg  6543  mulidnq  6545  recexnq  6546  ltmnqg  6557  nqnq0m  6611
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