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Theorem addpipqqs 6908
Description: Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
Assertion
Ref Expression
addpipqqs  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( [ <. A ,  B >. ]  ~Q  +Q  [ <. C ,  D >. ]  ~Q  )  =  [ <. (
( A  .N  D
)  +N  ( B  .N  C ) ) ,  ( B  .N  D ) >. ]  ~Q  )

Proof of Theorem addpipqqs
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 addpipqqslem 6907 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  <. ( ( A  .N  D )  +N  ( B  .N  C ) ) ,  ( B  .N  D
) >.  e.  ( N. 
X.  N. ) )
2 addpipqqslem 6907 . 2  |-  ( ( ( a  e.  N.  /\  b  e.  N. )  /\  ( g  e.  N.  /\  h  e.  N. )
)  ->  <. ( ( a  .N  h )  +N  ( b  .N  g ) ) ,  ( b  .N  h
) >.  e.  ( N. 
X.  N. ) )
3 addpipqqslem 6907 . 2  |-  ( ( ( c  e.  N.  /\  d  e.  N. )  /\  ( t  e.  N.  /\  s  e.  N. )
)  ->  <. ( ( c  .N  s )  +N  ( d  .N  t ) ) ,  ( d  .N  s
) >.  e.  ( N. 
X.  N. ) )
4 enqex 6898 . 2  |-  ~Q  e.  _V
5 enqer 6896 . 2  |-  ~Q  Er  ( N.  X.  N. )
6 df-enq 6885 . 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 ) ) ) }
7 oveq12 5643 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  .N  u
)  =  ( a  .N  d ) )
8 oveq12 5643 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  .N  v
)  =  ( b  .N  c ) )
97, 8eqeqan12d 2103 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
109an42s 556 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
11 oveq12 5643 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  .N  u
)  =  ( g  .N  s ) )
12 oveq12 5643 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  .N  v
)  =  ( h  .N  t ) )
1311, 12eqeqan12d 2103 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
1413an42s 556 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
15 dfplpq2 6892 . 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  f
)  +N  ( v  .N  u ) ) ,  ( v  .N  f ) >. )
) }
16 oveq12 5643 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .N  f
)  =  ( a  .N  h ) )
17 oveq12 5643 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .N  u
)  =  ( b  .N  g ) )
1816, 17oveqan12d 5653 . . . 4  |-  ( ( ( w  =  a  /\  f  =  h )  /\  ( v  =  b  /\  u  =  g ) )  ->  ( ( w  .N  f )  +N  ( v  .N  u
) )  =  ( ( a  .N  h
)  +N  ( b  .N  g ) ) )
1918an42s 556 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( ( w  .N  f )  +N  (
v  .N  u ) )  =  ( ( a  .N  h )  +N  ( b  .N  g ) ) )
20 oveq12 5643 . . . 4  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  .N  f
)  =  ( b  .N  h ) )
2120ad2ant2l 492 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( v  .N  f
)  =  ( b  .N  h ) )
2219, 21opeq12d 3625 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( ( w  .N  f )  +N  (
v  .N  u ) ) ,  ( v  .N  f ) >.  =  <. ( ( a  .N  h )  +N  ( b  .N  g
) ) ,  ( b  .N  h )
>. )
23 oveq12 5643 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .N  f
)  =  ( c  .N  s ) )
24 oveq12 5643 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .N  u
)  =  ( d  .N  t ) )
2523, 24oveqan12d 5653 . . . 4  |-  ( ( ( w  =  c  /\  f  =  s )  /\  ( v  =  d  /\  u  =  t ) )  ->  ( ( w  .N  f )  +N  ( v  .N  u
) )  =  ( ( c  .N  s
)  +N  ( d  .N  t ) ) )
2625an42s 556 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( ( w  .N  f )  +N  ( v  .N  u
) )  =  ( ( c  .N  s
)  +N  ( d  .N  t ) ) )
27 oveq12 5643 . . . 4  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  .N  f
)  =  ( d  .N  s ) )
2827ad2ant2l 492 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( v  .N  f )  =  ( d  .N  s ) )
2926, 28opeq12d 3625 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( ( w  .N  f )  +N  ( v  .N  u
) ) ,  ( v  .N  f )
>.  =  <. ( ( c  .N  s )  +N  ( d  .N  t ) ) ,  ( d  .N  s
) >. )
30 oveq12 5643 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .N  f
)  =  ( A  .N  D ) )
31 oveq12 5643 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .N  u
)  =  ( B  .N  C ) )
3230, 31oveqan12d 5653 . . . 4  |-  ( ( ( w  =  A  /\  f  =  D )  /\  ( v  =  B  /\  u  =  C ) )  -> 
( ( w  .N  f )  +N  (
v  .N  u ) )  =  ( ( A  .N  D )  +N  ( B  .N  C ) ) )
3332an42s 556 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( ( w  .N  f )  +N  (
v  .N  u ) )  =  ( ( A  .N  D )  +N  ( B  .N  C ) ) )
34 oveq12 5643 . . . 4  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  .N  f
)  =  ( B  .N  D ) )
3534ad2ant2l 492 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( v  .N  f
)  =  ( B  .N  D ) )
3633, 35opeq12d 3625 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( ( w  .N  f )  +N  (
v  .N  u ) ) ,  ( v  .N  f ) >.  =  <. ( ( A  .N  D )  +N  ( B  .N  C
) ) ,  ( B  .N  D )
>. )
37 df-plqqs 6887 . 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  )
) }
38 df-nqqs 6886 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
39 addcmpblnq 6905 . 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  h )  +N  ( b  .N  g ) ) ,  ( b  .N  h
) >.  ~Q  <. ( ( c  .N  s )  +N  ( d  .N  t ) ) ,  ( d  .N  s
) >. ) )
401, 2, 3, 4, 5, 6, 10, 14, 15, 22, 29, 36, 37, 38, 39oviec 6378 1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( [ <. A ,  B >. ]  ~Q  +Q  [ <. C ,  D >. ]  ~Q  )  =  [ <. (
( A  .N  D
)  +N  ( B  .N  C ) ) ,  ( B  .N  D ) >. ]  ~Q  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   <.cop 3444  (class class class)co 5634   [cec 6270   N.cnpi 6810    +N cpli 6811    .N cmi 6812    +pQ cplpq 6814    ~Q ceq 6817   Q.cnq 6818    +Q cplq 6820
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 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
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 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-plpq 6882  df-enq 6885  df-nqqs 6886  df-plqqs 6887
This theorem is referenced by:  addclnq  6913  addcomnqg  6919  addassnqg  6920  distrnqg  6925  ltanqg  6938  1lt2nq  6944  ltexnqq  6946  nqnq0a  6992  addpinq1  7002
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