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Theorem addpipqqs 7290
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 7289 . 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 7289 . 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 7289 . 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 7280 . 2  |-  ~Q  e.  _V
5 enqer 7278 . 2  |-  ~Q  Er  ( N.  X.  N. )
6 df-enq 7267 . 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 5833 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  .N  u
)  =  ( a  .N  d ) )
8 oveq12 5833 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  .N  v
)  =  ( b  .N  c ) )
97, 8eqeqan12d 2173 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
109an42s 579 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
11 oveq12 5833 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  .N  u
)  =  ( g  .N  s ) )
12 oveq12 5833 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  .N  v
)  =  ( h  .N  t ) )
1311, 12eqeqan12d 2173 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
1413an42s 579 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
15 dfplpq2 7274 . 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 5833 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .N  f
)  =  ( a  .N  h ) )
17 oveq12 5833 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .N  u
)  =  ( b  .N  g ) )
1816, 17oveqan12d 5843 . . . 4  |-  ( ( ( w  =  a  /\  f  =  h )  /\  ( v  =  b  /\  u  =  g ) )  ->  ( ( w  .N  f )  +N  ( v  .N  u
) )  =  ( ( a  .N  h
)  +N  ( b  .N  g ) ) )
1918an42s 579 . . 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 5833 . . . 4  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  .N  f
)  =  ( b  .N  h ) )
2120ad2ant2l 500 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( v  .N  f
)  =  ( b  .N  h ) )
2219, 21opeq12d 3749 . 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 5833 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .N  f
)  =  ( c  .N  s ) )
24 oveq12 5833 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .N  u
)  =  ( d  .N  t ) )
2523, 24oveqan12d 5843 . . . 4  |-  ( ( ( w  =  c  /\  f  =  s )  /\  ( v  =  d  /\  u  =  t ) )  ->  ( ( w  .N  f )  +N  ( v  .N  u
) )  =  ( ( c  .N  s
)  +N  ( d  .N  t ) ) )
2625an42s 579 . . 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 5833 . . . 4  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  .N  f
)  =  ( d  .N  s ) )
2827ad2ant2l 500 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( v  .N  f )  =  ( d  .N  s ) )
2926, 28opeq12d 3749 . 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 5833 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .N  f
)  =  ( A  .N  D ) )
31 oveq12 5833 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .N  u
)  =  ( B  .N  C ) )
3230, 31oveqan12d 5843 . . . 4  |-  ( ( ( w  =  A  /\  f  =  D )  /\  ( v  =  B  /\  u  =  C ) )  -> 
( ( w  .N  f )  +N  (
v  .N  u ) )  =  ( ( A  .N  D )  +N  ( B  .N  C ) ) )
3332an42s 579 . . 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 5833 . . . 4  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  .N  f
)  =  ( B  .N  D ) )
3534ad2ant2l 500 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( v  .N  f
)  =  ( B  .N  D ) )
3633, 35opeq12d 3749 . 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 7269 . 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 7268 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
39 addcmpblnq 7287 . 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 6586 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 103    <-> wb 104    = wceq 1335    e. wcel 2128   <.cop 3563  (class class class)co 5824   [cec 6478   N.cnpi 7192    +N cpli 7193    .N cmi 7194    +pQ cplpq 7196    ~Q ceq 7199   Q.cnq 7200    +Q cplq 7202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-plpq 7264  df-enq 7267  df-nqqs 7268  df-plqqs 7269
This theorem is referenced by:  addclnq  7295  addcomnqg  7301  addassnqg  7302  distrnqg  7307  ltanqg  7320  1lt2nq  7326  ltexnqq  7328  nqnq0a  7374  addpinq1  7384
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