ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addpipqqs Unicode version

Theorem addpipqqs 6622
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 6621 . 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 6621 . 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 6621 . 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 6612 . 2  |-  ~Q  e.  _V
5 enqer 6610 . 2  |-  ~Q  Er  ( N.  X.  N. )
6 df-enq 6599 . 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 5552 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  .N  u
)  =  ( a  .N  d ) )
8 oveq12 5552 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  .N  v
)  =  ( b  .N  c ) )
97, 8eqeqan12d 2097 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
109an42s 554 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( a  .N  d )  =  ( b  .N  c ) ) )
11 oveq12 5552 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  .N  u
)  =  ( g  .N  s ) )
12 oveq12 5552 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  .N  v
)  =  ( h  .N  t ) )
1311, 12eqeqan12d 2097 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
1413an42s 554 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  .N  u )  =  ( w  .N  v
)  <->  ( g  .N  s )  =  ( h  .N  t ) ) )
15 dfplpq2 6606 . 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 5552 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .N  f
)  =  ( a  .N  h ) )
17 oveq12 5552 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .N  u
)  =  ( b  .N  g ) )
1816, 17oveqan12d 5562 . . . 4  |-  ( ( ( w  =  a  /\  f  =  h )  /\  ( v  =  b  /\  u  =  g ) )  ->  ( ( w  .N  f )  +N  ( v  .N  u
) )  =  ( ( a  .N  h
)  +N  ( b  .N  g ) ) )
1918an42s 554 . . 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 5552 . . . 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 3586 . 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 5552 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .N  f
)  =  ( c  .N  s ) )
24 oveq12 5552 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .N  u
)  =  ( d  .N  t ) )
2523, 24oveqan12d 5562 . . . 4  |-  ( ( ( w  =  c  /\  f  =  s )  /\  ( v  =  d  /\  u  =  t ) )  ->  ( ( w  .N  f )  +N  ( v  .N  u
) )  =  ( ( c  .N  s
)  +N  ( d  .N  t ) ) )
2625an42s 554 . . 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 5552 . . . 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 3586 . 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 5552 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .N  f
)  =  ( A  .N  D ) )
31 oveq12 5552 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .N  u
)  =  ( B  .N  C ) )
3230, 31oveqan12d 5562 . . . 4  |-  ( ( ( w  =  A  /\  f  =  D )  /\  ( v  =  B  /\  u  =  C ) )  -> 
( ( w  .N  f )  +N  (
v  .N  u ) )  =  ( ( A  .N  D )  +N  ( B  .N  C ) ) )
3332an42s 554 . . 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 5552 . . . 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 3586 . 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 6601 . 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 6600 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
39 addcmpblnq 6619 . 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 6278 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 1285    e. wcel 1434   <.cop 3409  (class class class)co 5543   [cec 6170   N.cnpi 6524    +N cpli 6525    .N cmi 6526    +pQ cplpq 6528    ~Q ceq 6531   Q.cnq 6532    +Q cplq 6534
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-plpq 6596  df-enq 6599  df-nqqs 6600  df-plqqs 6601
This theorem is referenced by:  addclnq  6627  addcomnqg  6633  addassnqg  6634  distrnqg  6639  ltanqg  6652  1lt2nq  6658  ltexnqq  6660  nqnq0a  6706  addpinq1  6716
  Copyright terms: Public domain W3C validator