MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  genpss Unicode version

Theorem genpss 8837
Description: The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )
genp.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
Assertion
Ref Expression
genpss  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, w, v, G, y, z
Allowed substitution hints:    A( w, v)    B( w, v)    F( x, y, z, w, v)

Proof of Theorem genpss
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genp.1 . . . 4  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )
2 genp.2 . . . 4  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
31, 2genpelv 8833 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) ) )
4 elprnq 8824 . . . . . . . 8  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
54ex 424 . . . . . . 7  |-  ( A  e.  P.  ->  (
g  e.  A  -> 
g  e.  Q. )
)
6 elprnq 8824 . . . . . . . 8  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
76ex 424 . . . . . . 7  |-  ( B  e.  P.  ->  (
h  e.  B  ->  h  e.  Q. )
)
85, 7im2anan9 809 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
g  e.  Q.  /\  h  e.  Q. )
) )
92caovcl 6200 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
108, 9syl6 31 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
g G h )  e.  Q. ) )
11 eleq1a 2473 . . . . 5  |-  ( ( g G h )  e.  Q.  ->  (
f  =  ( g G h )  -> 
f  e.  Q. )
)
1210, 11syl6 31 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
f  =  ( g G h )  -> 
f  e.  Q. )
) )
1312rexlimdvv 2796 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. g  e.  A  E. h  e.  B  f  =  ( g G h )  ->  f  e.  Q. ) )
143, 13sylbid 207 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  f  e.  Q. ) )
1514ssrdv 3314 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667    C_ wss 3280  (class class class)co 6040    e. cmpt2 6042   Q.cnq 8683   P.cnp 8690
This theorem is referenced by:  genpcl  8841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-ni 8705  df-nq 8745  df-np 8814
  Copyright terms: Public domain W3C validator