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Theorem genpss 8881
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 8877 . . 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 8868 . . . . . . . 8  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
54ex 424 . . . . . . 7  |-  ( A  e.  P.  ->  (
g  e.  A  -> 
g  e.  Q. )
)
6 elprnq 8868 . . . . . . . 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 6241 . . . . . 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 2505 . . . . 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 2836 . . 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 3354 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 1652    e. wcel 1725   {cab 2422   E.wrex 2706    C_ wss 3320  (class class class)co 6081    e. cmpt2 6083   Q.cnq 8727   P.cnp 8734
This theorem is referenced by:  genpcl  8885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-ni 8749  df-nq 8789  df-np 8858
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