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Theorem genpss 9371
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 9367 . . 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 9358 . . . . . . . 8  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
54ex 432 . . . . . . 7  |-  ( A  e.  P.  ->  (
g  e.  A  -> 
g  e.  Q. )
)
6 elprnq 9358 . . . . . . . 8  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
76ex 432 . . . . . . 7  |-  ( B  e.  P.  ->  (
h  e.  B  ->  h  e.  Q. )
)
85, 7im2anan9 833 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
g  e.  Q.  /\  h  e.  Q. )
) )
92caovcl 6442 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
108, 9syl6 33 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
g G h )  e.  Q. ) )
11 eleq1a 2537 . . . . 5  |-  ( ( g G h )  e.  Q.  ->  (
f  =  ( g G h )  -> 
f  e.  Q. )
)
1210, 11syl6 33 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
f  =  ( g G h )  -> 
f  e.  Q. )
) )
1312rexlimdvv 2952 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. g  e.  A  E. h  e.  B  f  =  ( g G h )  ->  f  e.  Q. ) )
143, 13sylbid 215 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  f  e.  Q. ) )
1514ssrdv 3495 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   E.wrex 2805    C_ wss 3461  (class class class)co 6270    |-> cmpt2 6272   Q.cnq 9219   P.cnp 9226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-ni 9239  df-nq 9279  df-np 9348
This theorem is referenced by:  genpcl  9375
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