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Theorem genpss 8718
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 8714 . . 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 8705 . . . . . . . 8  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
54ex 423 . . . . . . 7  |-  ( A  e.  P.  ->  (
g  e.  A  -> 
g  e.  Q. )
)
6 elprnq 8705 . . . . . . . 8  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
76ex 423 . . . . . . 7  |-  ( B  e.  P.  ->  (
h  e.  B  ->  h  e.  Q. )
)
85, 7im2anan9 808 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
g  e.  Q.  /\  h  e.  Q. )
) )
92caovcl 6101 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
108, 9syl6 29 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
g G h )  e.  Q. ) )
11 eleq1a 2427 . . . . 5  |-  ( ( g G h )  e.  Q.  ->  (
f  =  ( g G h )  -> 
f  e.  Q. )
)
1210, 11syl6 29 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
f  =  ( g G h )  -> 
f  e.  Q. )
) )
1312rexlimdvv 2749 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. g  e.  A  E. h  e.  B  f  =  ( g G h )  ->  f  e.  Q. ) )
143, 13sylbid 206 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  f  e.  Q. ) )
1514ssrdv 3261 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   E.wrex 2620    C_ wss 3228  (class class class)co 5945    e. cmpt2 5947   Q.cnq 8564   P.cnp 8571
This theorem is referenced by:  genpcl  8722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-ni 8586  df-nq 8626  df-np 8695
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