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Theorem genpelv 8624
Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (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
genpelv  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  C  =  ( g G h ) ) )
Distinct variable groups:    x, y,
z, g, h, A   
x, B, y, z, g, h    x, w, v, G, y, z, g, h    g, F    C, g, h
Allowed substitution hints:    A( w, v)    B( w, v)    C( x, y, z, w, v)    F( x, y, z, w, v, h)

Proof of Theorem genpelv
Dummy variable  f is 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, 2genpv 8623 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  { f  |  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) } )
43eleq2d 2350 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( A F B )  <-> 
C  e.  { f  |  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) } ) )
5 id 19 . . . . . 6  |-  ( C  =  ( g G h )  ->  C  =  ( g G h ) )
6 ovex 5883 . . . . . 6  |-  ( g G h )  e. 
_V
75, 6syl6eqel 2371 . . . . 5  |-  ( C  =  ( g G h )  ->  C  e.  _V )
87rexlimivw 2663 . . . 4  |-  ( E. h  e.  B  C  =  ( g G h )  ->  C  e.  _V )
98rexlimivw 2663 . . 3  |-  ( E. g  e.  A  E. h  e.  B  C  =  ( g G h )  ->  C  e.  _V )
10 eqeq1 2289 . . . 4  |-  ( f  =  C  ->  (
f  =  ( g G h )  <->  C  =  ( g G h ) ) )
11102rexbidv 2586 . . 3  |-  ( f  =  C  ->  ( E. g  e.  A  E. h  e.  B  f  =  ( g G h )  <->  E. g  e.  A  E. h  e.  B  C  =  ( g G h ) ) )
129, 11elab3 2921 . 2  |-  ( C  e.  { f  |  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) }  <->  E. g  e.  A  E. h  e.  B  C  =  ( g G h ) )
134, 12syl6bb 252 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  C  =  ( g G h ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788  (class class class)co 5858    e. cmpt2 5860   Q.cnq 8474   P.cnp 8481
This theorem is referenced by:  genpprecl  8625  genpss  8628  genpnnp  8629  genpcd  8630  genpnmax  8631  genpass  8633  distrlem1pr  8649  distrlem5pr  8651  1idpr  8653  ltexprlem6  8665  reclem3pr  8673  reclem4pr  8674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ni 8496  df-nq 8536  df-np 8605
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