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Theorem genpelv 9165
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 9164 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  { f  |  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) } )
43eleq2d 2508 . 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 22 . . . . . 6  |-  ( C  =  ( g G h )  ->  C  =  ( g G h ) )
6 ovex 6115 . . . . . 6  |-  ( g G h )  e. 
_V
75, 6syl6eqel 2529 . . . . 5  |-  ( C  =  ( g G h )  ->  C  e.  _V )
87rexlimivw 2835 . . . 4  |-  ( E. h  e.  B  C  =  ( g G h )  ->  C  e.  _V )
98rexlimivw 2835 . . 3  |-  ( E. g  e.  A  E. h  e.  B  C  =  ( g G h )  ->  C  e.  _V )
10 eqeq1 2447 . . . 4  |-  ( f  =  C  ->  (
f  =  ( g G h )  <->  C  =  ( g G h ) ) )
11102rexbidv 2756 . . 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 3110 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970  (class class class)co 6090    e. cmpt2 6092   Q.cnq 9015   P.cnp 9022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-ni 9037  df-nq 9077  df-np 9146
This theorem is referenced by:  genpprecl  9166  genpss  9169  genpnnp  9170  genpcd  9171  genpnmax  9172  genpass  9174  distrlem1pr  9190  distrlem5pr  9192  1idpr  9194  ltexprlem6  9206  reclem3pr  9214  reclem4pr  9215
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