MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  genpelv Structured version   Unicode version

Theorem genpelv 9424
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 9423 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  { f  |  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) } )
43eleq2d 2499 . 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 23 . . . . . 6  |-  ( C  =  ( g G h )  ->  C  =  ( g G h ) )
6 ovex 6333 . . . . . 6  |-  ( g G h )  e. 
_V
75, 6syl6eqel 2525 . . . . 5  |-  ( C  =  ( g G h )  ->  C  e.  _V )
87rexlimivw 2921 . . . 4  |-  ( E. h  e.  B  C  =  ( g G h )  ->  C  e.  _V )
98rexlimivw 2921 . . 3  |-  ( E. g  e.  A  E. h  e.  B  C  =  ( g G h )  ->  C  e.  _V )
10 eqeq1 2433 . . . 4  |-  ( f  =  C  ->  (
f  =  ( g G h )  <->  C  =  ( g G h ) ) )
11102rexbidv 2953 . . 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 3231 . 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 264 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   {cab 2414   E.wrex 2783   _Vcvv 3087  (class class class)co 6305    |-> cmpt2 6307   Q.cnq 9276   P.cnp 9283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-ni 9296  df-nq 9336  df-np 9405
This theorem is referenced by:  genpprecl  9425  genpss  9428  genpnnp  9429  genpcd  9430  genpnmax  9431  genpass  9433  distrlem1pr  9449  distrlem5pr  9451  1idpr  9453  ltexprlem6  9465  reclem3pr  9473  reclem4pr  9474
  Copyright terms: Public domain W3C validator