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

Theorem genpprecl 8871
Description: Pre-closure law for general operation on positive reals. (Contributed by NM, 10-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
genpprecl  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C G D )  e.  ( A F B ) ) )
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)    C( x, y, z, w, v)    D( x, y, z, w, v)    F( x, y, z, w, v)

Proof of Theorem genpprecl
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( C G D )  =  ( C G D )
2 rspceov 6109 . . 3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( C G D )  =  ( C G D ) )  ->  E. g  e.  A  E. h  e.  B  ( C G D )  =  ( g G h ) )
31, 2mp3an3 1268 . 2  |-  ( ( C  e.  A  /\  D  e.  B )  ->  E. g  e.  A  E. h  e.  B  ( C G D )  =  ( g G h ) )
4 genp.1 . . 3  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )
5 genp.2 . . 3  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
64, 5genpelv 8870 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C G D )  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  ( C G D )  =  ( g G h ) ) )
73, 6syl5ibr 213 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C G D )  e.  ( A F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2699  (class class class)co 6074    e. cmpt2 6076   Q.cnq 8720   P.cnp 8727
This theorem is referenced by:  genpn0  8873  genpnmax  8877  addclprlem2  8887  mulclprlem  8889  distrlem1pr  8895  distrlem4pr  8896  ltaddpr  8904  ltexprlem7  8912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-ni 8742  df-nq 8782  df-np 8851
  Copyright terms: Public domain W3C validator