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Theorem genpprecl 9375
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 2467 . . 3  |-  ( C G D )  =  ( C G D )
2 rspceov 6319 . . 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 1313 . 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 9374 . 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 221 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C G D )  e.  ( A F B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815  (class class class)co 6282    |-> cmpt2 6284   Q.cnq 9226   P.cnp 9233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-ni 9246  df-nq 9286  df-np 9355
This theorem is referenced by:  genpn0  9377  genpnmax  9381  addclprlem2  9391  mulclprlem  9393  distrlem1pr  9399  distrlem4pr  9400  ltaddpr  9408  ltexprlem7  9416
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