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Theorem genpprecl 9408
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 2402 . . 3  |-  ( C G D )  =  ( C G D )
2 rspceov 6316 . . 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 1315 . 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 9407 . 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 367    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2754  (class class class)co 6277    |-> cmpt2 6279   Q.cnq 9259   P.cnp 9266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-ni 9279  df-nq 9319  df-np 9388
This theorem is referenced by:  genpn0  9410  genpnmax  9414  addclprlem2  9424  mulclprlem  9426  distrlem1pr  9432  distrlem4pr  9433  ltaddpr  9441  ltexprlem7  9449
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