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Theorem genpcd 8914
Description: Downward closure of an operation 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. )
genpcd.2  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g G h )  ->  x  e.  ( A F B ) ) )
Assertion
Ref Expression
genpcd  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
Distinct variable groups:    x, y,
z, f, g, h, A    x, B, y, z, f, g, h   
x, w, v, G, y, z, f, g, h    f, F, g, h
Allowed substitution hints:    A( w, v)    B( w, v)    F( x, y, z, w, v)

Proof of Theorem genpcd
StepHypRef Expression
1 ltrelnq 8834 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4955 . . . . . 6  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
32simpld 447 . . . . 5  |-  ( x 
<Q  f  ->  x  e. 
Q. )
4 genp.1 . . . . . . . . 9  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )
5 genp.2 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
64, 5genpelv 8908 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) ) )
76adantr 453 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( f  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) ) )
8 breq2 4241 . . . . . . . . . . . . 13  |-  ( f  =  ( g G h )  ->  (
x  <Q  f  <->  x  <Q  ( g G h ) ) )
98biimpd 200 . . . . . . . . . . . 12  |-  ( f  =  ( g G h )  ->  (
x  <Q  f  ->  x  <Q  ( g G h ) ) )
10 genpcd.2 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g G h )  ->  x  e.  ( A F B ) ) )
119, 10sylan9r 641 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  /\  f  =  (
g G h ) )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) )
1211exp31 589 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( x  e.  Q.  ->  ( f  =  ( g G h )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) ) )
1312an4s 801 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  A  /\  h  e.  B
) )  ->  (
x  e.  Q.  ->  ( f  =  ( g G h )  -> 
( x  <Q  f  ->  x  e.  ( A F B ) ) ) ) )
1413impancom 429 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( ( g  e.  A  /\  h  e.  B )  ->  (
f  =  ( g G h )  -> 
( x  <Q  f  ->  x  e.  ( A F B ) ) ) ) )
1514rexlimdvv 2842 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( E. g  e.  A  E. h  e.  B  f  =  ( g G h )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
167, 15sylbid 208 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
1716ex 425 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  Q.  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) ) )
183, 17syl5 31 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) ) )
1918com34 80 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( x  <Q  f  ->  ( f  e.  ( A F B )  ->  x  e.  ( A F B ) ) ) ) )
2019pm2.43d 47 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( f  e.  ( A F B )  ->  x  e.  ( A F B ) ) ) )
2120com23 75 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   {cab 2428   E.wrex 2712   class class class wbr 4237  (class class class)co 6110    e. cmpt2 6112   Q.cnq 8758    <Q cltq 8764   P.cnp 8765
This theorem is referenced by:  genpcl  8916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-ni 8780  df-nq 8820  df-ltnq 8826  df-np 8889
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