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Theorem genpcd 9281
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 9201 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4990 . . . . . 6  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
32simpld 459 . . . . 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 9275 . . . . . . . 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 465 . . . . . . 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 4399 . . . . . . . . . . . . 13  |-  ( f  =  ( g G h )  ->  (
x  <Q  f  <->  x  <Q  ( g G h ) ) )
98biimpd 207 . . . . . . . . . . . 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 658 . . . . . . . . . . 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 604 . . . . . . . . . 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 822 . . . . . . . . 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 440 . . . . . . . 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 2947 . . . . . . 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 215 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
1716ex 434 . . . . 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 32 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) ) )
1918com34 83 . . 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 48 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( f  e.  ( A F B )  ->  x  e.  ( A F B ) ) ) )
2120com23 78 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2437   E.wrex 2797   class class class wbr 4395  (class class class)co 6195    |-> cmpt2 6197   Q.cnq 9125    <Q cltq 9131   P.cnp 9132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-ni 9147  df-nq 9187  df-ltnq 9193  df-np 9256
This theorem is referenced by:  genpcl  9283
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