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Theorem genpcl 8779
Description: Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (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. )
genpcl.3  |-  ( h  e.  Q.  ->  (
f  <Q  g  <->  ( h G f )  <Q 
( h G g ) ) )
genpcl.4  |-  ( x G y )  =  ( y G x )
genpcl.5  |-  ( ( ( ( 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
genpcl  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
Distinct variable groups:    x, y,
z, f, g, h, A    x, B, y, z, f, g, h, w, v    x, G   
y, w, v, G, z, f, g, h   
f, F, g    w, A, v    w, B, v   
x, F, y, w, v, h
Allowed substitution hint:    F( z)

Proof of Theorem genpcl
StepHypRef Expression
1 genp.1 . . . 4  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )
2 genp.2 . . . 4  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
31, 2genpn0 8774 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
(/)  C.  ( A F B ) )
41, 2genpss 8775 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
5 vex 2876 . . . . . 6  |-  x  e. 
_V
6 vex 2876 . . . . . 6  |-  y  e. 
_V
7 genpcl.3 . . . . . 6  |-  ( h  e.  Q.  ->  (
f  <Q  g  <->  ( h G f )  <Q 
( h G g ) ) )
85, 6, 7caovord 6158 . . . . 5  |-  ( z  e.  Q.  ->  (
x  <Q  y  <->  ( z G x )  <Q 
( z G y ) ) )
9 genpcl.4 . . . . 5  |-  ( x G y )  =  ( y G x )
101, 2, 8, 9genpnnp 8776 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( A F B )  =  Q. )
11 dfpss2 3348 . . . 4  |-  ( ( A F B ) 
C.  Q.  <->  ( ( A F B )  C_  Q.  /\  -.  ( A F B )  =  Q. ) )
124, 10, 11sylanbrc 645 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C.  Q. )
13 genpcl.5 . . . . . . 7  |-  ( ( ( ( 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 ) ) )
141, 2, 13genpcd 8777 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
1514alrimdv 1638 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  A. x ( x 
<Q  f  ->  x  e.  ( A F B ) ) ) )
16 vex 2876 . . . . . . 7  |-  z  e. 
_V
17 vex 2876 . . . . . . 7  |-  w  e. 
_V
1816, 17, 7caovord 6158 . . . . . 6  |-  ( v  e.  Q.  ->  (
z  <Q  w  <->  ( v G z )  <Q 
( v G w ) ) )
1916, 17, 9caovcom 6144 . . . . . 6  |-  ( z G w )  =  ( w G z )
201, 2, 18, 19genpnmax 8778 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  E. x  e.  ( A F B ) f  <Q  x )
)
2115, 20jcad 519 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( A. x
( x  <Q  f  ->  x  e.  ( A F B ) )  /\  E. x  e.  ( A F B ) f  <Q  x
) ) )
2221ralrimiv 2710 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. f  e.  ( A F B ) ( A. x ( x  <Q  f  ->  x  e.  ( A F B ) )  /\  E. x  e.  ( A F B ) f 
<Q  x ) )
233, 12, 22jca31 520 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( (/)  C.  ( A F B )  /\  ( A F B ) 
C.  Q. )  /\  A. f  e.  ( A F B ) ( A. x ( x  <Q  f  ->  x  e.  ( A F B ) )  /\  E. x  e.  ( A F B ) f  <Q  x
) ) )
24 elnp 8758 . 2  |-  ( ( A F B )  e.  P.  <->  ( ( (/)  C.  ( A F B )  /\  ( A F B )  C.  Q. )  /\  A. f  e.  ( A F B ) ( A. x
( x  <Q  f  ->  x  e.  ( A F B ) )  /\  E. x  e.  ( A F B ) f  <Q  x
) ) )
2523, 24sylibr 203 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1545    = wceq 1647    e. wcel 1715   {cab 2352   A.wral 2628   E.wrex 2629    C_ wss 3238    C. wpss 3239   (/)c0 3543   class class class wbr 4125  (class class class)co 5981    e. cmpt2 5983   Q.cnq 8621    <Q cltq 8627   P.cnp 8628
This theorem is referenced by:  addclpr  8789  mulclpr  8791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-oadd 6625  df-omul 6626  df-er 6802  df-ni 8643  df-mi 8645  df-lti 8646  df-ltpq 8681  df-enq 8682  df-nq 8683  df-ltnq 8689  df-np 8752
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