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Theorem genpcl 9292
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 9287 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
(/)  C.  ( A F B ) )
41, 2genpss 9288 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
5 vex 3081 . . . . . 6  |-  x  e. 
_V
6 vex 3081 . . . . . 6  |-  y  e. 
_V
7 genpcl.3 . . . . . 6  |-  ( h  e.  Q.  ->  (
f  <Q  g  <->  ( h G f )  <Q 
( h G g ) ) )
85, 6, 7caovord 6387 . . . . 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 9289 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( A F B )  =  Q. )
11 dfpss2 3552 . . . 4  |-  ( ( A F B ) 
C.  Q.  <->  ( ( A F B )  C_  Q.  /\  -.  ( A F B )  =  Q. ) )
124, 10, 11sylanbrc 664 . . 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 9290 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
1514alrimdv 1688 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  A. x ( x 
<Q  f  ->  x  e.  ( A F B ) ) ) )
16 vex 3081 . . . . . . 7  |-  z  e. 
_V
17 vex 3081 . . . . . . 7  |-  w  e. 
_V
1816, 17, 7caovord 6387 . . . . . 6  |-  ( v  e.  Q.  ->  (
z  <Q  w  <->  ( v G z )  <Q 
( v G w ) ) )
1916, 17, 9caovcom 6373 . . . . . 6  |-  ( z G w )  =  ( w G z )
201, 2, 18, 19genpnmax 9291 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  E. x  e.  ( A F B ) f  <Q  x )
)
2115, 20jcad 533 . . . 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 2828 . . 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 534 . 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 9271 . 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 212 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800    C_ wss 3439    C. wpss 3440   (/)c0 3748   class class class wbr 4403  (class class class)co 6203    |-> cmpt2 6205   Q.cnq 9134    <Q cltq 9140   P.cnp 9141
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-oadd 7037  df-omul 7038  df-er 7214  df-ni 9156  df-mi 9158  df-lti 9159  df-ltpq 9194  df-enq 9195  df-nq 9196  df-ltnq 9202  df-np 9265
This theorem is referenced by:  addclpr  9302  mulclpr  9304
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