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Theorem genpcl 8890
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 8885 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
(/)  C.  ( A F B ) )
41, 2genpss 8886 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
5 vex 2961 . . . . . 6  |-  x  e. 
_V
6 vex 2961 . . . . . 6  |-  y  e. 
_V
7 genpcl.3 . . . . . 6  |-  ( h  e.  Q.  ->  (
f  <Q  g  <->  ( h G f )  <Q 
( h G g ) ) )
85, 6, 7caovord 6261 . . . . 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 8887 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( A F B )  =  Q. )
11 dfpss2 3434 . . . 4  |-  ( ( A F B ) 
C.  Q.  <->  ( ( A F B )  C_  Q.  /\  -.  ( A F B )  =  Q. ) )
124, 10, 11sylanbrc 647 . . 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 8888 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
1514alrimdv 1644 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  A. x ( x 
<Q  f  ->  x  e.  ( A F B ) ) ) )
16 vex 2961 . . . . . . 7  |-  z  e. 
_V
17 vex 2961 . . . . . . 7  |-  w  e. 
_V
1816, 17, 7caovord 6261 . . . . . 6  |-  ( v  e.  Q.  ->  (
z  <Q  w  <->  ( v G z )  <Q 
( v G w ) ) )
1916, 17, 9caovcom 6247 . . . . . 6  |-  ( z G w )  =  ( w G z )
201, 2, 18, 19genpnmax 8889 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  E. x  e.  ( A F B ) f  <Q  x )
)
2115, 20jcad 521 . . . 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 2790 . . 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 522 . 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 8869 . 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 205 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 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708    C_ wss 3322    C. wpss 3323   (/)c0 3630   class class class wbr 4215  (class class class)co 6084    e. cmpt2 6086   Q.cnq 8732    <Q cltq 8738   P.cnp 8739
This theorem is referenced by:  addclpr  8900  mulclpr  8902
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 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-oadd 6731  df-omul 6732  df-er 6908  df-ni 8754  df-mi 8756  df-lti 8757  df-ltpq 8792  df-enq 8793  df-nq 8794  df-ltnq 8800  df-np 8863
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