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Theorem genpcl 8845
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 8840 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
(/)  C.  ( A F B ) )
41, 2genpss 8841 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
5 vex 2923 . . . . . 6  |-  x  e. 
_V
6 vex 2923 . . . . . 6  |-  y  e. 
_V
7 genpcl.3 . . . . . 6  |-  ( h  e.  Q.  ->  (
f  <Q  g  <->  ( h G f )  <Q 
( h G g ) ) )
85, 6, 7caovord 6221 . . . . 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 8842 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( A F B )  =  Q. )
11 dfpss2 3396 . . . 4  |-  ( ( A F B ) 
C.  Q.  <->  ( ( A F B )  C_  Q.  /\  -.  ( A F B )  =  Q. ) )
124, 10, 11sylanbrc 646 . . 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 8843 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
1514alrimdv 1640 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  A. x ( x 
<Q  f  ->  x  e.  ( A F B ) ) ) )
16 vex 2923 . . . . . . 7  |-  z  e. 
_V
17 vex 2923 . . . . . . 7  |-  w  e. 
_V
1816, 17, 7caovord 6221 . . . . . 6  |-  ( v  e.  Q.  ->  (
z  <Q  w  <->  ( v G z )  <Q 
( v G w ) ) )
1916, 17, 9caovcom 6207 . . . . . 6  |-  ( z G w )  =  ( w G z )
201, 2, 18, 19genpnmax 8844 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  E. x  e.  ( A F B ) f  <Q  x )
)
2115, 20jcad 520 . . . 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 2752 . . 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 521 . 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 8824 . 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 204 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 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   {cab 2394   A.wral 2670   E.wrex 2671    C_ wss 3284    C. wpss 3285   (/)c0 3592   class class class wbr 4176  (class class class)co 6044    e. cmpt2 6046   Q.cnq 8687    <Q cltq 8693   P.cnp 8694
This theorem is referenced by:  addclpr  8855  mulclpr  8857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-recs 6596  df-rdg 6631  df-oadd 6691  df-omul 6692  df-er 6868  df-ni 8709  df-mi 8711  df-lti 8712  df-ltpq 8747  df-enq 8748  df-nq 8749  df-ltnq 8755  df-np 8818
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