MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  genpcl Structured version   Visualization version   Unicode version

Theorem genpcl 9458
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 9453 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
(/)  C.  ( A F B ) )
41, 2genpss 9454 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
5 vex 3059 . . . . . 6  |-  x  e. 
_V
6 vex 3059 . . . . . 6  |-  y  e. 
_V
7 genpcl.3 . . . . . 6  |-  ( h  e.  Q.  ->  (
f  <Q  g  <->  ( h G f )  <Q 
( h G g ) ) )
85, 6, 7caovord 6506 . . . . 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 9455 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( A F B )  =  Q. )
11 dfpss2 3529 . . . 4  |-  ( ( A F B ) 
C.  Q.  <->  ( ( A F B )  C_  Q.  /\  -.  ( A F B )  =  Q. ) )
124, 10, 11sylanbrc 675 . . 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 9456 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f  ->  x  e.  ( A F B ) ) ) )
1514alrimdv 1785 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  A. x ( x 
<Q  f  ->  x  e.  ( A F B ) ) ) )
16 vex 3059 . . . . . . 7  |-  z  e. 
_V
17 vex 3059 . . . . . . 7  |-  w  e. 
_V
1816, 17, 7caovord 6506 . . . . . 6  |-  ( v  e.  Q.  ->  (
z  <Q  w  <->  ( v G z )  <Q 
( v G w ) ) )
1916, 17, 9caovcom 6492 . . . . . 6  |-  ( z G w )  =  ( w G z )
201, 2, 18, 19genpnmax 9457 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  E. x  e.  ( A F B ) f  <Q  x )
)
2115, 20jcad 540 . . . 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 2811 . . 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 541 . 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 9437 . 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 217 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 189    /\ wa 375   A.wal 1452    = wceq 1454    e. wcel 1897   {cab 2447   A.wral 2748   E.wrex 2749    C_ wss 3415    C. wpss 3416   (/)c0 3742   class class class wbr 4415  (class class class)co 6314    |-> cmpt2 6316   Q.cnq 9302    <Q cltq 9308   P.cnp 9309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-oadd 7211  df-omul 7212  df-er 7388  df-ni 9322  df-mi 9324  df-lti 9325  df-ltpq 9360  df-enq 9361  df-nq 9362  df-ltnq 9368  df-np 9431
This theorem is referenced by:  addclpr  9468  mulclpr  9470
  Copyright terms: Public domain W3C validator