ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genpelvu Unicode version

Theorem genpelvu 6669
Description: Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)
Hypotheses
Ref Expression
genpelvl.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
genpelvl.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
Assertion
Ref Expression
genpelvu  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) C  =  ( g G h ) ) )
Distinct variable groups:    x, y, z, g, h, w, v, A    x, B, y, z, g, h, w, v    x, G, y, z, g, h, w, v    g, F    C, g, h
Allowed substitution hints:    C( x, y, z, w, v)    F( x, y, z, w, v, h)

Proof of Theorem genpelvu
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . 7  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
2 genpelvl.2 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
31, 2genipv 6665 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  <. { f  e.  Q.  |  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B
) f  =  ( g G h ) } ,  { f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) } >. )
43fveq2d 5210 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 2nd `  ( A F B ) )  =  ( 2nd `  <. { f  e.  Q.  |  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) f  =  ( g G h ) } ,  {
f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) f  =  ( g G h ) } >. )
)
5 nqex 6519 . . . . . . 7  |-  Q.  e.  _V
65rabex 3929 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B
) f  =  ( g G h ) }  e.  _V
75rabex 3929 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) }  e.  _V
86, 7op2nd 5802 . . . . 5  |-  ( 2nd `  <. { f  e. 
Q.  |  E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  B
) f  =  ( g G h ) } ,  { f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) } >. )  =  {
f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) f  =  ( g G h ) }
94, 8syl6eq 2104 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 2nd `  ( A F B ) )  =  { f  e. 
Q.  |  E. g  e.  ( 2nd `  A
) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) } )
109eleq2d 2123 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 2nd `  ( A F B ) )  <-> 
C  e.  { f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) } ) )
11 elrabi 2718 . . 3  |-  ( C  e.  { f  e. 
Q.  |  E. g  e.  ( 2nd `  A
) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) }  ->  C  e.  Q. )
1210, 11syl6bi 156 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 2nd `  ( A F B ) )  ->  C  e.  Q. ) )
13 prop 6631 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
14 elprnqu 6638 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  g  e.  ( 2nd `  A ) )  -> 
g  e.  Q. )
1513, 14sylan 271 . . . . . 6  |-  ( ( A  e.  P.  /\  g  e.  ( 2nd `  A ) )  -> 
g  e.  Q. )
16 prop 6631 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
17 elprnqu 6638 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  h  e.  ( 2nd `  B ) )  ->  h  e.  Q. )
1816, 17sylan 271 . . . . . 6  |-  ( ( B  e.  P.  /\  h  e.  ( 2nd `  B ) )  ->  h  e.  Q. )
192caovcl 5683 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
2015, 18, 19syl2an 277 . . . . 5  |-  ( ( ( A  e.  P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B ) ) )  ->  ( g G h )  e.  Q. )
2120an4s 530 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 2nd `  A )  /\  h  e.  ( 2nd `  B ) ) )  ->  (
g G h )  e.  Q. )
22 eleq1 2116 . . . 4  |-  ( C  =  ( g G h )  ->  ( C  e.  Q.  <->  ( g G h )  e. 
Q. ) )
2321, 22syl5ibrcom 150 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 2nd `  A )  /\  h  e.  ( 2nd `  B ) ) )  ->  ( C  =  ( g G h )  ->  C  e.  Q. )
)
2423rexlimdvva 2457 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. g  e.  ( 2nd `  A
) E. h  e.  ( 2nd `  B
) C  =  ( g G h )  ->  C  e.  Q. ) )
25 eqeq1 2062 . . . . . 6  |-  ( f  =  C  ->  (
f  =  ( g G h )  <->  C  =  ( g G h ) ) )
26252rexbidv 2366 . . . . 5  |-  ( f  =  C  ->  ( E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) f  =  ( g G h )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) C  =  ( g G h ) ) )
2726elrab3 2722 . . . 4  |-  ( C  e.  Q.  ->  ( C  e.  { f  e.  Q.  |  E. g  e.  ( 2nd `  A
) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) }  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) C  =  ( g G h ) ) )
2810, 27sylan9bb 443 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  C  e.  Q. )  ->  ( C  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) C  =  ( g G h ) ) )
2928ex 112 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  Q.  ->  ( C  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) C  =  ( g G h ) ) ) )
3012, 24, 29pm5.21ndd 631 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) C  =  ( g G h ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   {crab 2327   <.cop 3406   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   P.cnp 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-qs 6143  df-ni 6460  df-nqqs 6504  df-inp 6622
This theorem is referenced by:  genppreclu  6671  genpcuu  6676  genprndu  6678  genpdisj  6679  genpassu  6681  addnqprlemru  6714  mulnqprlemru  6730  distrlem1pru  6739  distrlem5pru  6743  1idpru  6747  ltexprlemfu  6767  recexprlem1ssu  6790  recexprlemss1u  6792  cauappcvgprlemladdfu  6810  caucvgprlemladdfu  6833
  Copyright terms: Public domain W3C validator