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Theorem genpelvl 6668
Description: Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-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
genpelvl  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  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 genpelvl
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. )  ->  ( 1st `  ( A F B ) )  =  ( 1st `  <. { 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, 7op1st 5801 . . . . 5  |-  ( 1st `  <. { 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.  ( 1st `  A ) E. h  e.  ( 1st `  B ) f  =  ( g G h ) }
94, 8syl6eq 2104 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 1st `  ( A F B ) )  =  { f  e. 
Q.  |  E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  B
) f  =  ( g G h ) } )
109eleq2d 2123 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 1st `  ( A F B ) )  <-> 
C  e.  { f  e.  Q.  |  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B
) f  =  ( g G h ) } ) )
11 elrabi 2718 . . 3  |-  ( C  e.  { f  e. 
Q.  |  E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  B
) f  =  ( g G h ) }  ->  C  e.  Q. )
1210, 11syl6bi 156 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 1st `  ( A F B ) )  ->  C  e.  Q. ) )
13 prop 6631 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
14 elprnql 6637 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  g  e.  ( 1st `  A ) )  -> 
g  e.  Q. )
1513, 14sylan 271 . . . . . 6  |-  ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  -> 
g  e.  Q. )
16 prop 6631 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
17 elprnql 6637 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  h  e.  ( 1st `  B ) )  ->  h  e.  Q. )
1816, 17sylan 271 . . . . . 6  |-  ( ( B  e.  P.  /\  h  e.  ( 1st `  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.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B ) ) )  ->  ( g G h )  e.  Q. )
2120an4s 530 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 1st `  A )  /\  h  e.  ( 1st `  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.  ( 1st `  A )  /\  h  e.  ( 1st `  B ) ) )  ->  ( C  =  ( g G h )  ->  C  e.  Q. )
)
2423rexlimdvva 2457 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  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.  ( 1st `  A ) E. h  e.  ( 1st `  B ) f  =  ( g G h )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) C  =  ( g G h ) ) )
2726elrab3 2722 . . . 4  |-  ( C  e.  Q.  ->  ( C  e.  { f  e.  Q.  |  E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  B
) f  =  ( g G h ) }  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) C  =  ( g G h ) ) )
2810, 27sylan9bb 443 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  C  e.  Q. )  ->  ( C  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) C  =  ( g G h ) ) )
2928ex 112 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  Q.  ->  ( C  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) C  =  ( g G h ) ) ) )
3012, 24, 29pm5.21ndd 631 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  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:  genpprecll  6670  genpcdl  6675  genprndl  6677  genpdisj  6679  genpassl  6680  addnqprlemrl  6713  mulnqprlemrl  6729  distrlem1prl  6738  distrlem5prl  6742  1idprl  6746  ltexprlemfl  6765  recexprlem1ssl  6789  recexprlemss1l  6791  cauappcvgprlemladdfl  6811
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