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Theorem genpelvl 7463
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 7460 . . . . . 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 5498 . . . . 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 7314 . . . . . . 7  |-  Q.  e.  _V
65rabex 4131 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B
) f  =  ( g G h ) }  e.  _V
75rabex 4131 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) }  e.  _V
86, 7op1st 6123 . . . . 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, 8eqtrdi 2219 . . . 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 2240 . . 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 2883 . . 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 162 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 1st `  ( A F B ) )  ->  C  e.  Q. ) )
13 prop 7426 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
14 elprnql 7432 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  g  e.  ( 1st `  A ) )  -> 
g  e.  Q. )
1513, 14sylan 281 . . . . . 6  |-  ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  -> 
g  e.  Q. )
16 prop 7426 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
17 elprnql 7432 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  h  e.  ( 1st `  B ) )  ->  h  e.  Q. )
1816, 17sylan 281 . . . . . 6  |-  ( ( B  e.  P.  /\  h  e.  ( 1st `  B ) )  ->  h  e.  Q. )
192caovcl 6005 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
2015, 18, 19syl2an 287 . . . . 5  |-  ( ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B ) ) )  ->  ( g G h )  e.  Q. )
2120an4s 583 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 1st `  A )  /\  h  e.  ( 1st `  B ) ) )  ->  (
g G h )  e.  Q. )
22 eleq1 2233 . . . 4  |-  ( C  =  ( g G h )  ->  ( C  e.  Q.  <->  ( g G h )  e. 
Q. ) )
2321, 22syl5ibrcom 156 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 1st `  A )  /\  h  e.  ( 1st `  B ) ) )  ->  ( C  =  ( g G h )  ->  C  e.  Q. )
)
2423rexlimdvva 2595 . 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 2177 . . . . . 6  |-  ( f  =  C  ->  (
f  =  ( g G h )  <->  C  =  ( g G h ) ) )
26252rexbidv 2495 . . . . 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 2887 . . . 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 459 . . 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 114 . 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 700 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 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3584   ` cfv 5196  (class class class)co 5851    e. cmpo 5853   1stc1st 6115   2ndc2nd 6116   Q.cnq 7231   P.cnp 7242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-qs 6516  df-ni 7255  df-nqqs 7299  df-inp 7417
This theorem is referenced by:  genpprecll  7465  genpcdl  7470  genprndl  7472  genpdisj  7474  genpassl  7475  addnqprlemrl  7508  mulnqprlemrl  7524  distrlem1prl  7533  distrlem5prl  7537  1idprl  7541  ltexprlemfl  7560  recexprlem1ssl  7584  recexprlemss1l  7586  cauappcvgprlemladdfl  7606
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