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Theorem genpelvl 7513
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 7510 . . . . . 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 5521 . . . . 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 7364 . . . . . . 7  |-  Q.  e.  _V
65rabex 4149 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B
) f  =  ( g G h ) }  e.  _V
75rabex 4149 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) }  e.  _V
86, 7op1st 6149 . . . . 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 2226 . . . 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 2247 . . 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 2892 . . 3  |-  ( C  e.  { f  e. 
Q.  |  E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  B
) f  =  ( g G h ) }  ->  C  e.  Q. )
1210, 11biimtrdi 163 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 1st `  ( A F B ) )  ->  C  e.  Q. ) )
13 prop 7476 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
14 elprnql 7482 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  g  e.  ( 1st `  A ) )  -> 
g  e.  Q. )
1513, 14sylan 283 . . . . . 6  |-  ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  -> 
g  e.  Q. )
16 prop 7476 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
17 elprnql 7482 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  h  e.  ( 1st `  B ) )  ->  h  e.  Q. )
1816, 17sylan 283 . . . . . 6  |-  ( ( B  e.  P.  /\  h  e.  ( 1st `  B ) )  ->  h  e.  Q. )
192caovcl 6031 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
2015, 18, 19syl2an 289 . . . . 5  |-  ( ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B ) ) )  ->  ( g G h )  e.  Q. )
2120an4s 588 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 1st `  A )  /\  h  e.  ( 1st `  B ) ) )  ->  (
g G h )  e.  Q. )
22 eleq1 2240 . . . 4  |-  ( C  =  ( g G h )  ->  ( C  e.  Q.  <->  ( g G h )  e. 
Q. ) )
2321, 22syl5ibrcom 157 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 1st `  A )  /\  h  e.  ( 1st `  B ) ) )  ->  ( C  =  ( g G h )  ->  C  e.  Q. )
)
2423rexlimdvva 2602 . 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 2184 . . . . . 6  |-  ( f  =  C  ->  (
f  =  ( g G h )  <->  C  =  ( g G h ) ) )
26252rexbidv 2502 . . . . 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 2896 . . . 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 462 . . 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 115 . 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 705 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3597   ` cfv 5218  (class class class)co 5877    e. cmpo 5879   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   P.cnp 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-qs 6543  df-ni 7305  df-nqqs 7349  df-inp 7467
This theorem is referenced by:  genpprecll  7515  genpcdl  7520  genprndl  7522  genpdisj  7524  genpassl  7525  addnqprlemrl  7558  mulnqprlemrl  7574  distrlem1prl  7583  distrlem5prl  7587  1idprl  7591  ltexprlemfl  7610  recexprlem1ssl  7634  recexprlemss1l  7636  cauappcvgprlemladdfl  7656
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