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Theorem genpelvl 7313
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 7310 . . . . . 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 5418 . . . . 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 7164 . . . . . . 7  |-  Q.  e.  _V
65rabex 4067 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B
) f  =  ( g G h ) }  e.  _V
75rabex 4067 . . . . . 6  |-  { f  e.  Q.  |  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B
) f  =  ( g G h ) }  e.  _V
86, 7op1st 6037 . . . . 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 2186 . . . 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 2207 . . 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 2832 . . 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 7276 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
14 elprnql 7282 . . . . . . 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 7276 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
17 elprnql 7282 . . . . . . 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 5918 . . . . . 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 577 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 1st `  A )  /\  h  e.  ( 1st `  B ) ) )  ->  (
g G h )  e.  Q. )
22 eleq1 2200 . . . 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 2555 . 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 2144 . . . . . 6  |-  ( f  =  C  ->  (
f  =  ( g G h )  <->  C  =  ( g G h ) ) )
26252rexbidv 2458 . . . . 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 2836 . . . 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 457 . . 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 694 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 962    = wceq 1331    e. wcel 1480   E.wrex 2415   {crab 2418   <.cop 3525   ` cfv 5118  (class class class)co 5767    e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   P.cnp 7092
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-qs 6428  df-ni 7105  df-nqqs 7149  df-inp 7267
This theorem is referenced by:  genpprecll  7315  genpcdl  7320  genprndl  7322  genpdisj  7324  genpassl  7325  addnqprlemrl  7358  mulnqprlemrl  7374  distrlem1prl  7383  distrlem5prl  7387  1idprl  7391  ltexprlemfl  7410  recexprlem1ssl  7434  recexprlemss1l  7436  cauappcvgprlemladdfl  7456
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