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Theorem genppreclu 7545
Description: Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-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
genppreclu  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  ( 2nd `  A
)  /\  D  e.  ( 2nd `  B ) )  ->  ( C G D )  e.  ( 2nd `  ( A F B ) ) ) )
Distinct variable groups:    x, y, z, w, v, A    x, B, y, z, w, v   
x, G, y, z, w, v
Allowed substitution hints:    C( x, y, z, w, v)    D( x, y, z, w, v)    F( x, y, z, w, v)

Proof of Theorem genppreclu
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2189 . . 3  |-  ( C G D )  =  ( C G D )
2 rspceov 5939 . . 3  |-  ( ( C  e.  ( 2nd `  A )  /\  D  e.  ( 2nd `  B
)  /\  ( C G D )  =  ( C G D ) )  ->  E. g  e.  ( 2nd `  A
) E. h  e.  ( 2nd `  B
) ( C G D )  =  ( g G h ) )
31, 2mp3an3 1337 . 2  |-  ( ( C  e.  ( 2nd `  A )  /\  D  e.  ( 2nd `  B
) )  ->  E. g  e.  ( 2nd `  A
) E. h  e.  ( 2nd `  B
) ( C G D )  =  ( g G h ) )
4 genpelvl.1 . . 3  |-  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 ) ) } >. )
5 genpelvl.2 . . 3  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
64, 5genpelvu 7543 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C G D )  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) ( C G D )  =  ( g G h ) ) )
73, 6imbitrrid 156 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  ( 2nd `  A
)  /\  D  e.  ( 2nd `  B ) )  ->  ( C G D )  e.  ( 2nd `  ( A F B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   E.wrex 2469   {crab 2472   <.cop 3610   ` cfv 5235  (class class class)co 5897    e. cmpo 5899   1stc1st 6164   2ndc2nd 6165   Q.cnq 7310   P.cnp 7321
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-qs 6566  df-ni 7334  df-nqqs 7378  df-inp 7496
This theorem is referenced by:  genpmu  7548  genprndu  7552  addnqpru  7560  mulnqpru  7599  distrlem1pru  7613  distrlem4pru  7615  ltexprlemru  7642  addcanprleml  7644  addcanprlemu  7645
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