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Theorem genppreclu 7330
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 2139 . . 3  |-  ( C G D )  =  ( C G D )
2 rspceov 5813 . . 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 1304 . 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 7328 . 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, 6syl5ibr 155 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417   {crab 2420   <.cop 3530   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095   P.cnp 7106
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 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7119  df-nqqs 7163  df-inp 7281
This theorem is referenced by:  genpmu  7333  genprndu  7337  addnqpru  7345  mulnqpru  7384  distrlem1pru  7398  distrlem4pru  7400  ltexprlemru  7427  addcanprleml  7429  addcanprlemu  7430
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