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Theorem genppreclu 7291
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 2117 . . 3  |-  ( C G D )  =  ( C G D )
2 rspceov 5781 . . 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 1289 . 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 7289 . 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 947    = wceq 1316    e. wcel 1465   E.wrex 2394   {crab 2397   <.cop 3500   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   P.cnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-qs 6403  df-ni 7080  df-nqqs 7124  df-inp 7242
This theorem is referenced by:  genpmu  7294  genprndu  7298  addnqpru  7306  mulnqpru  7345  distrlem1pru  7359  distrlem4pru  7361  ltexprlemru  7388  addcanprleml  7390  addcanprlemu  7391
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