ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genpprecll Unicode version

Theorem genpprecll 7223
Description: Pre-closure law for general operation on lower cuts. (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
genpprecll  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  ( 1st `  A
)  /\  D  e.  ( 1st `  B ) )  ->  ( C G D )  e.  ( 1st `  ( 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 genpprecll
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2100 . . 3  |-  ( C G D )  =  ( C G D )
2 rspceov 5745 . . 3  |-  ( ( C  e.  ( 1st `  A )  /\  D  e.  ( 1st `  B
)  /\  ( C G D )  =  ( C G D ) )  ->  E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  B
) ( C G D )  =  ( g G h ) )
31, 2mp3an3 1272 . 2  |-  ( ( C  e.  ( 1st `  A )  /\  D  e.  ( 1st `  B
) )  ->  E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  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, 5genpelvl 7221 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C G D )  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) ( C G D )  =  ( g G h ) ) )
73, 6syl5ibr 155 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  ( 1st `  A
)  /\  D  e.  ( 1st `  B ) )  ->  ( C G D )  e.  ( 1st `  ( A F B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 930    = wceq 1299    e. wcel 1448   E.wrex 2376   {crab 2379   <.cop 3477   ` cfv 5059  (class class class)co 5706    e. cmpo 5708   1stc1st 5967   2ndc2nd 5968   Q.cnq 6989   P.cnp 7000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-qs 6365  df-ni 7013  df-nqqs 7057  df-inp 7175
This theorem is referenced by:  genpml  7226  genprndl  7230  addnqprl  7238  mulnqprl  7277  distrlem1prl  7291  distrlem4prl  7293  ltaddpr  7306  ltexprlemrl  7319  addcanprleml  7323  addcanprlemu  7324
  Copyright terms: Public domain W3C validator