Theorem genpprecll 7322

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.

Step	Hyp	Ref	Expression
1		eqid 2139	. . 3 |- ( C G D ) = ( C G D )
2		rspceov 5813	. . 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 ) )
3	1, 2	mp3an3 1304	. 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. )
6	4, 5	genpelvl 7320	. 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 ) ) )
7	3, 6	syl5ibr 155	1 |- ( ( A e. P. /\ B e. P. ) -> ( ( C e. ( 1st ` A ) /\ D e. ( 1st ` B ) ) -> ( C G D ) e. ( 1st ` ( A F B ) ) ) )

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 7088   P.cnp 7099

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 7112  df-nqqs 7156  df-inp 7274

This theorem is referenced by:  genpml  7325  genprndl  7329  addnqprl  7337  mulnqprl  7376  distrlem1prl  7390  distrlem4prl  7392  ltaddpr  7405  ltexprlemrl  7418  addcanprleml  7422  addcanprlemu  7423