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.

Step	Hyp	Ref	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 ) )
3	1, 2	mp3an3 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. )
6	4, 5	genpelvl 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 ) ) )
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 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