Theorem genppreclu 6705

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.

Step	Hyp	Ref	Expression
1		eqid 2081	. . 3 |- ( C G D ) = ( C G D )
2		rspceov 5567	. . 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 ) )
3	1, 2	mp3an3 1257	. 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. )
6	4, 5	genpelvu 6703	. 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 ) ) )
7	3, 6	syl5ibr 154	1 |- ( ( A e. P. /\ B e. P. ) -> ( ( C e. ( 2nd ` A ) /\ D e. ( 2nd ` B ) ) -> ( C G D ) e. ( 2nd ` ( A F B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   E.wrex 2349   {crab 2352   <.cop 3401   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   P.cnp 6481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-qs 6135  df-ni 6494  df-nqqs 6538  df-inp 6656

This theorem is referenced by:  genpmu  6708  genprndu  6712  addnqpru  6720  mulnqpru  6759  distrlem1pru  6773  distrlem4pru  6775  ltexprlemru  6802  addcanprleml  6804  addcanprlemu  6805