Theorem genppreclu 7665

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 2207	. . 3 |- ( C G D ) = ( C G D )
2		rspceov 6012	. . 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 1339	. 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 7663	. 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	imbitrrid 156	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 104   /\ w3a 981   = wceq 1373   e. wcel 2178   E.wrex 2487   {crab 2490   <.cop 3647   ` cfv 5291  (class class class)co 5969   e. cmpo 5971   1stc1st 6249   2ndc2nd 6250   Q.cnq 7430   P.cnp 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4176  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-iinf 4655

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-qs 6651  df-ni 7454  df-nqqs 7498  df-inp 7616

This theorem is referenced by:  genpmu  7668  genprndu  7672  addnqpru  7680  mulnqpru  7719  distrlem1pru  7733  distrlem4pru  7735  ltexprlemru  7762  addcanprleml  7764  addcanprlemu  7765