Theorem genpprecll 7574

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 2193	. . 3 |- ( C G D ) = ( C G D )
2		rspceov 5960	. . 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 1337	. 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 7572	. 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	imbitrrid 156	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 104   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   {crab 2476   <.cop 3621   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   P.cnp 7351

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-qs 6593  df-ni 7364  df-nqqs 7408  df-inp 7526

This theorem is referenced by:  genpml  7577  genprndl  7581  addnqprl  7589  mulnqprl  7628  distrlem1prl  7642  distrlem4prl  7644  ltaddpr  7657  ltexprlemrl  7670  addcanprleml  7674  addcanprlemu  7675