Theorem genpcuu 7519

Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-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. )
genpcuu.2	|- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )

Assertion

Ref	Expression
genpcuu	|- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )

Distinct variable groups:   x, y, z, f, g, h, w, v, A   x, B, y, z, f, g, h, w, v   x, G, y, z, f, g, h, w, v   f, F, g, h

Allowed substitution hints:   F( x, y, z, w, v)

Proof of Theorem genpcuu

Step	Hyp	Ref	Expression
1		ltrelnq 7364	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2	1	brel 4679	. . . . . 6 |- ( f <Q x -> ( f e. Q. /\ x e. Q. ) )
3	2	simprd 114	. . . . 5 |- ( f <Q x -> x e. Q. )
4		genpelvl.1	. . . . . . . . 9 |- 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	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
6	4, 5	genpelvu 7512	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) ) )
7	6	adantr 276	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( f e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) ) )
8		breq1 4007	. . . . . . . . . . . . 13 |- ( f = ( g G h ) -> ( f <Q x <-> ( g G h ) <Q x ) )
9	8	biimpd 144	. . . . . . . . . . . 12 |- ( f = ( g G h ) -> ( f <Q x -> ( g G h ) <Q x ) )
10		genpcuu.2	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )
11	9, 10	sylan9r 410	. . . . . . . . . . 11 |- ( ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) /\ f = ( g G h ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) )
12	11	exp31 364	. . . . . . . . . 10 |- ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) -> ( x e. Q. -> ( f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
13	12	an4s 588	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 2nd ` A ) /\ h e. ( 2nd ` B ) ) ) -> ( x e. Q. -> ( f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
14	13	impancom 260	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( ( g e. ( 2nd ` A ) /\ h e. ( 2nd ` B ) ) -> ( f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
15	14	rexlimdvv 2601	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )
16	7, 15	sylbid 150	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )
17	16	ex 115	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( x e. Q. -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
18	3, 17	syl5 32	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( f <Q x -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
19	18	com34 83	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( f <Q x -> ( f <Q x -> ( f e. ( 2nd ` ( A F B ) ) -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
20	19	pm2.43d 50	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( f <Q x -> ( f e. ( 2nd ` ( A F B ) ) -> x e. ( 2nd ` ( A F B ) ) ) ) )
21	20	com23 78	1 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3596   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   e. cmpo 5877   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   <Q cltq 7284   P.cnp 7290

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-qs 6541  df-ni 7303  df-nqqs 7347  df-ltnqqs 7352  df-inp 7465

This theorem is referenced by:  genprndu  7521