Theorem genpcuu 7703

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 7548	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2	1	brel 4770	. . . . . 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 7696	. . . . . . . 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 4085	. . . . . . . . . . . . 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 590	. . . . . . . . 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 2655	. . . . . . 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 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   {crab 2512   <.cop 3669   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   e. cmpo 6002   1stc1st 6282   2ndc2nd 6283   Q.cnq 7463   <Q cltq 7468   P.cnp 7474

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-qs 6684  df-ni 7487  df-nqqs 7531  df-ltnqqs 7536  df-inp 7649

This theorem is referenced by:  genprndu  7705