Theorem genpcdl 6617

Description: Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-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. )
genpcdl.2	|- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) )

Assertion

Ref	Expression
genpcdl	|- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 1st ` ( A F B ) ) -> ( x <Q f -> x e. ( 1st ` ( 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 genpcdl

Step	Hyp	Ref	Expression
1		ltrelnq 6463	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2	1	brel 4392	. . . . . 6 |- ( x <Q f -> ( x e. Q. /\ f e. Q. ) )
3	2	simpld 105	. . . . 5 |- ( x <Q f -> 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	genpelvl 6610	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 1st ` ( A F B ) ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) ) )
7	6	adantr 261	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( f e. ( 1st ` ( A F B ) ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) ) )
8		breq2 3768	. . . . . . . . . . . . 13 |- ( f = ( g G h ) -> ( x <Q f <-> x <Q ( g G h ) ) )
9	8	biimpd 132	. . . . . . . . . . . 12 |- ( f = ( g G h ) -> ( x <Q f -> x <Q ( g G h ) ) )
10		genpcdl.2	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) )
11	9, 10	sylan9r 390	. . . . . . . . . . 11 |- ( ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) /\ f = ( g G h ) ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) )
12	11	exp31 346	. . . . . . . . . 10 |- ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) -> ( x e. Q. -> ( f = ( g G h ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) ) )
13	12	an4s 522	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 1st ` A ) /\ h e. ( 1st ` B ) ) ) -> ( x e. Q. -> ( f = ( g G h ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) ) )
14	13	impancom 247	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( ( g e. ( 1st ` A ) /\ h e. ( 1st ` B ) ) -> ( f = ( g G h ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) ) )
15	14	rexlimdvv 2439	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) )
16	7, 15	sylbid 139	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( f e. ( 1st ` ( A F B ) ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) )
17	16	ex 108	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( x e. Q. -> ( f e. ( 1st ` ( A F B ) ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) ) )
18	3, 17	syl5 28	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( x <Q f -> ( f e. ( 1st ` ( A F B ) ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) ) )
19	18	com34 77	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( x <Q f -> ( x <Q f -> ( f e. ( 1st ` ( A F B ) ) -> x e. ( 1st ` ( A F B ) ) ) ) ) )
20	19	pm2.43d 44	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( x <Q f -> ( f e. ( 1st ` ( A F B ) ) -> x e. ( 1st ` ( A F B ) ) ) ) )
21	20	com23 72	1 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 1st ` ( A F B ) ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   <Q cltq 6383   P.cnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-qs 6112  df-ni 6402  df-nqqs 6446  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  genprndl  6619