Theorem genpcdl 7320

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 7166	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2	1	brel 4586	. . . . . 6 |- ( x <Q f -> ( x e. Q. /\ f e. Q. ) )
3	2	simpld 111	. . . . 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 7313	. . . . . . . 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 274	. . . . . . 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 3928	. . . . . . . . . . . . 13 |- ( f = ( g G h ) -> ( x <Q f <-> x <Q ( g G h ) ) )
9	8	biimpd 143	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 361	. . . . . . . . . 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 577	. . . . . . . . 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 258	. . . . . . . 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 2554	. . . . . . 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 149	. . . . . 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 114	. . . . 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 32	. . . 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 83	. . 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 50	. 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 78	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 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   <Q cltq 7086   P.cnp 7092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-qs 6428  df-ni 7105  df-nqqs 7149  df-ltnqqs 7154  df-inp 7267

This theorem is referenced by:  genprndl  7322