Theorem genpcdl 7439

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 7285	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2	1	brel 4638	. . . . . 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 7432	. . . . . . . 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 3969	. . . . . . . . . . . . 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 408	. . . . . . . . . . 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 362	. . . . . . . . . 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 578	. . . . . . . . 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 2581	. . . . . . 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 963   = wceq 1335   e. wcel 2128   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   e. cmpo 5826   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200   <Q cltq 7205   P.cnp 7211

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-qs 6486  df-ni 7224  df-nqqs 7268  df-ltnqqs 7273  df-inp 7386

This theorem is referenced by:  genprndl  7441