Theorem genpcdl 7782

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 7628	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2	1	brel 4784	. . . . . 6 |- ( x <Q f -> ( x e. Q. /\ f e. Q. ) )
3	2	simpld 112	. . . . 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 7775	. . . . . . . 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 276	. . . . . . 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 4097	. . . . . . . . . . . . 13 |- ( f = ( g G h ) -> ( x <Q f <-> x <Q ( g G h ) ) )
9	8	biimpd 144	. . . . . . . . . . . 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 410	. . . . . . . . . . 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 364	. . . . . . . . . 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 592	. . . . . . . . 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 260	. . . . . . . 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 2658	. . . . . . 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 150	. . . . . 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 115	. . . . 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 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   {crab 2515   <.cop 3676   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311   Q.cnq 7543   <Q cltq 7548   P.cnp 7554

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-qs 6751  df-ni 7567  df-nqqs 7611  df-ltnqqs 7616  df-inp 7729

This theorem is referenced by:  genprndl  7784