Theorem genpcd 3903

Description: Downward closure of an operation on positive reals.

Hypotheses

Ref	Expression
genp.1	|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}
genpcd.2	|- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (gGh) -> x e. (AFB)))

Assertion

Ref	Expression
genpcd	|- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> (x <Q f -> x e. (AFB))))

Distinct variable group(s):   x,y,z,f,g,h,A   x,B,y,z,f,g,h   x,w,v,u,G,y,z,f,g,h   f,F,g,h

Proof of Theorem genpcd

Step	Hyp	Ref	Expression
1		genp.1	. . . . . . . . 9 |- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}
2		visset 1350	. . . . . . . . 9 |- f e. V
3	1, 2	genpelv 3897	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> (f e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ f = (gGh))))
4	3	adantr 306	. . . . . . 7 |- (((A e. P. /\ B e. P.) /\ x e. Q.) -> (f e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ f = (gGh))))
5		breq2 2066	. . . . . . . . . . . . . . 15 |- (f = (gGh) -> (x <Q f <-> x <Q (gGh)))
6	5	biimpd 135	. . . . . . . . . . . . . 14 |- (f = (gGh) -> (x <Q f -> x <Q (gGh)))
7		genpcd.2	. . . . . . . . . . . . . 14 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (gGh) -> x e. (AFB)))
8	6, 7	sylan9r 360	. . . . . . . . . . . . 13 |- (((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) /\ f = (gGh)) -> (x <Q f -> x e. (AFB)))
9	8	exp31 293	. . . . . . . . . . . 12 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (x e. Q. -> (f = (gGh) -> (x <Q f -> x e. (AFB)))))
10	9	an4s 390	. . . . . . . . . . 11 |- (((A e. P. /\ B e. P.) /\ (g e. A /\ h e. B)) -> (x e. Q. -> (f = (gGh) -> (x <Q f -> x e. (AFB)))))
11	10	exp 291	. . . . . . . . . 10 |- ((A e. P. /\ B e. P.) -> ((g e. A /\ h e. B) -> (x e. Q. -> (f = (gGh) -> (x <Q f -> x e. (AFB))))))
12	11	com23 32	. . . . . . . . 9 |- ((A e. P. /\ B e. P.) -> (x e. Q. -> ((g e. A /\ h e. B) -> (f = (gGh) -> (x <Q f -> x e. (AFB))))))
13	12	imp4b 283	. . . . . . . 8 |- (((A e. P. /\ B e. P.) /\ x e. Q.) -> (((g e. A /\ h e. B) /\ f = (gGh)) -> (x <Q f -> x e. (AFB))))
14	13	19.23advv 955	. . . . . . 7 |- (((A e. P. /\ B e. P.) /\ x e. Q.) -> (E.gE.h((g e. A /\ h e. B) /\ f = (gGh)) -> (x <Q f -> x e. (AFB))))
15	4, 14	sylbid 178	. . . . . 6 |- (((A e. P. /\ B e. P.) /\ x e. Q.) -> (f e. (AFB) -> (x <Q f -> x e. (AFB))))
16	15	exp 291	. . . . 5 |- ((A e. P. /\ B e. P.) -> (x e. Q. -> (f e. (AFB) -> (x <Q f -> x e. (AFB)))))
17		ltrelpq 3845	. . . . . . 7 |- <Q (_ (Q. X. Q.)
18	2, 17	brel 2459	. . . . . 6 |- (x <Q f -> (x e. Q. /\ f e. Q.))
19	18	pm3.26d 258	. . . . 5 |- (x <Q f -> x e. Q.)
20	16, 19	syl5 22	. . . 4 |- ((A e. P. /\ B e. P.) -> (x <Q f -> (f e. (AFB) -> (x <Q f -> x e. (AFB)))))
21	20	com34 36	. . 3 |- ((A e. P. /\ B e. P.) -> (x <Q f -> (x <Q f -> (f e. (AFB) -> x e. (AFB)))))
22	21	pm2.43d 59	. 2 |- ((A e. P. /\ B e. P.) -> (x <Q f -> (f e. (AFB) -> x e. (AFB))))
23	22	com23 32	1 |- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> (x <Q f -> x e. (AFB))))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202   class class class wbr 2054  (class class class)co 3001  {copab2 3002  Q.cnq 3773   <Q cltq 3778  P.cnp 3779

This theorem is referenced by:  genpcl 3905

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004  df-ltq 3836

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