Theorem genpcd 5109

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 groups:   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 1813	. . . . . . . . 9 |- f e. V
3	1, 2	genpelv 5103	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> (f e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ f = (gGh))))
4	3	adantr 389	. . . . . . 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 2623	. . . . . . . . . . . . . . 15 |- (f = (gGh) -> (x <Q f <-> x <Q (gGh)))
6	5	biimpd 153	. . . . . . . . . . . . . 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 469	. . . . . . . . . . . . 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 376	. . . . . . . . . . . 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 508	. . . . . . . . . . 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	ex 373	. . . . . . . . . 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 365	. . . . . . . 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 1297	. . . . . . 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 203	. . . . . 6 |- (((A e. P. /\ B e. P.) /\ x e. Q.) -> (f e. (AFB) -> (x <Q f -> x e. (AFB))))
16	15	ex 373	. . . . 5 |- ((A e. P. /\ B e. P.) -> (x e. Q. -> (f e. (AFB) -> (x <Q f -> x e. (AFB)))))
17		ltrelpq 5051	. . . . . . 7 |- <Q (_ (Q. X. Q.)
18	2, 17	brel 3223	. . . . . 6 |- (x <Q f -> (x e. Q. /\ f e. Q.))
19	18	pm3.26d 321	. . . . 5 |- (x <Q f -> x e. Q.)
20	16, 19	syl5 21	. . . 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 65	. 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 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646   class class class wbr 2619  (class class class)co 3963  {copab2 3964  Q.cnq 4979   <Q cltq 4984  P.cnp 4985

This theorem is referenced by:  genpcl 5111

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966  df-ltq 5042

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