Theorem genpnmax 6262

Description: An operation on positive reals has no largest member.

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)})}
genpnmax.2	|- (v e. Q. -> (z <Q w <-> (vGz) <Q (vGw)))
genpnmax.3	|- (zGw) = (wGz)

Assertion

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

Distinct variable groups:   x,y,z,f,A   x,B,y,z,f   x,w,v,u,G,y,z,f   f,F,x,y

Proof of Theorem genpnmax

Step	Hyp	Ref	Expression
1		genp.1	. . . 4 |- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}
2	1	genpv 6254	. . 3 |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})
3	2	abeq2d 2003	. 2 |- ((A e. P. /\ B e. P.) -> (f e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ f = (gGh))))
4		breq1 3341	. . . . . . . . 9 |- (f = (gGh) -> (f <Q x <-> (gGh) <Q x))
5	4	anbi2d 678	. . . . . . . 8 |- (f = (gGh) -> ((x e. (AFB) /\ f <Q x) <-> (x e. (AFB) /\ (gGh) <Q x)))
6	5	exbidv 1657	. . . . . . 7 |- (f = (gGh) -> (E.x(x e. (AFB) /\ f <Q x) <-> E.x(x e. (AFB) /\ (gGh) <Q x)))
7		prnmax 6251	. . . . . . . . . 10 |- ((A e. P. /\ g e. A) -> E.y(y e. A /\ g <Q y))
8	7	adantr 425	. . . . . . . . 9 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> E.y(y e. A /\ g <Q y))
9	1	genpprecl 6256	. . . . . . . . . . . . . . . 16 |- ((A e. P. /\ B e. P.) -> ((y e. A /\ h e. B) -> (yGh) e. (AFB)))
10	9	exp4b 410	. . . . . . . . . . . . . . 15 |- (A e. P. -> (B e. P. -> (y e. A -> (h e. B -> (yGh) e. (AFB)))))
11	10	com34 40	. . . . . . . . . . . . . 14 |- (A e. P. -> (B e. P. -> (h e. B -> (y e. A -> (yGh) e. (AFB)))))
12	11	imp32 390	. . . . . . . . . . . . 13 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> (y e. A -> (yGh) e. (AFB)))
13		elprpq 6247	. . . . . . . . . . . . . . 15 |- ((B e. P. /\ h e. B) -> h e. Q.)
14		visset 2295	. . . . . . . . . . . . . . . . 17 |- g e. _V
15		visset 2295	. . . . . . . . . . . . . . . . 17 |- y e. _V
16		genpnmax.2	. . . . . . . . . . . . . . . . 17 |- (v e. Q. -> (z <Q w <-> (vGz) <Q (vGw)))
17		visset 2295	. . . . . . . . . . . . . . . . 17 |- h e. _V
18		genpnmax.3	. . . . . . . . . . . . . . . . 17 |- (zGw) = (wGz)
19	14, 15, 16, 17, 18	caoprord2 4990	. . . . . . . . . . . . . . . 16 |- (h e. Q. -> (g <Q y <-> (gGh) <Q (yGh)))
20	19	biimpd 170	. . . . . . . . . . . . . . 15 |- (h e. Q. -> (g <Q y -> (gGh) <Q (yGh)))
21	13, 20	syl 12	. . . . . . . . . . . . . 14 |- ((B e. P. /\ h e. B) -> (g <Q y -> (gGh) <Q (yGh)))
22	21	adantl 424	. . . . . . . . . . . . 13 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> (g <Q y -> (gGh) <Q (yGh)))
23	12, 22	anim12d 617	. . . . . . . . . . . 12 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> ((y e. A /\ g <Q y) -> ((yGh) e. (AFB) /\ (gGh) <Q (yGh))))
24		oprex 4907	. . . . . . . . . . . . 13 |- (yGh) e. _V
25		eleq1 1957	. . . . . . . . . . . . . 14 |- (x = (yGh) -> (x e. (AFB) <-> (yGh) e. (AFB)))
26		breq2 3342	. . . . . . . . . . . . . 14 |- (x = (yGh) -> ((gGh) <Q x <-> (gGh) <Q (yGh)))
27	25, 26	anbi12d 690	. . . . . . . . . . . . 13 |- (x = (yGh) -> ((x e. (AFB) /\ (gGh) <Q x) <-> ((yGh) e. (AFB) /\ (gGh) <Q (yGh))))
28	24, 27	cla4ev 2371	. . . . . . . . . . . 12 |- (((yGh) e. (AFB) /\ (gGh) <Q (yGh)) -> E.x(x e. (AFB) /\ (gGh) <Q x))
29	23, 28	syl6 25	. . . . . . . . . . 11 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> ((y e. A /\ g <Q y) -> E.x(x e. (AFB) /\ (gGh) <Q x)))
30	29	adantlr 429	. . . . . . . . . 10 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> ((y e. A /\ g <Q y) -> E.x(x e. (AFB) /\ (gGh) <Q x)))
31	30	19.23adv 1584	. . . . . . . . 9 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (E.y(y e. A /\ g <Q y) -> E.x(x e. (AFB) /\ (gGh) <Q x)))
32	8, 31	mpd 29	. . . . . . . 8 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> E.x(x e. (AFB) /\ (gGh) <Q x))
33	32	an4s 566	. . . . . . 7 |- (((A e. P. /\ B e. P.) /\ (g e. A /\ h e. B)) -> E.x(x e. (AFB) /\ (gGh) <Q x))
34	6, 33	syl5bir 227	. . . . . 6 |- (f = (gGh) -> (((A e. P. /\ B e. P.) /\ (g e. A /\ h e. B)) -> E.x(x e. (AFB) /\ f <Q x)))
35	34	exp3a 405	. . . . 5 |- (f = (gGh) -> ((A e. P. /\ B e. P.) -> ((g e. A /\ h e. B) -> E.x(x e. (AFB) /\ f <Q x))))
36	35	com3l 38	. . . 4 |- ((A e. P. /\ B e. P.) -> ((g e. A /\ h e. B) -> (f = (gGh) -> E.x(x e. (AFB) /\ f <Q x))))
37	36	imp3a 388	. . 3 |- ((A e. P. /\ B e. P.) -> (((g e. A /\ h e. B) /\ f = (gGh)) -> E.x(x e. (AFB) /\ f <Q x)))
38	37	19.23advv 1676	. 2 |- ((A e. P. /\ B e. P.) -> (E.gE.h((g e. A /\ h e. B) /\ f = (gGh)) -> E.x(x e. (AFB) /\ f <Q x)))
39	3, 38	sylbid 220	1 |- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> E.x(x e. (AFB) /\ f <Q x)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106   class class class wbr 3338  (class class class)co 4884  {copab2 4885  Q.cnq 6131   <Q cltq 6136  P.cnp 6137

This theorem is referenced by:  genpcl 6263

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-qs 5323  df-ni 6152  df-nq 6190  df-np 6238

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