Theorem genpnmax 5090

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 5082	. . 3 |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})
3	2	abeq2d 1569	. 2 |- ((A e. P. /\ B e. P.) -> (f e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ f = (gGh))))
4		breq1 2617	. . . . . . . . 9 |- (f = (gGh) -> (f <Q x <-> (gGh) <Q x))
5	4	anbi2d 615	. . . . . . . 8 |- (f = (gGh) -> ((x e. (AFB) /\ f <Q x) <-> (x e. (AFB) /\ (gGh) <Q x)))
6	5	exbidv 1277	. . . . . . 7 |- (f = (gGh) -> (E.x(x e. (AFB) /\ f <Q x) <-> E.x(x e. (AFB) /\ (gGh) <Q x)))
7		prnmax 5079	. . . . . . . . . 10 |- ((A e. P. /\ g e. A) -> E.y(y e. A /\ g <Q y))
8	7	adantr 389	. . . . . . . . 9 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> E.y(y e. A /\ g <Q y))
9	1	genpprecl 5084	. . . . . . . . . . . . . . . 16 |- ((A e. P. /\ B e. P.) -> ((y e. A /\ h e. B) -> (yGh) e. (AFB)))
10	9	exp4b 379	. . . . . . . . . . . . . . 15 |- (A e. P. -> (B e. P. -> (y e. A -> (h e. B -> (yGh) e. (AFB)))))
11	10	com34 36	. . . . . . . . . . . . . 14 |- (A e. P. -> (B e. P. -> (h e. B -> (y e. A -> (yGh) e. (AFB)))))
12	11	imp32 363	. . . . . . . . . . . . 13 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> (y e. A -> (yGh) e. (AFB)))
13		elprpq 5075	. . . . . . . . . . . . . . 15 |- ((B e. P. /\ h e. B) -> h e. Q.)
14		visset 1809	. . . . . . . . . . . . . . . . 17 |- g e. V
15		visset 1809	. . . . . . . . . . . . . . . . 17 |- y e. V
16		genpnmax.2	. . . . . . . . . . . . . . . . 17 |- (v e. Q. -> (z <Q w <-> (vGz) <Q (vGw)))
17		visset 1809	. . . . . . . . . . . . . . . . 17 |- h e. V
18		genpnmax.3	. . . . . . . . . . . . . . . . 17 |- (zGw) = (wGz)
19	14, 15, 16, 17, 18	caoprord2 4049	. . . . . . . . . . . . . . . 16 |- (h e. Q. -> (g <Q y <-> (gGh) <Q (yGh)))
20	19	biimpd 153	. . . . . . . . . . . . . . 15 |- (h e. Q. -> (g <Q y -> (gGh) <Q (yGh)))
21	13, 20	syl 10	. . . . . . . . . . . . . 14 |- ((B e. P. /\ h e. B) -> (g <Q y -> (gGh) <Q (yGh)))
22	21	adantl 388	. . . . . . . . . . . . 13 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> (g <Q y -> (gGh) <Q (yGh)))
23	12, 22	anim12d 557	. . . . . . . . . . . 12 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> ((y e. A /\ g <Q y) -> ((yGh) e. (AFB) /\ (gGh) <Q (yGh))))
24		oprex 3974	. . . . . . . . . . . . 13 |- (yGh) e. V
25		eleq1 1531	. . . . . . . . . . . . . 14 |- (x = (yGh) -> (x e. (AFB) <-> (yGh) e. (AFB)))
26		breq2 2618	. . . . . . . . . . . . . 14 |- (x = (yGh) -> ((gGh) <Q x <-> (gGh) <Q (yGh)))
27	25, 26	anbi12d 627	. . . . . . . . . . . . 13 |- (x = (yGh) -> ((x e. (AFB) /\ (gGh) <Q x) <-> ((yGh) e. (AFB) /\ (gGh) <Q (yGh))))
28	24, 27	cla4ev 1865	. . . . . . . . . . . 12 |- (((yGh) e. (AFB) /\ (gGh) <Q (yGh)) -> E.x(x e. (AFB) /\ (gGh) <Q x))
29	23, 28	syl6 22	. . . . . . . . . . 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 393	. . . . . . . . . 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 1212	. . . . . . . . 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 26	. . . . . . . 8 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> E.x(x e. (AFB) /\ (gGh) <Q x))
33	32	an4s 508	. . . . . . 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 210	. . . . . 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 375	. . . . 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 34	. . . 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 361	. . 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 1295	. 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 203	1 |- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> E.x(x e. (AFB) /\ f <Q x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  E.wrex 1643   class class class wbr 2614  (class class class)co 3954  {copab2 3955  Q.cnq 4959   <Q cltq 4964  P.cnp 4965

This theorem is referenced by:  genpcl 5091

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-opr 3956  df-oprab 3957  df-qs 4256  df-ni 4980  df-nq 5018  df-np 5066

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