Theorem amgmwlem 42548

Description: Weighted version of amgmlem 24716. (Contributed by Kunhao Zheng, 19-Jun-2021.)

Hypotheses

Ref	Expression
amgmwlem.0	|- M = (mulGrp ` ℂfld )
amgmwlem.1	|- ( ph -> A e. Fin )
amgmwlem.2	|- ( ph -> A =/= (/) )
amgmwlem.3	|- ( ph -> F : A --> RR+ )
amgmwlem.4	|- ( ph -> W : A --> RR+ )
amgmwlem.5	|- ( ph -> (ℂfld gsumg W ) = 1 )

Assertion

Ref	Expression
amgmwlem	|- ( ph -> ( M gsumg ( F oF ^c W ) ) <_ (ℂfld gsumg ( F oF x. W ) ) )

Proof of Theorem amgmwlem

Dummy variables x k a b s u v y w t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		amgmwlem.1	. . . . . . . 8 |- ( ph -> A e. Fin )
2		amgmwlem.3	. . . . . . . . . . . 12 |- ( ph -> F : A --> RR+ )
3	2	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. RR+ )
4		amgmwlem.4	. . . . . . . . . . . . 13 |- ( ph -> W : A --> RR+ )
5	4	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> ( W ` k ) e. RR+ )
6	5	rpred 11872	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( W ` k ) e. RR )
7	3, 6	rpcxpcld 24476	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( ( F ` k ) ^c ( W ` k ) ) e. RR+ )
8	7	relogcld 24369	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) e. RR )
9	8	recnd 10068	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) e. CC )
10	1, 9	gsumfsum 19813	. . . . . . 7 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) ) ) = sum_ k e. A ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) )
11	9	negnegd 10383	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> -u -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) )
12	11	sumeq2dv 14433	. . . . . . 7 |- ( ph -> sum_ k e. A -u -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = sum_ k e. A ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) )
13	8	renegcld 10457	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) e. RR )
14	13	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) e. CC )
15	1, 14	fsumneg 14519	. . . . . . . 8 |- ( ph -> sum_ k e. A -u -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = -u sum_ k e. A -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) )
16	3, 6	logcxpd 24477	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = ( ( W ` k ) x. ( log ` ( F ` k ) ) ) )
17	16	negeqd 10275	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = -u ( ( W ` k ) x. ( log ` ( F ` k ) ) ) )
18	17	sumeq2dv 14433	. . . . . . . . 9 |- ( ph -> sum_ k e. A -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = sum_ k e. A -u ( ( W ` k ) x. ( log ` ( F ` k ) ) ) )
19	18	negeqd 10275	. . . . . . . 8 |- ( ph -> -u sum_ k e. A -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = -u sum_ k e. A -u ( ( W ` k ) x. ( log ` ( F ` k ) ) ) )
20	5	rpcnd 11874	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> ( W ` k ) e. CC )
21	3	relogcld 24369	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. A ) -> ( log ` ( F ` k ) ) e. RR )
22	21	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> ( log ` ( F ` k ) ) e. CC )
23	20, 22	mulneg2d 10484	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) = -u ( ( W ` k ) x. ( log ` ( F ` k ) ) ) )
24	23	eqcomd 2628	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> -u ( ( W ` k ) x. ( log ` ( F ` k ) ) ) = ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) )
25	24	sumeq2dv 14433	. . . . . . . . 9 |- ( ph -> sum_ k e. A -u ( ( W ` k ) x. ( log ` ( F ` k ) ) ) = sum_ k e. A ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) )
26	25	negeqd 10275	. . . . . . . 8 |- ( ph -> -u sum_ k e. A -u ( ( W ` k ) x. ( log ` ( F ` k ) ) ) = -u sum_ k e. A ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) )
27	15, 19, 26	3eqtrd 2660	. . . . . . 7 |- ( ph -> sum_ k e. A -u -u ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) = -u sum_ k e. A ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) )
28	10, 12, 27	3eqtr2rd 2663	. . . . . 6 |- ( ph -> -u sum_ k e. A ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) = (ℂfld gsumg ( k e. A |-> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) ) ) )
29		negex 10279	. . . . . . . . . . 11 |- -u ( log ` ( F ` k ) ) e. _V
30	29	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> -u ( log ` ( F ` k ) ) e. _V )
31	4	feqmptd 6249	. . . . . . . . . 10 |- ( ph -> W = ( k e. A |-> ( W ` k ) ) )
32		eqidd 2623	. . . . . . . . . 10 |- ( ph -> ( k e. A |-> -u ( log ` ( F ` k ) ) ) = ( k e. A |-> -u ( log ` ( F ` k ) ) ) )
33	1, 5, 30, 31, 32	offval2 6914	. . . . . . . . 9 |- ( ph -> ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) = ( k e. A |-> ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) ) )
34	33	oveq2d 6666	. . . . . . . 8 |- ( ph -> (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) = (ℂfld gsumg ( k e. A |-> ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) ) ) )
35	22	negcld 10379	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> -u ( log ` ( F ` k ) ) e. CC )
36	20, 35	mulcld 10060	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) e. CC )
37	1, 36	gsumfsum 19813	. . . . . . . 8 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) ) ) = sum_ k e. A ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) )
38	34, 37	eqtrd 2656	. . . . . . 7 |- ( ph -> (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) = sum_ k e. A ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) )
39	38	negeqd 10275	. . . . . 6 |- ( ph -> -u (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) = -u sum_ k e. A ( ( W ` k ) x. -u ( log ` ( F ` k ) ) ) )
40		relogf1o 24313	. . . . . . . . . 10 |- ( log |` RR+ ) : RR+ -1-1-onto-> RR
41		f1of 6137	. . . . . . . . . 10 |- ( ( log |` RR+ ) : RR+ -1-1-onto-> RR -> ( log |` RR+ ) : RR+ --> RR )
42	40, 41	ax-mp 5	. . . . . . . . 9 |- ( log |` RR+ ) : RR+ --> RR
43		rpre 11839	. . . . . . . . . . . . 13 |- ( y e. RR+ -> y e. RR )
44	43	anim2i 593	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ y e. RR+ ) -> ( x e. RR+ /\ y e. RR ) )
45	44	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( x e. RR+ /\ y e. RR ) )
46		rpcxpcl 24422	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ y e. RR ) -> ( x ^c y ) e. RR+ )
47	45, 46	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( x ^c y ) e. RR+ )
48		inidm 3822	. . . . . . . . . 10 |- ( A i^i A ) = A
49	47, 2, 4, 1, 1, 48	off 6912	. . . . . . . . 9 |- ( ph -> ( F oF ^c W ) : A --> RR+ )
50		fcompt 6400	. . . . . . . . 9 |- ( ( ( log |` RR+ ) : RR+ --> RR /\ ( F oF ^c W ) : A --> RR+ ) -> ( ( log |` RR+ ) o. ( F oF ^c W ) ) = ( k e. A |-> ( ( log |` RR+ ) ` ( ( F oF ^c W ) ` k ) ) ) )
51	42, 49, 50	sylancr 695	. . . . . . . 8 |- ( ph -> ( ( log |` RR+ ) o. ( F oF ^c W ) ) = ( k e. A |-> ( ( log |` RR+ ) ` ( ( F oF ^c W ) ` k ) ) ) )
52	49	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( ( F oF ^c W ) ` k ) e. RR+ )
53		fvres 6207	. . . . . . . . . . 11 |- ( ( ( F oF ^c W ) ` k ) e. RR+ -> ( ( log |` RR+ ) ` ( ( F oF ^c W ) ` k ) ) = ( log ` ( ( F oF ^c W ) ` k ) ) )
54	52, 53	syl 17	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( ( log |` RR+ ) ` ( ( F oF ^c W ) ` k ) ) = ( log ` ( ( F oF ^c W ) ` k ) ) )
55	2	ffnd 6046	. . . . . . . . . . . 12 |- ( ph -> F Fn A )
56	4	ffnd 6046	. . . . . . . . . . . 12 |- ( ph -> W Fn A )
57		eqidd 2623	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> ( F ` k ) = ( F ` k ) )
58		eqidd 2623	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> ( W ` k ) = ( W ` k ) )
59	55, 56, 1, 1, 48, 57, 58	ofval 6906	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( ( F oF ^c W ) ` k ) = ( ( F ` k ) ^c ( W ` k ) ) )
60	59	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( log ` ( ( F oF ^c W ) ` k ) ) = ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) )
61	54, 60	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> ( ( log |` RR+ ) ` ( ( F oF ^c W ) ` k ) ) = ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) )
62	61	mpteq2dva 4744	. . . . . . . 8 |- ( ph -> ( k e. A |-> ( ( log |` RR+ ) ` ( ( F oF ^c W ) ` k ) ) ) = ( k e. A |-> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) ) )
63	51, 62	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( log |` RR+ ) o. ( F oF ^c W ) ) = ( k e. A |-> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) ) )
64	63	oveq2d 6666	. . . . . 6 |- ( ph -> (ℂfld gsumg ( ( log |` RR+ ) o. ( F oF ^c W ) ) ) = (ℂfld gsumg ( k e. A |-> ( log ` ( ( F ` k ) ^c ( W ` k ) ) ) ) ) )
65	28, 39, 64	3eqtr4d 2666	. . . . 5 |- ( ph -> -u (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) = (ℂfld gsumg ( ( log |` RR+ ) o. ( F oF ^c W ) ) ) )
66		amgmwlem.0	. . . . . . . . . . . . 13 |- M = (mulGrp ` ℂfld )
67	66	oveq1i 6660	. . . . . . . . . . . 12 |- ( M ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
68	67	rpmsubg 19810	. . . . . . . . . . 11 |- RR+ e. (SubGrp ` ( M ↾s ( CC \ { 0 } ) ) )
69		subgsubm 17616	. . . . . . . . . . 11 |- ( RR+ e. (SubGrp ` ( M ↾s ( CC \ { 0 } ) ) ) -> RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) )
70	68, 69	ax-mp 5	. . . . . . . . . 10 |- RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) )
71		cnring 19768	. . . . . . . . . . 11 |- ℂfld e. Ring
72		cnfldbas 19750	. . . . . . . . . . . . 13 |- CC = ( Base ` ℂfld )
73		cnfld0 19770	. . . . . . . . . . . . 13 |- 0 = ( 0g ` ℂfld )
74		cndrng 19775	. . . . . . . . . . . . 13 |- ℂfld e. DivRing
75	72, 73, 74	drngui 18753	. . . . . . . . . . . 12 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
76	75, 66	unitsubm 18670	. . . . . . . . . . 11 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` M ) )
77		eqid 2622	. . . . . . . . . . . 12 |- ( M ↾s ( CC \ { 0 } ) ) = ( M ↾s ( CC \ { 0 } ) )
78	77	subsubm 17357	. . . . . . . . . . 11 |- ( ( CC \ { 0 } ) e. (SubMnd ` M ) -> ( RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) <-> ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) ) ) )
79	71, 76, 78	mp2b 10	. . . . . . . . . 10 |- ( RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) <-> ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) ) )
80	70, 79	mpbi 220	. . . . . . . . 9 |- ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) )
81	80	simpli 474	. . . . . . . 8 |- RR+ e. (SubMnd ` M )
82		eqid 2622	. . . . . . . . 9 |- ( M ↾s RR+ ) = ( M ↾s RR+ )
83	82	submbas 17355	. . . . . . . 8 |- ( RR+ e. (SubMnd ` M ) -> RR+ = ( Base ` ( M ↾s RR+ ) ) )
84	81, 83	ax-mp 5	. . . . . . 7 |- RR+ = ( Base ` ( M ↾s RR+ ) )
85		cnfld1 19771	. . . . . . . . 9 |- 1 = ( 1r ` ℂfld )
86	66, 85	ringidval 18503	. . . . . . . 8 |- 1 = ( 0g ` M )
87		eqid 2622	. . . . . . . . . 10 |- ( 0g ` M ) = ( 0g ` M )
88	82, 87	subm0 17356	. . . . . . . . 9 |- ( RR+ e. (SubMnd ` M ) -> ( 0g ` M ) = ( 0g ` ( M ↾s RR+ ) ) )
89	81, 88	ax-mp 5	. . . . . . . 8 |- ( 0g ` M ) = ( 0g ` ( M ↾s RR+ ) )
90	86, 89	eqtri 2644	. . . . . . 7 |- 1 = ( 0g ` ( M ↾s RR+ ) )
91		cncrng 19767	. . . . . . . . 9 |- ℂfld e. CRing
92	66	crngmgp 18555	. . . . . . . . 9 |- (ℂfld e. CRing -> M e. CMnd )
93	91, 92	mp1i 13	. . . . . . . 8 |- ( ph -> M e. CMnd )
94	82	submmnd 17354	. . . . . . . . 9 |- ( RR+ e. (SubMnd ` M ) -> ( M ↾s RR+ ) e. Mnd )
95	81, 94	mp1i 13	. . . . . . . 8 |- ( ph -> ( M ↾s RR+ ) e. Mnd )
96	82	subcmn 18242	. . . . . . . 8 |- ( ( M e. CMnd /\ ( M ↾s RR+ ) e. Mnd ) -> ( M ↾s RR+ ) e. CMnd )
97	93, 95, 96	syl2anc 693	. . . . . . 7 |- ( ph -> ( M ↾s RR+ ) e. CMnd )
98		resubdrg 19954	. . . . . . . . . 10 |- ( RR e. (SubRing ` ℂfld ) /\ RRfld e. DivRing )
99	98	simpli 474	. . . . . . . . 9 |- RR e. (SubRing ` ℂfld )
100		df-refld 19951	. . . . . . . . . 10 |- RRfld = (ℂfld ↾s RR )
101	100	subrgring 18783	. . . . . . . . 9 |- ( RR e. (SubRing ` ℂfld ) -> RRfld e. Ring )
102	99, 101	ax-mp 5	. . . . . . . 8 |- RRfld e. Ring
103		ringmnd 18556	. . . . . . . 8 |- (RRfld e. Ring -> RRfld e. Mnd )
104	102, 103	mp1i 13	. . . . . . 7 |- ( ph -> RRfld e. Mnd )
105	66	oveq1i 6660	. . . . . . . . . 10 |- ( M ↾s RR+ ) = ( (mulGrp ` ℂfld ) ↾s RR+ )
106	105	reloggim 24345	. . . . . . . . 9 |- ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpIso RRfld )
107		gimghm 17706	. . . . . . . . 9 |- ( ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpIso RRfld ) -> ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpHom RRfld ) )
108	106, 107	ax-mp 5	. . . . . . . 8 |- ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpHom RRfld )
109		ghmmhm 17670	. . . . . . . 8 |- ( ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpHom RRfld ) -> ( log |` RR+ ) e. ( ( M ↾s RR+ ) MndHom RRfld ) )
110	108, 109	mp1i 13	. . . . . . 7 |- ( ph -> ( log |` RR+ ) e. ( ( M ↾s RR+ ) MndHom RRfld ) )
111		1red 10055	. . . . . . . 8 |- ( ph -> 1 e. RR )
112	49, 1, 111	fdmfifsupp 8285	. . . . . . 7 |- ( ph -> ( F oF ^c W ) finSupp 1 )
113	84, 90, 97, 104, 1, 110, 49, 112	gsummhm 18338	. . . . . 6 |- ( ph -> (RRfld gsumg ( ( log |` RR+ ) o. ( F oF ^c W ) ) ) = ( ( log |` RR+ ) ` ( ( M ↾s RR+ ) gsumg ( F oF ^c W ) ) ) )
114		subrgsubg 18786	. . . . . . . . . 10 |- ( RR e. (SubRing ` ℂfld ) -> RR e. (SubGrp ` ℂfld ) )
115	99, 114	ax-mp 5	. . . . . . . . 9 |- RR e. (SubGrp ` ℂfld )
116		subgsubm 17616	. . . . . . . . 9 |- ( RR e. (SubGrp ` ℂfld ) -> RR e. (SubMnd ` ℂfld ) )
117	115, 116	ax-mp 5	. . . . . . . 8 |- RR e. (SubMnd ` ℂfld )
118	117	a1i 11	. . . . . . 7 |- ( ph -> RR e. (SubMnd ` ℂfld ) )
119	40, 41	mp1i 13	. . . . . . . 8 |- ( ph -> ( log |` RR+ ) : RR+ --> RR )
120		fco 6058	. . . . . . . 8 |- ( ( ( log |` RR+ ) : RR+ --> RR /\ ( F oF ^c W ) : A --> RR+ ) -> ( ( log |` RR+ ) o. ( F oF ^c W ) ) : A --> RR )
121	119, 49, 120	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( log |` RR+ ) o. ( F oF ^c W ) ) : A --> RR )
122	1, 118, 121, 100	gsumsubm 17373	. . . . . 6 |- ( ph -> (ℂfld gsumg ( ( log |` RR+ ) o. ( F oF ^c W ) ) ) = (RRfld gsumg ( ( log |` RR+ ) o. ( F oF ^c W ) ) ) )
123	81	a1i 11	. . . . . . . 8 |- ( ph -> RR+ e. (SubMnd ` M ) )
124	1, 123, 49, 82	gsumsubm 17373	. . . . . . 7 |- ( ph -> ( M gsumg ( F oF ^c W ) ) = ( ( M ↾s RR+ ) gsumg ( F oF ^c W ) ) )
125	124	fveq2d 6195	. . . . . 6 |- ( ph -> ( ( log |` RR+ ) ` ( M gsumg ( F oF ^c W ) ) ) = ( ( log |` RR+ ) ` ( ( M ↾s RR+ ) gsumg ( F oF ^c W ) ) ) )
126	113, 122, 125	3eqtr4d 2666	. . . . 5 |- ( ph -> (ℂfld gsumg ( ( log |` RR+ ) o. ( F oF ^c W ) ) ) = ( ( log |` RR+ ) ` ( M gsumg ( F oF ^c W ) ) ) )
127	86, 93, 1, 123, 49, 112	gsumsubmcl 18319	. . . . . 6 |- ( ph -> ( M gsumg ( F oF ^c W ) ) e. RR+ )
128		fvres 6207	. . . . . 6 |- ( ( M gsumg ( F oF ^c W ) ) e. RR+ -> ( ( log |` RR+ ) ` ( M gsumg ( F oF ^c W ) ) ) = ( log ` ( M gsumg ( F oF ^c W ) ) ) )
129	127, 128	syl 17	. . . . 5 |- ( ph -> ( ( log |` RR+ ) ` ( M gsumg ( F oF ^c W ) ) ) = ( log ` ( M gsumg ( F oF ^c W ) ) ) )
130	65, 126, 129	3eqtrd 2660	. . . 4 |- ( ph -> -u (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) = ( log ` ( M gsumg ( F oF ^c W ) ) ) )
131		simprl 794	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> x e. RR+ )
132	131	rpcnd 11874	. . . . . . . . . 10 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> x e. CC )
133		simprr 796	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> y e. RR+ )
134	133	rpcnd 11874	. . . . . . . . . 10 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> y e. CC )
135	132, 134	mulcomd 10061	. . . . . . . . 9 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( x x. y ) = ( y x. x ) )
136	1, 4, 2, 135	caofcom 6929	. . . . . . . 8 |- ( ph -> ( W oF x. F ) = ( F oF x. W ) )
137	136	oveq2d 6666	. . . . . . 7 |- ( ph -> (ℂfld gsumg ( W oF x. F ) ) = (ℂfld gsumg ( F oF x. W ) ) )
138	2	feqmptd 6249	. . . . . . . . . . 11 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
139	1, 5, 3, 31, 138	offval2 6914	. . . . . . . . . 10 |- ( ph -> ( W oF x. F ) = ( k e. A |-> ( ( W ` k ) x. ( F ` k ) ) ) )
140	139	oveq2d 6666	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( W oF x. F ) ) = (ℂfld gsumg ( k e. A |-> ( ( W ` k ) x. ( F ` k ) ) ) ) )
141	5, 3	rpmulcld 11888	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( ( W ` k ) x. ( F ` k ) ) e. RR+ )
142	141	rpcnd 11874	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( ( W ` k ) x. ( F ` k ) ) e. CC )
143	1, 142	gsumfsum 19813	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( ( W ` k ) x. ( F ` k ) ) ) ) = sum_ k e. A ( ( W ` k ) x. ( F ` k ) ) )
144	140, 143	eqtrd 2656	. . . . . . . 8 |- ( ph -> (ℂfld gsumg ( W oF x. F ) ) = sum_ k e. A ( ( W ` k ) x. ( F ` k ) ) )
145		amgmwlem.2	. . . . . . . . 9 |- ( ph -> A =/= (/) )
146	1, 145, 141	fsumrpcl 14468	. . . . . . . 8 |- ( ph -> sum_ k e. A ( ( W ` k ) x. ( F ` k ) ) e. RR+ )
147	144, 146	eqeltrd 2701	. . . . . . 7 |- ( ph -> (ℂfld gsumg ( W oF x. F ) ) e. RR+ )
148	137, 147	eqeltrrd 2702	. . . . . 6 |- ( ph -> (ℂfld gsumg ( F oF x. W ) ) e. RR+ )
149	148	relogcld 24369	. . . . 5 |- ( ph -> ( log ` (ℂfld gsumg ( F oF x. W ) ) ) e. RR )
150		ringcmn 18581	. . . . . . 7 |- (ℂfld e. Ring -> ℂfld e. CMnd )
151	71, 150	mp1i 13	. . . . . 6 |- ( ph -> ℂfld e. CMnd )
152		remulcl 10021	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
153	152	adantl 482	. . . . . . 7 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
154		rpssre 11843	. . . . . . . 8 |- RR+ C_ RR
155		fss 6056	. . . . . . . 8 |- ( ( W : A --> RR+ /\ RR+ C_ RR ) -> W : A --> RR )
156	4, 154, 155	sylancl 694	. . . . . . 7 |- ( ph -> W : A --> RR )
157	21	renegcld 10457	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> -u ( log ` ( F ` k ) ) e. RR )
158		eqid 2622	. . . . . . . 8 |- ( k e. A |-> -u ( log ` ( F ` k ) ) ) = ( k e. A |-> -u ( log ` ( F ` k ) ) )
159	157, 158	fmptd 6385	. . . . . . 7 |- ( ph -> ( k e. A |-> -u ( log ` ( F ` k ) ) ) : A --> RR )
160	153, 156, 159, 1, 1, 48	off 6912	. . . . . 6 |- ( ph -> ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) : A --> RR )
161		0red 10041	. . . . . . 7 |- ( ph -> 0 e. RR )
162	160, 1, 161	fdmfifsupp 8285	. . . . . 6 |- ( ph -> ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) finSupp 0 )
163	73, 151, 1, 118, 160, 162	gsumsubmcl 18319	. . . . 5 |- ( ph -> (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) e. RR )
164	154	a1i 11	. . . . . . . 8 |- ( ph -> RR+ C_ RR )
165		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ w e. RR+ ) -> w e. RR+ )
166	165	relogcld 24369	. . . . . . . . . 10 |- ( ( ph /\ w e. RR+ ) -> ( log ` w ) e. RR )
167	166	renegcld 10457	. . . . . . . . 9 |- ( ( ph /\ w e. RR+ ) -> -u ( log ` w ) e. RR )
168		eqid 2622	. . . . . . . . 9 |- ( w e. RR+ |-> -u ( log ` w ) ) = ( w e. RR+ |-> -u ( log ` w ) )
169	167, 168	fmptd 6385	. . . . . . . 8 |- ( ph -> ( w e. RR+ |-> -u ( log ` w ) ) : RR+ --> RR )
170		simpl 473	. . . . . . . . . . . 12 |- ( ( a e. RR+ /\ b e. RR+ ) -> a e. RR+ )
171		ioorp 12251	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) = RR+
172	170, 171	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( a e. RR+ /\ b e. RR+ ) -> a e. ( 0 (,) +oo ) )
173		simpr 477	. . . . . . . . . . . 12 |- ( ( a e. RR+ /\ b e. RR+ ) -> b e. RR+ )
174	173, 171	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( a e. RR+ /\ b e. RR+ ) -> b e. ( 0 (,) +oo ) )
175		iccssioo2 12246	. . . . . . . . . . 11 |- ( ( a e. ( 0 (,) +oo ) /\ b e. ( 0 (,) +oo ) ) -> ( a [,] b ) C_ ( 0 (,) +oo ) )
176	172, 174, 175	syl2anc 693	. . . . . . . . . 10 |- ( ( a e. RR+ /\ b e. RR+ ) -> ( a [,] b ) C_ ( 0 (,) +oo ) )
177	176, 171	syl6sseq 3651	. . . . . . . . 9 |- ( ( a e. RR+ /\ b e. RR+ ) -> ( a [,] b ) C_ RR+ )
178	177	adantl 482	. . . . . . . 8 |- ( ( ph /\ ( a e. RR+ /\ b e. RR+ ) ) -> ( a [,] b ) C_ RR+ )
179		ioossico 12262	. . . . . . . . . 10 |- ( 0 (,) +oo ) C_ ( 0 [,) +oo )
180	171, 179	eqsstr3i 3636	. . . . . . . . 9 |- RR+ C_ ( 0 [,) +oo )
181		fss 6056	. . . . . . . . 9 |- ( ( W : A --> RR+ /\ RR+ C_ ( 0 [,) +oo ) ) -> W : A --> ( 0 [,) +oo ) )
182	4, 180, 181	sylancl 694	. . . . . . . 8 |- ( ph -> W : A --> ( 0 [,) +oo ) )
183		0lt1 10550	. . . . . . . . 9 |- 0 < 1
184		amgmwlem.5	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg W ) = 1 )
185	183, 184	syl5breqr 4691	. . . . . . . 8 |- ( ph -> 0 < (ℂfld gsumg W ) )
186		logccv 24409	. . . . . . . . . . . 12 |- ( ( ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) < ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
187	186	3adant1 1079	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) < ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
188		elioore 12205	. . . . . . . . . . . . . . 15 |- ( t e. ( 0 (,) 1 ) -> t e. RR )
189	188	3ad2ant3 1084	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> t e. RR )
190		simp21 1094	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> x e. RR+ )
191	190	relogcld 24369	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` x ) e. RR )
192	189, 191	remulcld 10070	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. ( log ` x ) ) e. RR )
193		1red 10055	. . . . . . . . . . . . . . . 16 |- ( t e. ( 0 (,) 1 ) -> 1 e. RR )
194	193, 188	resubcld 10458	. . . . . . . . . . . . . . 15 |- ( t e. ( 0 (,) 1 ) -> ( 1 - t ) e. RR )
195	194	3ad2ant3 1084	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( 1 - t ) e. RR )
196		simp22 1095	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> y e. RR+ )
197	196	relogcld 24369	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` y ) e. RR )
198	195, 197	remulcld 10070	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. ( log ` y ) ) e. RR )
199	192, 198	readdcld 10069	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) e. RR )
200		eliooord 12233	. . . . . . . . . . . . . . . . . 18 |- ( t e. ( 0 (,) 1 ) -> ( 0 < t /\ t < 1 ) )
201	200	simpld 475	. . . . . . . . . . . . . . . . 17 |- ( t e. ( 0 (,) 1 ) -> 0 < t )
202	188, 201	elrpd 11869	. . . . . . . . . . . . . . . 16 |- ( t e. ( 0 (,) 1 ) -> t e. RR+ )
203	202	3ad2ant3 1084	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> t e. RR+ )
204	203, 190	rpmulcld 11888	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. x ) e. RR+ )
205		0red 10041	. . . . . . . . . . . . . . . . . 18 |- ( t e. ( 0 (,) 1 ) -> 0 e. RR )
206	200	simprd 479	. . . . . . . . . . . . . . . . . . 19 |- ( t e. ( 0 (,) 1 ) -> t < 1 )
207		1m0e1 11131	. . . . . . . . . . . . . . . . . . 19 |- ( 1 - 0 ) = 1
208	206, 207	syl6breqr 4695	. . . . . . . . . . . . . . . . . 18 |- ( t e. ( 0 (,) 1 ) -> t < ( 1 - 0 ) )
209	188, 193, 205, 208	ltsub13d 10633	. . . . . . . . . . . . . . . . 17 |- ( t e. ( 0 (,) 1 ) -> 0 < ( 1 - t ) )
210	194, 209	elrpd 11869	. . . . . . . . . . . . . . . 16 |- ( t e. ( 0 (,) 1 ) -> ( 1 - t ) e. RR+ )
211	210	3ad2ant3 1084	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( 1 - t ) e. RR+ )
212	211, 196	rpmulcld 11888	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. y ) e. RR+ )
213		rpaddcl 11854	. . . . . . . . . . . . . 14 |- ( ( ( t x. x ) e. RR+ /\ ( ( 1 - t ) x. y ) e. RR+ ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. RR+ )
214	204, 212, 213	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. RR+ )
215	214	relogcld 24369	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) e. RR )
216	199, 215	ltnegd 10605	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) < ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) <-> -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) < -u ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) ) )
217	187, 216	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) < -u ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) )
218		eqidd 2623	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( w e. RR+ |-> -u ( log ` w ) ) = ( w e. RR+ |-> -u ( log ` w ) ) )
219		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = ( ( t x. x ) + ( ( 1 - t ) x. y ) ) -> ( log ` w ) = ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
220	219	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) /\ w = ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) -> ( log ` w ) = ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
221	220	negeqd 10275	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) /\ w = ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) -> -u ( log ` w ) = -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
222		negex 10279	. . . . . . . . . . . 12 |- -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) e. _V
223	222	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) e. _V )
224	218, 221, 214, 223	fvmptd 6288	. . . . . . . . . 10 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) = -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
225		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( w = x -> ( log ` w ) = ( log ` x ) )
226	225	negeqd 10275	. . . . . . . . . . . . . . . 16 |- ( w = x -> -u ( log ` w ) = -u ( log ` x ) )
227		negex 10279	. . . . . . . . . . . . . . . 16 |- -u ( log ` w ) e. _V
228	226, 168, 227	fvmpt3i 6287	. . . . . . . . . . . . . . 15 |- ( x e. RR+ -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) = -u ( log ` x ) )
229	190, 228	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) = -u ( log ` x ) )
230	229	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) ) = ( t x. -u ( log ` x ) ) )
231	189	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> t e. CC )
232	191	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` x ) e. CC )
233	231, 232	mulneg2d 10484	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. -u ( log ` x ) ) = -u ( t x. ( log ` x ) ) )
234	230, 233	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) ) = -u ( t x. ( log ` x ) ) )
235		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( w = y -> ( log ` w ) = ( log ` y ) )
236	235	negeqd 10275	. . . . . . . . . . . . . . . 16 |- ( w = y -> -u ( log ` w ) = -u ( log ` y ) )
237	236, 168, 227	fvmpt3i 6287	. . . . . . . . . . . . . . 15 |- ( y e. RR+ -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) = -u ( log ` y ) )
238	196, 237	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) = -u ( log ` y ) )
239	238	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) ) = ( ( 1 - t ) x. -u ( log ` y ) ) )
240	211	rpcnd 11874	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( 1 - t ) e. CC )
241	197	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` y ) e. CC )
242	240, 241	mulneg2d 10484	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. -u ( log ` y ) ) = -u ( ( 1 - t ) x. ( log ` y ) ) )
243	239, 242	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) ) = -u ( ( 1 - t ) x. ( log ` y ) ) )
244	234, 243	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( t x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) ) + ( ( 1 - t ) x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) ) ) = ( -u ( t x. ( log ` x ) ) + -u ( ( 1 - t ) x. ( log ` y ) ) ) )
245	192	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. ( log ` x ) ) e. CC )
246	198	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. ( log ` y ) ) e. CC )
247	245, 246	negdid 10405	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> -u ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) = ( -u ( t x. ( log ` x ) ) + -u ( ( 1 - t ) x. ( log ` y ) ) ) )
248	244, 247	eqtr4d 2659	. . . . . . . . . 10 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( t x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) ) + ( ( 1 - t ) x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) ) ) = -u ( ( t x. ( log ` x ) ) + ( ( 1 - t ) x. ( log ` y ) ) ) )
249	217, 224, 248	3brtr4d 4685	. . . . . . . . 9 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) < ( ( t x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) ) + ( ( 1 - t ) x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) ) ) )
250	164, 169, 178, 249	scvxcvx 24712	. . . . . . . 8 |- ( ( ph /\ ( u e. RR+ /\ v e. RR+ /\ s e. ( 0 [,] 1 ) ) ) -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( ( s x. u ) + ( ( 1 - s ) x. v ) ) ) <_ ( ( s x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` u ) ) + ( ( 1 - s ) x. ( ( w e. RR+ |-> -u ( log ` w ) ) ` v ) ) ) )
251	164, 169, 178, 1, 182, 2, 185, 250	jensen 24715	. . . . . . 7 |- ( ph -> ( ( (ℂfld gsumg ( W oF x. F ) ) / (ℂfld gsumg W ) ) e. RR+ /\ ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( W oF x. F ) ) / (ℂfld gsumg W ) ) ) <_ ( (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg W ) ) ) )
252	251	simprd 479	. . . . . 6 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( W oF x. F ) ) / (ℂfld gsumg W ) ) ) <_ ( (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg W ) ) )
253	184	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( (ℂfld gsumg ( W oF x. F ) ) / (ℂfld gsumg W ) ) = ( (ℂfld gsumg ( W oF x. F ) ) / 1 ) )
254	253	fveq2d 6195	. . . . . . 7 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( W oF x. F ) ) / (ℂfld gsumg W ) ) ) = ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( W oF x. F ) ) / 1 ) ) )
255	147	rpcnd 11874	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( W oF x. F ) ) e. CC )
256	255	div1d 10793	. . . . . . . 8 |- ( ph -> ( (ℂfld gsumg ( W oF x. F ) ) / 1 ) = (ℂfld gsumg ( W oF x. F ) ) )
257	256	fveq2d 6195	. . . . . . 7 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( W oF x. F ) ) / 1 ) ) = ( ( w e. RR+ |-> -u ( log ` w ) ) ` (ℂfld gsumg ( W oF x. F ) ) ) )
258		fveq2 6191	. . . . . . . . . . 11 |- ( w = (ℂfld gsumg ( W oF x. F ) ) -> ( log ` w ) = ( log ` (ℂfld gsumg ( W oF x. F ) ) ) )
259	258	negeqd 10275	. . . . . . . . . 10 |- ( w = (ℂfld gsumg ( W oF x. F ) ) -> -u ( log ` w ) = -u ( log ` (ℂfld gsumg ( W oF x. F ) ) ) )
260	259, 168, 227	fvmpt3i 6287	. . . . . . . . 9 |- ( (ℂfld gsumg ( W oF x. F ) ) e. RR+ -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` (ℂfld gsumg ( W oF x. F ) ) ) = -u ( log ` (ℂfld gsumg ( W oF x. F ) ) ) )
261	147, 260	syl 17	. . . . . . . 8 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` (ℂfld gsumg ( W oF x. F ) ) ) = -u ( log ` (ℂfld gsumg ( W oF x. F ) ) ) )
262	137	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( log ` (ℂfld gsumg ( W oF x. F ) ) ) = ( log ` (ℂfld gsumg ( F oF x. W ) ) ) )
263	262	negeqd 10275	. . . . . . . 8 |- ( ph -> -u ( log ` (ℂfld gsumg ( W oF x. F ) ) ) = -u ( log ` (ℂfld gsumg ( F oF x. W ) ) ) )
264	261, 263	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` (ℂfld gsumg ( W oF x. F ) ) ) = -u ( log ` (ℂfld gsumg ( F oF x. W ) ) ) )
265	254, 257, 264	3eqtrd 2660	. . . . . 6 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( W oF x. F ) ) / (ℂfld gsumg W ) ) ) = -u ( log ` (ℂfld gsumg ( F oF x. W ) ) ) )
266	184	oveq2d 6666	. . . . . . 7 |- ( ph -> ( (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg W ) ) = ( (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / 1 ) )
267		ringmnd 18556	. . . . . . . . . . 11 |- (ℂfld e. Ring -> ℂfld e. Mnd )
268	71, 267	ax-mp 5	. . . . . . . . . 10 |- ℂfld e. Mnd
269	72	submid 17351	. . . . . . . . . 10 |- (ℂfld e. Mnd -> CC e. (SubMnd ` ℂfld ) )
270	268, 269	mp1i 13	. . . . . . . . 9 |- ( ph -> CC e. (SubMnd ` ℂfld ) )
271		mulcl 10020	. . . . . . . . . . 11 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
272	271	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
273		rpcn 11841	. . . . . . . . . . . . 13 |- ( x e. RR+ -> x e. CC )
274	273	ssriv 3607	. . . . . . . . . . . 12 |- RR+ C_ CC
275	274	a1i 11	. . . . . . . . . . 11 |- ( ph -> RR+ C_ CC )
276	4, 275	fssd 6057	. . . . . . . . . 10 |- ( ph -> W : A --> CC )
277	166	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ph /\ w e. RR+ ) -> ( log ` w ) e. CC )
278	277	negcld 10379	. . . . . . . . . . . 12 |- ( ( ph /\ w e. RR+ ) -> -u ( log ` w ) e. CC )
279	278, 168	fmptd 6385	. . . . . . . . . . 11 |- ( ph -> ( w e. RR+ |-> -u ( log ` w ) ) : RR+ --> CC )
280		fco 6058	. . . . . . . . . . 11 |- ( ( ( w e. RR+ |-> -u ( log ` w ) ) : RR+ --> CC /\ F : A --> RR+ ) -> ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) : A --> CC )
281	279, 2, 280	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) : A --> CC )
282	272, 276, 281, 1, 1, 48	off 6912	. . . . . . . . 9 |- ( ph -> ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) : A --> CC )
283	282, 1, 161	fdmfifsupp 8285	. . . . . . . . 9 |- ( ph -> ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) finSupp 0 )
284	73, 151, 1, 270, 282, 283	gsumsubmcl 18319	. . . . . . . 8 |- ( ph -> (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) e. CC )
285	284	div1d 10793	. . . . . . 7 |- ( ph -> ( (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / 1 ) = (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) )
286		eqidd 2623	. . . . . . . . . 10 |- ( ph -> ( w e. RR+ |-> -u ( log ` w ) ) = ( w e. RR+ |-> -u ( log ` w ) ) )
287		fveq2 6191	. . . . . . . . . . 11 |- ( w = ( F ` k ) -> ( log ` w ) = ( log ` ( F ` k ) ) )
288	287	negeqd 10275	. . . . . . . . . 10 |- ( w = ( F ` k ) -> -u ( log ` w ) = -u ( log ` ( F ` k ) ) )
289	3, 138, 286, 288	fmptco 6396	. . . . . . . . 9 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) = ( k e. A |-> -u ( log ` ( F ` k ) ) ) )
290	289	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) = ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) )
291	290	oveq2d 6666	. . . . . . 7 |- ( ph -> (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) = (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) )
292	266, 285, 291	3eqtrd 2660	. . . . . 6 |- ( ph -> ( (ℂfld gsumg ( W oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg W ) ) = (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) )
293	252, 265, 292	3brtr3d 4684	. . . . 5 |- ( ph -> -u ( log ` (ℂfld gsumg ( F oF x. W ) ) ) <_ (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) )
294	149, 163, 293	lenegcon1d 10609	. . . 4 |- ( ph -> -u (ℂfld gsumg ( W oF x. ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) <_ ( log ` (ℂfld gsumg ( F oF x. W ) ) ) )
295	130, 294	eqbrtrrd 4677	. . 3 |- ( ph -> ( log ` ( M gsumg ( F oF ^c W ) ) ) <_ ( log ` (ℂfld gsumg ( F oF x. W ) ) ) )
296	127	relogcld 24369	. . . 4 |- ( ph -> ( log ` ( M gsumg ( F oF ^c W ) ) ) e. RR )
297		efle 14848	. . . 4 |- ( ( ( log ` ( M gsumg ( F oF ^c W ) ) ) e. RR /\ ( log ` (ℂfld gsumg ( F oF x. W ) ) ) e. RR ) -> ( ( log ` ( M gsumg ( F oF ^c W ) ) ) <_ ( log ` (ℂfld gsumg ( F oF x. W ) ) ) <-> ( exp ` ( log ` ( M gsumg ( F oF ^c W ) ) ) ) <_ ( exp ` ( log ` (ℂfld gsumg ( F oF x. W ) ) ) ) ) )
298	296, 149, 297	syl2anc 693	. . 3 |- ( ph -> ( ( log ` ( M gsumg ( F oF ^c W ) ) ) <_ ( log ` (ℂfld gsumg ( F oF x. W ) ) ) <-> ( exp ` ( log ` ( M gsumg ( F oF ^c W ) ) ) ) <_ ( exp ` ( log ` (ℂfld gsumg ( F oF x. W ) ) ) ) ) )
299	295, 298	mpbid 222	. 2 |- ( ph -> ( exp ` ( log ` ( M gsumg ( F oF ^c W ) ) ) ) <_ ( exp ` ( log ` (ℂfld gsumg ( F oF x. W ) ) ) ) )
300	127	reeflogd 24370	. . 3 |- ( ph -> ( exp ` ( log ` ( M gsumg ( F oF ^c W ) ) ) ) = ( M gsumg ( F oF ^c W ) ) )
301	300	eqcomd 2628	. 2 |- ( ph -> ( M gsumg ( F oF ^c W ) ) = ( exp ` ( log ` ( M gsumg ( F oF ^c W ) ) ) ) )
302	148	reeflogd 24370	. . 3 |- ( ph -> ( exp ` ( log ` (ℂfld gsumg ( F oF x. W ) ) ) ) = (ℂfld gsumg ( F oF x. W ) ) )
303	302	eqcomd 2628	. 2 |- ( ph -> (ℂfld gsumg ( F oF x. W ) ) = ( exp ` ( log ` (ℂfld gsumg ( F oF x. W ) ) ) ) )
304	299, 301, 303	3brtr4d 4685	1 |- ( ph -> ( M gsumg ( F oF ^c W ) ) <_ (ℂfld gsumg ( F oF x. W ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   RR+crp 11832   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   sum_csu 14416   expce 14792   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   MndHom cmhm 17333  SubMndcsubmnd 17334  SubGrpcsubg 17588   GrpHom cghm 17657   GrpIso cgim 17699  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548   DivRingcdr 18747  SubRingcsubrg 18776  ℂ_fldccnfld 19746  RRfldcrefld 19950   logclog 24301   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-subrg 18778  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-refld 19951  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  amgmlemALT  42549  amgmw2d  42550