Theorem amgmlem 24716

Description: Lemma for amgm 24717. (Contributed by Mario Carneiro, 21-Jun-2015.)

Hypotheses

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

Assertion

Ref	Expression
amgmlem	|- ( ph -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )

Proof of Theorem amgmlem

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

Step	Hyp	Ref	Expression
1		cnfld0 19770	. . . . . . . 8 |- 0 = ( 0g ` ℂfld )
2		cnring 19768	. . . . . . . . 9 |- ℂfld e. Ring
3		ringabl 18580	. . . . . . . . 9 |- (ℂfld e. Ring -> ℂfld e. Abel )
4	2, 3	mp1i 13	. . . . . . . 8 |- ( ph -> ℂfld e. Abel )
5		amgm.2	. . . . . . . 8 |- ( ph -> A e. Fin )
6		resubdrg 19954	. . . . . . . . . 10 |- ( RR e. (SubRing ` ℂfld ) /\ RRfld e. DivRing )
7	6	simpli 474	. . . . . . . . 9 |- RR e. (SubRing ` ℂfld )
8		subrgsubg 18786	. . . . . . . . 9 |- ( RR e. (SubRing ` ℂfld ) -> RR e. (SubGrp ` ℂfld ) )
9	7, 8	mp1i 13	. . . . . . . 8 |- ( ph -> RR e. (SubGrp ` ℂfld ) )
10		amgm.4	. . . . . . . . . . . 12 |- ( ph -> F : A --> RR+ )
11	10	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. RR+ )
12	11	relogcld 24369	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( log ` ( F ` k ) ) e. RR )
13	12	renegcld 10457	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> -u ( log ` ( F ` k ) ) e. RR )
14		eqid 2622	. . . . . . . . 9 |- ( k e. A |-> -u ( log ` ( F ` k ) ) ) = ( k e. A |-> -u ( log ` ( F ` k ) ) )
15	13, 14	fmptd 6385	. . . . . . . 8 |- ( ph -> ( k e. A |-> -u ( log ` ( F ` k ) ) ) : A --> RR )
16		c0ex 10034	. . . . . . . . . 10 |- 0 e. _V
17	16	a1i 11	. . . . . . . . 9 |- ( ph -> 0 e. _V )
18	15, 5, 17	fdmfifsupp 8285	. . . . . . . 8 |- ( ph -> ( k e. A |-> -u ( log ` ( F ` k ) ) ) finSupp 0 )
19	1, 4, 5, 9, 15, 18	gsumsubgcl 18320	. . . . . . 7 |- ( ph -> (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) e. RR )
20	19	recnd 10068	. . . . . 6 |- ( ph -> (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) e. CC )
21		amgm.3	. . . . . . . 8 |- ( ph -> A =/= (/) )
22		hashnncl 13157	. . . . . . . . 9 |- ( A e. Fin -> ( ( # ` A ) e. NN <-> A =/= (/) ) )
23	5, 22	syl 17	. . . . . . . 8 |- ( ph -> ( ( # ` A ) e. NN <-> A =/= (/) ) )
24	21, 23	mpbird 247	. . . . . . 7 |- ( ph -> ( # ` A ) e. NN )
25	24	nncnd 11036	. . . . . 6 |- ( ph -> ( # ` A ) e. CC )
26	24	nnne0d 11065	. . . . . 6 |- ( ph -> ( # ` A ) =/= 0 )
27	20, 25, 26	divnegd 10814	. . . . 5 |- ( ph -> -u ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) = ( -u (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) )
28	12	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( log ` ( F ` k ) ) e. CC )
29	5, 28	gsumfsum 19813	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( log ` ( F ` k ) ) ) ) = sum_ k e. A ( log ` ( F ` k ) ) )
30	28	negnegd 10383	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> -u -u ( log ` ( F ` k ) ) = ( log ` ( F ` k ) ) )
31	30	sumeq2dv 14433	. . . . . . . . 9 |- ( ph -> sum_ k e. A -u -u ( log ` ( F ` k ) ) = sum_ k e. A ( log ` ( F ` k ) ) )
32	13	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> -u ( log ` ( F ` k ) ) e. CC )
33	5, 32	fsumneg 14519	. . . . . . . . 9 |- ( ph -> sum_ k e. A -u -u ( log ` ( F ` k ) ) = -u sum_ k e. A -u ( log ` ( F ` k ) ) )
34	29, 31, 33	3eqtr2rd 2663	. . . . . . . 8 |- ( ph -> -u sum_ k e. A -u ( log ` ( F ` k ) ) = (ℂfld gsumg ( k e. A |-> ( log ` ( F ` k ) ) ) ) )
35	5, 32	gsumfsum 19813	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) = sum_ k e. A -u ( log ` ( F ` k ) ) )
36	35	negeqd 10275	. . . . . . . 8 |- ( ph -> -u (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) = -u sum_ k e. A -u ( log ` ( F ` k ) ) )
37	10	feqmptd 6249	. . . . . . . . . 10 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
38		relogf1o 24313	. . . . . . . . . . . . 13 |- ( log |` RR+ ) : RR+ -1-1-onto-> RR
39		f1of 6137	. . . . . . . . . . . . 13 |- ( ( log |` RR+ ) : RR+ -1-1-onto-> RR -> ( log |` RR+ ) : RR+ --> RR )
40	38, 39	mp1i 13	. . . . . . . . . . . 12 |- ( ph -> ( log |` RR+ ) : RR+ --> RR )
41	40	feqmptd 6249	. . . . . . . . . . 11 |- ( ph -> ( log |` RR+ ) = ( x e. RR+ |-> ( ( log |` RR+ ) ` x ) ) )
42		fvres 6207	. . . . . . . . . . . 12 |- ( x e. RR+ -> ( ( log |` RR+ ) ` x ) = ( log ` x ) )
43	42	mpteq2ia 4740	. . . . . . . . . . 11 |- ( x e. RR+ |-> ( ( log |` RR+ ) ` x ) ) = ( x e. RR+ |-> ( log ` x ) )
44	41, 43	syl6eq 2672	. . . . . . . . . 10 |- ( ph -> ( log |` RR+ ) = ( x e. RR+ |-> ( log ` x ) ) )
45		fveq2 6191	. . . . . . . . . 10 |- ( x = ( F ` k ) -> ( log ` x ) = ( log ` ( F ` k ) ) )
46	11, 37, 44, 45	fmptco 6396	. . . . . . . . 9 |- ( ph -> ( ( log |` RR+ ) o. F ) = ( k e. A |-> ( log ` ( F ` k ) ) ) )
47	46	oveq2d 6666	. . . . . . . 8 |- ( ph -> (ℂfld gsumg ( ( log |` RR+ ) o. F ) ) = (ℂfld gsumg ( k e. A |-> ( log ` ( F ` k ) ) ) ) )
48	34, 36, 47	3eqtr4d 2666	. . . . . . 7 |- ( ph -> -u (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) = (ℂfld gsumg ( ( log |` RR+ ) o. F ) ) )
49		amgm.1	. . . . . . . . . . . . . . 15 |- M = (mulGrp ` ℂfld )
50	49	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( M ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
51	50	rpmsubg 19810	. . . . . . . . . . . . 13 |- RR+ e. (SubGrp ` ( M ↾s ( CC \ { 0 } ) ) )
52		subgsubm 17616	. . . . . . . . . . . . 13 |- ( RR+ e. (SubGrp ` ( M ↾s ( CC \ { 0 } ) ) ) -> RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) )
53	51, 52	ax-mp 5	. . . . . . . . . . . 12 |- RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) )
54		cnfldbas 19750	. . . . . . . . . . . . . . 15 |- CC = ( Base ` ℂfld )
55		cndrng 19775	. . . . . . . . . . . . . . 15 |- ℂfld e. DivRing
56	54, 1, 55	drngui 18753	. . . . . . . . . . . . . 14 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
57	56, 49	unitsubm 18670	. . . . . . . . . . . . 13 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` M ) )
58		eqid 2622	. . . . . . . . . . . . . 14 |- ( M ↾s ( CC \ { 0 } ) ) = ( M ↾s ( CC \ { 0 } ) )
59	58	subsubm 17357	. . . . . . . . . . . . 13 |- ( ( CC \ { 0 } ) e. (SubMnd ` M ) -> ( RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) <-> ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) ) ) )
60	2, 57, 59	mp2b 10	. . . . . . . . . . . 12 |- ( RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) <-> ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) ) )
61	53, 60	mpbi 220	. . . . . . . . . . 11 |- ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) )
62	61	simpli 474	. . . . . . . . . 10 |- RR+ e. (SubMnd ` M )
63		eqid 2622	. . . . . . . . . . 11 |- ( M ↾s RR+ ) = ( M ↾s RR+ )
64	63	submbas 17355	. . . . . . . . . 10 |- ( RR+ e. (SubMnd ` M ) -> RR+ = ( Base ` ( M ↾s RR+ ) ) )
65	62, 64	ax-mp 5	. . . . . . . . 9 |- RR+ = ( Base ` ( M ↾s RR+ ) )
66		cnfld1 19771	. . . . . . . . . . . 12 |- 1 = ( 1r ` ℂfld )
67	49, 66	ringidval 18503	. . . . . . . . . . 11 |- 1 = ( 0g ` M )
68	63, 67	subm0 17356	. . . . . . . . . 10 |- ( RR+ e. (SubMnd ` M ) -> 1 = ( 0g ` ( M ↾s RR+ ) ) )
69	62, 68	ax-mp 5	. . . . . . . . 9 |- 1 = ( 0g ` ( M ↾s RR+ ) )
70		cncrng 19767	. . . . . . . . . . 11 |- ℂfld e. CRing
71	49	crngmgp 18555	. . . . . . . . . . 11 |- (ℂfld e. CRing -> M e. CMnd )
72	70, 71	mp1i 13	. . . . . . . . . 10 |- ( ph -> M e. CMnd )
73	63	submmnd 17354	. . . . . . . . . . 11 |- ( RR+ e. (SubMnd ` M ) -> ( M ↾s RR+ ) e. Mnd )
74	62, 73	mp1i 13	. . . . . . . . . 10 |- ( ph -> ( M ↾s RR+ ) e. Mnd )
75	63	subcmn 18242	. . . . . . . . . 10 |- ( ( M e. CMnd /\ ( M ↾s RR+ ) e. Mnd ) -> ( M ↾s RR+ ) e. CMnd )
76	72, 74, 75	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( M ↾s RR+ ) e. CMnd )
77		df-refld 19951	. . . . . . . . . . . 12 |- RRfld = (ℂfld ↾s RR )
78	77	subrgring 18783	. . . . . . . . . . 11 |- ( RR e. (SubRing ` ℂfld ) -> RRfld e. Ring )
79	7, 78	ax-mp 5	. . . . . . . . . 10 |- RRfld e. Ring
80		ringmnd 18556	. . . . . . . . . 10 |- (RRfld e. Ring -> RRfld e. Mnd )
81	79, 80	mp1i 13	. . . . . . . . 9 |- ( ph -> RRfld e. Mnd )
82	49	oveq1i 6660	. . . . . . . . . . . 12 |- ( M ↾s RR+ ) = ( (mulGrp ` ℂfld ) ↾s RR+ )
83	82	reloggim 24345	. . . . . . . . . . 11 |- ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpIso RRfld )
84		gimghm 17706	. . . . . . . . . . 11 |- ( ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpIso RRfld ) -> ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpHom RRfld ) )
85	83, 84	ax-mp 5	. . . . . . . . . 10 |- ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpHom RRfld )
86		ghmmhm 17670	. . . . . . . . . 10 |- ( ( log |` RR+ ) e. ( ( M ↾s RR+ ) GrpHom RRfld ) -> ( log |` RR+ ) e. ( ( M ↾s RR+ ) MndHom RRfld ) )
87	85, 86	mp1i 13	. . . . . . . . 9 |- ( ph -> ( log |` RR+ ) e. ( ( M ↾s RR+ ) MndHom RRfld ) )
88		1ex 10035	. . . . . . . . . . 11 |- 1 e. _V
89	88	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. _V )
90	10, 5, 89	fdmfifsupp 8285	. . . . . . . . 9 |- ( ph -> F finSupp 1 )
91	65, 69, 76, 81, 5, 87, 10, 90	gsummhm 18338	. . . . . . . 8 |- ( ph -> (RRfld gsumg ( ( log |` RR+ ) o. F ) ) = ( ( log |` RR+ ) ` ( ( M ↾s RR+ ) gsumg F ) ) )
92		subgsubm 17616	. . . . . . . . . 10 |- ( RR e. (SubGrp ` ℂfld ) -> RR e. (SubMnd ` ℂfld ) )
93	9, 92	syl 17	. . . . . . . . 9 |- ( ph -> RR e. (SubMnd ` ℂfld ) )
94		fco 6058	. . . . . . . . . 10 |- ( ( ( log |` RR+ ) : RR+ --> RR /\ F : A --> RR+ ) -> ( ( log |` RR+ ) o. F ) : A --> RR )
95	40, 10, 94	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( log |` RR+ ) o. F ) : A --> RR )
96	5, 93, 95, 77	gsumsubm 17373	. . . . . . . 8 |- ( ph -> (ℂfld gsumg ( ( log |` RR+ ) o. F ) ) = (RRfld gsumg ( ( log |` RR+ ) o. F ) ) )
97	62	a1i 11	. . . . . . . . . 10 |- ( ph -> RR+ e. (SubMnd ` M ) )
98	5, 97, 10, 63	gsumsubm 17373	. . . . . . . . 9 |- ( ph -> ( M gsumg F ) = ( ( M ↾s RR+ ) gsumg F ) )
99	98	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( ( log |` RR+ ) ` ( M gsumg F ) ) = ( ( log |` RR+ ) ` ( ( M ↾s RR+ ) gsumg F ) ) )
100	91, 96, 99	3eqtr4d 2666	. . . . . . 7 |- ( ph -> (ℂfld gsumg ( ( log |` RR+ ) o. F ) ) = ( ( log |` RR+ ) ` ( M gsumg F ) ) )
101	67, 72, 5, 97, 10, 90	gsumsubmcl 18319	. . . . . . . 8 |- ( ph -> ( M gsumg F ) e. RR+ )
102		fvres 6207	. . . . . . . 8 |- ( ( M gsumg F ) e. RR+ -> ( ( log |` RR+ ) ` ( M gsumg F ) ) = ( log ` ( M gsumg F ) ) )
103	101, 102	syl 17	. . . . . . 7 |- ( ph -> ( ( log |` RR+ ) ` ( M gsumg F ) ) = ( log ` ( M gsumg F ) ) )
104	48, 100, 103	3eqtrd 2660	. . . . . 6 |- ( ph -> -u (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) = ( log ` ( M gsumg F ) ) )
105	104	oveq1d 6665	. . . . 5 |- ( ph -> ( -u (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) = ( ( log ` ( M gsumg F ) ) / ( # ` A ) ) )
106	101	relogcld 24369	. . . . . . 7 |- ( ph -> ( log ` ( M gsumg F ) ) e. RR )
107	106	recnd 10068	. . . . . 6 |- ( ph -> ( log ` ( M gsumg F ) ) e. CC )
108	107, 25, 26	divrec2d 10805	. . . . 5 |- ( ph -> ( ( log ` ( M gsumg F ) ) / ( # ` A ) ) = ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) )
109	27, 105, 108	3eqtrd 2660	. . . 4 |- ( ph -> -u ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) = ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) )
110	37	oveq2d 6666	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg F ) = (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) )
111	11	rpcnd 11874	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. CC )
112	5, 111	gsumfsum 19813	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) = sum_ k e. A ( F ` k ) )
113	110, 112	eqtrd 2656	. . . . . . . 8 |- ( ph -> (ℂfld gsumg F ) = sum_ k e. A ( F ` k ) )
114	5, 21, 11	fsumrpcl 14468	. . . . . . . 8 |- ( ph -> sum_ k e. A ( F ` k ) e. RR+ )
115	113, 114	eqeltrd 2701	. . . . . . 7 |- ( ph -> (ℂfld gsumg F ) e. RR+ )
116	24	nnrpd 11870	. . . . . . 7 |- ( ph -> ( # ` A ) e. RR+ )
117	115, 116	rpdivcld 11889	. . . . . 6 |- ( ph -> ( (ℂfld gsumg F ) / ( # ` A ) ) e. RR+ )
118	117	relogcld 24369	. . . . 5 |- ( ph -> ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) e. RR )
119	19, 24	nndivred 11069	. . . . 5 |- ( ph -> ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) e. RR )
120		rpssre 11843	. . . . . . . . 9 |- RR+ C_ RR
121	120	a1i 11	. . . . . . . 8 |- ( ph -> RR+ C_ RR )
122		relogcl 24322	. . . . . . . . . . 11 |- ( w e. RR+ -> ( log ` w ) e. RR )
123	122	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ w e. RR+ ) -> ( log ` w ) e. RR )
124	123	renegcld 10457	. . . . . . . . 9 |- ( ( ph /\ w e. RR+ ) -> -u ( log ` w ) e. RR )
125		eqid 2622	. . . . . . . . 9 |- ( w e. RR+ |-> -u ( log ` w ) ) = ( w e. RR+ |-> -u ( log ` w ) )
126	124, 125	fmptd 6385	. . . . . . . 8 |- ( ph -> ( w e. RR+ |-> -u ( log ` w ) ) : RR+ --> RR )
127		ioorp 12251	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) = RR+
128	127	eleq2i 2693	. . . . . . . . . . 11 |- ( a e. ( 0 (,) +oo ) <-> a e. RR+ )
129	127	eleq2i 2693	. . . . . . . . . . 11 |- ( b e. ( 0 (,) +oo ) <-> b e. RR+ )
130		iccssioo2 12246	. . . . . . . . . . 11 |- ( ( a e. ( 0 (,) +oo ) /\ b e. ( 0 (,) +oo ) ) -> ( a [,] b ) C_ ( 0 (,) +oo ) )
131	128, 129, 130	syl2anbr 497	. . . . . . . . . 10 |- ( ( a e. RR+ /\ b e. RR+ ) -> ( a [,] b ) C_ ( 0 (,) +oo ) )
132	131, 127	syl6sseq 3651	. . . . . . . . 9 |- ( ( a e. RR+ /\ b e. RR+ ) -> ( a [,] b ) C_ RR+ )
133	132	adantl 482	. . . . . . . 8 |- ( ( ph /\ ( a e. RR+ /\ b e. RR+ ) ) -> ( a [,] b ) C_ RR+ )
134	24	nnrecred 11066	. . . . . . . . . 10 |- ( ph -> ( 1 / ( # ` A ) ) e. RR )
135	116	rpreccld 11882	. . . . . . . . . . 11 |- ( ph -> ( 1 / ( # ` A ) ) e. RR+ )
136	135	rpge0d 11876	. . . . . . . . . 10 |- ( ph -> 0 <_ ( 1 / ( # ` A ) ) )
137		elrege0 12278	. . . . . . . . . 10 |- ( ( 1 / ( # ` A ) ) e. ( 0 [,) +oo ) <-> ( ( 1 / ( # ` A ) ) e. RR /\ 0 <_ ( 1 / ( # ` A ) ) ) )
138	134, 136, 137	sylanbrc 698	. . . . . . . . 9 |- ( ph -> ( 1 / ( # ` A ) ) e. ( 0 [,) +oo ) )
139		fconst6g 6094	. . . . . . . . 9 |- ( ( 1 / ( # ` A ) ) e. ( 0 [,) +oo ) -> ( A X. { ( 1 / ( # ` A ) ) } ) : A --> ( 0 [,) +oo ) )
140	138, 139	syl 17	. . . . . . . 8 |- ( ph -> ( A X. { ( 1 / ( # ` A ) ) } ) : A --> ( 0 [,) +oo ) )
141		0lt1 10550	. . . . . . . . 9 |- 0 < 1
142		fconstmpt 5163	. . . . . . . . . . 11 |- ( A X. { ( 1 / ( # ` A ) ) } ) = ( k e. A |-> ( 1 / ( # ` A ) ) )
143	142	oveq2i 6661	. . . . . . . . . 10 |- (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) = (ℂfld gsumg ( k e. A |-> ( 1 / ( # ` A ) ) ) )
144		ringmnd 18556	. . . . . . . . . . . . 13 |- (ℂfld e. Ring -> ℂfld e. Mnd )
145	2, 144	mp1i 13	. . . . . . . . . . . 12 |- ( ph -> ℂfld e. Mnd )
146	134	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> ( 1 / ( # ` A ) ) e. CC )
147		eqid 2622	. . . . . . . . . . . . 13 |- (.g ` ℂfld ) = (.g ` ℂfld )
148	54, 147	gsumconst 18334	. . . . . . . . . . . 12 |- ( (ℂfld e. Mnd /\ A e. Fin /\ ( 1 / ( # ` A ) ) e. CC ) -> (ℂfld gsumg ( k e. A |-> ( 1 / ( # ` A ) ) ) ) = ( ( # ` A ) (.g ` ℂfld ) ( 1 / ( # ` A ) ) ) )
149	145, 5, 146, 148	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( 1 / ( # ` A ) ) ) ) = ( ( # ` A ) (.g ` ℂfld ) ( 1 / ( # ` A ) ) ) )
150	24	nnzd 11481	. . . . . . . . . . . 12 |- ( ph -> ( # ` A ) e. ZZ )
151		cnfldmulg 19778	. . . . . . . . . . . 12 |- ( ( ( # ` A ) e. ZZ /\ ( 1 / ( # ` A ) ) e. CC ) -> ( ( # ` A ) (.g ` ℂfld ) ( 1 / ( # ` A ) ) ) = ( ( # ` A ) x. ( 1 / ( # ` A ) ) ) )
152	150, 146, 151	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( ( # ` A ) (.g ` ℂfld ) ( 1 / ( # ` A ) ) ) = ( ( # ` A ) x. ( 1 / ( # ` A ) ) ) )
153	25, 26	recidd 10796	. . . . . . . . . . 11 |- ( ph -> ( ( # ` A ) x. ( 1 / ( # ` A ) ) ) = 1 )
154	149, 152, 153	3eqtrd 2660	. . . . . . . . . 10 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( 1 / ( # ` A ) ) ) ) = 1 )
155	143, 154	syl5eq 2668	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) = 1 )
156	141, 155	syl5breqr 4691	. . . . . . . 8 |- ( ph -> 0 < (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) )
157		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 ) ) ) )
158	157	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 ) ) ) )
159		ioossre 12235	. . . . . . . . . . . . . . 15 |- ( 0 (,) 1 ) C_ RR
160		simp3 1063	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> t e. ( 0 (,) 1 ) )
161	159, 160	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> t e. RR )
162		simp21 1094	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> x e. RR+ )
163	162	relogcld 24369	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` x ) e. RR )
164	161, 163	remulcld 10070	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. ( log ` x ) ) e. RR )
165		1re 10039	. . . . . . . . . . . . . . 15 |- 1 e. RR
166		resubcl 10345	. . . . . . . . . . . . . . 15 |- ( ( 1 e. RR /\ t e. RR ) -> ( 1 - t ) e. RR )
167	165, 161, 166	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( 1 - t ) e. RR )
168		simp22 1095	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> y e. RR+ )
169	168	relogcld 24369	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` y ) e. RR )
170	167, 169	remulcld 10070	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. ( log ` y ) ) e. RR )
171	164, 170	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 )
172		simp1 1061	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ph )
173		ioossicc 12259	. . . . . . . . . . . . . . 15 |- ( 0 (,) 1 ) C_ ( 0 [,] 1 )
174	173, 160	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> t e. ( 0 [,] 1 ) )
175	121, 133	cvxcl 24711	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ t e. ( 0 [,] 1 ) ) ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. RR+ )
176	172, 162, 168, 174, 175	syl13anc 1328	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. RR+ )
177	176	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 )
178	171, 177	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 ) ) ) ) )
179	158, 178	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 ) ) ) )
180		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = ( ( t x. x ) + ( ( 1 - t ) x. y ) ) -> ( log ` w ) = ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
181	180	negeqd 10275	. . . . . . . . . . . 12 |- ( w = ( ( t x. x ) + ( ( 1 - t ) x. y ) ) -> -u ( log ` w ) = -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
182		negex 10279	. . . . . . . . . . . 12 |- -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) e. _V
183	181, 125, 182	fvmpt 6282	. . . . . . . . . . 11 |- ( ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. RR+ -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) = -u ( log ` ( ( t x. x ) + ( ( 1 - t ) x. y ) ) ) )
184	176, 183	syl 17	. . . . . . . . . 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 ) ) ) )
185		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( w = x -> ( log ` w ) = ( log ` x ) )
186	185	negeqd 10275	. . . . . . . . . . . . . . . 16 |- ( w = x -> -u ( log ` w ) = -u ( log ` x ) )
187		negex 10279	. . . . . . . . . . . . . . . 16 |- -u ( log ` x ) e. _V
188	186, 125, 187	fvmpt 6282	. . . . . . . . . . . . . . 15 |- ( x e. RR+ -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` x ) = -u ( log ` x ) )
189	162, 188	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 ) )
190	189	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 ) ) )
191	161	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> t e. CC )
192	163	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` x ) e. CC )
193	191, 192	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 ) ) )
194	190, 193	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 ) ) )
195		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( w = y -> ( log ` w ) = ( log ` y ) )
196	195	negeqd 10275	. . . . . . . . . . . . . . . 16 |- ( w = y -> -u ( log ` w ) = -u ( log ` y ) )
197		negex 10279	. . . . . . . . . . . . . . . 16 |- -u ( log ` y ) e. _V
198	196, 125, 197	fvmpt 6282	. . . . . . . . . . . . . . 15 |- ( y e. RR+ -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` y ) = -u ( log ` y ) )
199	168, 198	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 ) )
200	199	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 ) ) )
201	167	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( 1 - t ) e. CC )
202	169	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( log ` y ) e. CC )
203	201, 202	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 ) ) )
204	200, 203	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 ) ) )
205	194, 204	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 ) ) ) )
206	164	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( t x. ( log ` x ) ) e. CC )
207	170	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ /\ x < y ) /\ t e. ( 0 (,) 1 ) ) -> ( ( 1 - t ) x. ( log ` y ) ) e. CC )
208	206, 207	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 ) ) ) )
209	205, 208	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 ) ) ) )
210	179, 184, 209	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 ) ) ) )
211	121, 126, 133, 210	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 ) ) ) )
212	121, 126, 133, 5, 140, 10, 156, 211	jensen 24715	. . . . . . 7 |- ( ph -> ( ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) e. RR+ /\ ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) ) <_ ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) ) )
213	212	simprd 479	. . . . . 6 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) ) <_ ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) )
214	134	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. A ) -> ( 1 / ( # ` A ) ) e. RR )
215	142	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( A X. { ( 1 / ( # ` A ) ) } ) = ( k e. A |-> ( 1 / ( # ` A ) ) ) )
216	5, 214, 11, 215, 37	offval2 6914	. . . . . . . . . . . 12 |- ( ph -> ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) = ( k e. A |-> ( ( 1 / ( # ` A ) ) x. ( F ` k ) ) ) )
217	216	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) = (ℂfld gsumg ( k e. A |-> ( ( 1 / ( # ` A ) ) x. ( F ` k ) ) ) ) )
218		cnfldadd 19751	. . . . . . . . . . . 12 |- + = ( +g ` ℂfld )
219		cnfldmul 19752	. . . . . . . . . . . 12 |- x. = ( .r ` ℂfld )
220	2	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ℂfld e. Ring )
221		eqid 2622	. . . . . . . . . . . . . 14 |- ( k e. A |-> ( F ` k ) ) = ( k e. A |-> ( F ` k ) )
222	111, 221	fmptd 6385	. . . . . . . . . . . . 13 |- ( ph -> ( k e. A |-> ( F ` k ) ) : A --> CC )
223	222, 5, 17	fdmfifsupp 8285	. . . . . . . . . . . 12 |- ( ph -> ( k e. A |-> ( F ` k ) ) finSupp 0 )
224	54, 1, 218, 219, 220, 5, 146, 111, 223	gsummulc2 18607	. . . . . . . . . . 11 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( ( 1 / ( # ` A ) ) x. ( F ` k ) ) ) ) = ( ( 1 / ( # ` A ) ) x. (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) ) )
225		fss 6056	. . . . . . . . . . . . . . . 16 |- ( ( F : A --> RR+ /\ RR+ C_ RR ) -> F : A --> RR )
226	10, 120, 225	sylancl 694	. . . . . . . . . . . . . . 15 |- ( ph -> F : A --> RR )
227	10, 5, 17	fdmfifsupp 8285	. . . . . . . . . . . . . . 15 |- ( ph -> F finSupp 0 )
228	1, 4, 5, 9, 226, 227	gsumsubgcl 18320	. . . . . . . . . . . . . 14 |- ( ph -> (ℂfld gsumg F ) e. RR )
229	228	recnd 10068	. . . . . . . . . . . . 13 |- ( ph -> (ℂfld gsumg F ) e. CC )
230	229, 25, 26	divrec2d 10805	. . . . . . . . . . . 12 |- ( ph -> ( (ℂfld gsumg F ) / ( # ` A ) ) = ( ( 1 / ( # ` A ) ) x. (ℂfld gsumg F ) ) )
231	110	oveq2d 6666	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 / ( # ` A ) ) x. (ℂfld gsumg F ) ) = ( ( 1 / ( # ` A ) ) x. (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) ) )
232	230, 231	eqtr2d 2657	. . . . . . . . . . 11 |- ( ph -> ( ( 1 / ( # ` A ) ) x. (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) ) = ( (ℂfld gsumg F ) / ( # ` A ) ) )
233	217, 224, 232	3eqtrd 2660	. . . . . . . . . 10 |- ( ph -> (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) = ( (ℂfld gsumg F ) / ( # ` A ) ) )
234	233, 155	oveq12d 6668	. . . . . . . . 9 |- ( ph -> ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = ( ( (ℂfld gsumg F ) / ( # ` A ) ) / 1 ) )
235	228, 24	nndivred 11069	. . . . . . . . . . 11 |- ( ph -> ( (ℂfld gsumg F ) / ( # ` A ) ) e. RR )
236	235	recnd 10068	. . . . . . . . . 10 |- ( ph -> ( (ℂfld gsumg F ) / ( # ` A ) ) e. CC )
237	236	div1d 10793	. . . . . . . . 9 |- ( ph -> ( ( (ℂfld gsumg F ) / ( # ` A ) ) / 1 ) = ( (ℂfld gsumg F ) / ( # ` A ) ) )
238	234, 237	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = ( (ℂfld gsumg F ) / ( # ` A ) ) )
239	238	fveq2d 6195	. . . . . . 7 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) ) = ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
240		fveq2 6191	. . . . . . . . . 10 |- ( w = ( (ℂfld gsumg F ) / ( # ` A ) ) -> ( log ` w ) = ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
241	240	negeqd 10275	. . . . . . . . 9 |- ( w = ( (ℂfld gsumg F ) / ( # ` A ) ) -> -u ( log ` w ) = -u ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
242		negex 10279	. . . . . . . . 9 |- -u ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) e. _V
243	241, 125, 242	fvmpt 6282	. . . . . . . 8 |- ( ( (ℂfld gsumg F ) / ( # ` A ) ) e. RR+ -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) = -u ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
244	117, 243	syl 17	. . . . . . 7 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) = -u ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
245	239, 244	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) ` ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. F ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) ) = -u ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
246	54, 1, 218, 219, 220, 5, 146, 32, 18	gsummulc2 18607	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( ( 1 / ( # ` A ) ) x. -u ( log ` ( F ` k ) ) ) ) ) = ( ( 1 / ( # ` A ) ) x. (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) )
247		negex 10279	. . . . . . . . . . . 12 |- -u ( log ` ( F ` k ) ) e. _V
248	247	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> -u ( log ` ( F ` k ) ) e. _V )
249		eqidd 2623	. . . . . . . . . . . 12 |- ( ph -> ( w e. RR+ |-> -u ( log ` w ) ) = ( w e. RR+ |-> -u ( log ` w ) ) )
250		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = ( F ` k ) -> ( log ` w ) = ( log ` ( F ` k ) ) )
251	250	negeqd 10275	. . . . . . . . . . . 12 |- ( w = ( F ` k ) -> -u ( log ` w ) = -u ( log ` ( F ` k ) ) )
252	11, 37, 249, 251	fmptco 6396	. . . . . . . . . . 11 |- ( ph -> ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) = ( k e. A |-> -u ( log ` ( F ` k ) ) ) )
253	5, 214, 248, 215, 252	offval2 6914	. . . . . . . . . 10 |- ( ph -> ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) = ( k e. A |-> ( ( 1 / ( # ` A ) ) x. -u ( log ` ( F ` k ) ) ) ) )
254	253	oveq2d 6666	. . . . . . . . 9 |- ( ph -> (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) = (ℂfld gsumg ( k e. A |-> ( ( 1 / ( # ` A ) ) x. -u ( log ` ( F ` k ) ) ) ) ) )
255	20, 25, 26	divrec2d 10805	. . . . . . . . 9 |- ( ph -> ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) = ( ( 1 / ( # ` A ) ) x. (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) ) )
256	246, 254, 255	3eqtr4d 2666	. . . . . . . 8 |- ( ph -> (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) = ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) )
257	256, 155	oveq12d 6668	. . . . . . 7 |- ( ph -> ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = ( ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) / 1 ) )
258	119	recnd 10068	. . . . . . . 8 |- ( ph -> ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) e. CC )
259	258	div1d 10793	. . . . . . 7 |- ( ph -> ( ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) / 1 ) = ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) )
260	257, 259	eqtrd 2656	. . . . . 6 |- ( ph -> ( (ℂfld gsumg ( ( A X. { ( 1 / ( # ` A ) ) } ) oF x. ( ( w e. RR+ |-> -u ( log ` w ) ) o. F ) ) ) / (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) )
261	213, 245, 260	3brtr3d 4684	. . . . 5 |- ( ph -> -u ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) <_ ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) )
262	118, 119, 261	lenegcon1d 10609	. . . 4 |- ( ph -> -u ( (ℂfld gsumg ( k e. A |-> -u ( log ` ( F ` k ) ) ) ) / ( # ` A ) ) <_ ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
263	109, 262	eqbrtrrd 4677	. . 3 |- ( ph -> ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) <_ ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
264	134, 106	remulcld 10070	. . . 4 |- ( ph -> ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) e. RR )
265		efle 14848	. . . 4 |- ( ( ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) e. RR /\ ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) e. RR ) -> ( ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) <_ ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) <-> ( exp ` ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) ) <_ ( exp ` ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) ) ) )
266	264, 118, 265	syl2anc 693	. . 3 |- ( ph -> ( ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) <_ ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) <-> ( exp ` ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) ) <_ ( exp ` ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) ) ) )
267	263, 266	mpbid 222	. 2 |- ( ph -> ( exp ` ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) ) <_ ( exp ` ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) ) )
268	101	rpcnd 11874	. . 3 |- ( ph -> ( M gsumg F ) e. CC )
269	101	rpne0d 11877	. . 3 |- ( ph -> ( M gsumg F ) =/= 0 )
270	268, 269, 146	cxpefd 24458	. 2 |- ( ph -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) = ( exp ` ( ( 1 / ( # ` A ) ) x. ( log ` ( M gsumg F ) ) ) ) )
271	117	reeflogd 24370	. . 3 |- ( ph -> ( exp ` ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) ) = ( (ℂfld gsumg F ) / ( # ` A ) ) )
272	271	eqcomd 2628	. 2 |- ( ph -> ( (ℂfld gsumg F ) / ( # ` A ) ) = ( exp ` ( log ` ( (ℂfld gsumg F ) / ( # ` A ) ) ) ) )
273	267, 270, 272	3brtr4d 4685	1 |- ( ph -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )

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   X. cxp 5112   |` 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   NNcn 11020   ZZcz 11377   RR+crp 11832   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   #chash 13117   sum_csu 14416   expce 14792   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   MndHom cmhm 17333  SubMndcsubmnd 17334  ._gcmg 17540  SubGrpcsubg 17588   GrpHom cghm 17657   GrpIso cgim 17699  CMndccmn 18193   Abelcabl 18194  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:  amgm  24717  amgm2d  38501  amgm3d  38502  amgm4d  38503