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Theorem amgmlem 20300
Description: Lemma for amgm 20301. (Contributed by Mario Carneiro, 21-Jun-2015.)
Hypotheses
Ref Expression
amgm.1  |-  M  =  (mulGrp ` fld )
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
) ) )  <_ 
( (fld 
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.
StepHypRef Expression
1 cnfld0 16414 . . . . . . . 8  |-  0  =  ( 0g ` fld )
2 cnrng 16412 . . . . . . . . 9  |-fld  e.  Ring
3 rngabl 15386 . . . . . . . . 9  |-  (fld  e.  Ring  ->fld  e.  Abel )
42, 3mp1i 11 . . . . . . . 8  |-  ( ph  ->fld  e. 
Abel )
5 amgm.2 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
6 resubdrg 16439 . . . . . . . . . 10  |-  ( RR  e.  (SubRing ` fld )  /\  (flds  RR )  e.  DivRing )
76simpli 444 . . . . . . . . 9  |-  RR  e.  (SubRing ` fld )
8 subrgsubg 15567 . . . . . . . . 9  |-  ( RR  e.  (SubRing ` fld )  ->  RR  e.  (SubGrp ` fld ) )
97, 8mp1i 11 . . . . . . . 8  |-  ( ph  ->  RR  e.  (SubGrp ` fld )
)
10 amgm.4 . . . . . . . . . . . 12  |-  ( ph  ->  F : A --> RR+ )
11 ffvelrn 5679 . . . . . . . . . . . 12  |-  ( ( F : A --> RR+  /\  k  e.  A )  ->  ( F `  k )  e.  RR+ )
1210, 11sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  RR+ )
1312relogcld 19990 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  RR )
1413renegcld 9226 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  RR )
15 eqid 2296 . . . . . . . . 9  |-  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )  =  ( k  e.  A  |->  -u ( log `  ( F `  k )
) )
1614, 15fmptd 5700 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) : A --> RR )
175, 16fisuppfi 14466 . . . . . . . 8  |-  ( ph  ->  ( `' ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )
" ( _V  \  { 0 } ) )  e.  Fin )
181, 4, 5, 9, 16, 17gsumsubgcl 15218 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  RR )
1918recnd 8877 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  CC )
20 amgm.3 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
21 hashnncl 11370 . . . . . . . . 9  |-  ( A  e.  Fin  ->  (
( # `  A )  e.  NN  <->  A  =/=  (/) ) )
225, 21syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( # `  A
)  e.  NN  <->  A  =/=  (/) ) )
2320, 22mpbird 223 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  NN )
2423nncnd 9778 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  CC )
2523nnne0d 9806 . . . . . 6  |-  ( ph  ->  ( # `  A
)  =/=  0 )
2619, 24, 25divnegd 9565 . . . . 5  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( -u (fld  gsumg  (
k  e.  A  |->  -u ( log `  ( F `
 k ) ) ) )  /  ( # `
 A ) ) )
2713recnd 8877 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  CC )
285, 27gsumfsum 16455 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  ( log `  ( F `
 k ) ) )
2927negnegd 9164 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u -u ( log `  ( F `  k ) )  =  ( log `  ( F `  k )
) )
3029sumeq2dv 12192 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  sum_ k  e.  A  ( log `  ( F `  k
) ) )
3114recnd 8877 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  CC )
325, 31fsumneg 12265 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  -u sum_ k  e.  A  -u ( log `  ( F `  k
) ) )
3328, 30, 323eqtr2rd 2335 . . . . . . . 8  |-  ( ph  -> 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
)  =  (fld  gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
345, 31gsumfsum 16455 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3534negeqd 9062 . . . . . . . 8  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3610feqmptd 5591 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
37 relogf1o 19940 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
38 f1of 5488 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
3937, 38mp1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( log  |`  RR+ ) : RR+ --> RR )
4039feqmptd 5591 . . . . . . . . . . 11  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x
) ) )
41 fvres 5558 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
4241mpteq2ia 4118 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
4340, 42syl6eq 2344 . . . . . . . . . 10  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x
) ) )
44 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  ( F `  k )  ->  ( log `  x )  =  ( log `  ( F `  k )
) )
4512, 36, 43, 44fmptco 5707 . . . . . . . . 9  |-  ( ph  ->  ( ( log  |`  RR+ )  o.  F )  =  ( k  e.  A  |->  ( log `  ( F `
 k ) ) ) )
4645oveq2d 5890 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
4733, 35, 463eqtr4d 2338 . . . . . . 7  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  =  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) ) )
48 amgm.1 . . . . . . . . . . . . . . 15  |-  M  =  (mulGrp ` fld )
4948oveq1i 5884 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
5049rpmsubg 16451 . . . . . . . . . . . . 13  |-  RR+  e.  (SubGrp `  ( Ms  ( CC 
\  { 0 } ) ) )
51 subgsubm 14655 . . . . . . . . . . . . 13  |-  ( RR+  e.  (SubGrp `  ( Ms  ( CC  \  { 0 } ) ) )  ->  RR+ 
e.  (SubMnd `  ( Ms  ( CC  \  { 0 } ) ) ) )
5250, 51ax-mp 8 . . . . . . . . . . . 12  |-  RR+  e.  (SubMnd `  ( Ms  ( CC 
\  { 0 } ) ) )
53 cnfldbas 16399 . . . . . . . . . . . . . . 15  |-  CC  =  ( Base ` fld )
54 cndrng 16419 . . . . . . . . . . . . . . 15  |-fld  e.  DivRing
5553, 1, 54drngui 15534 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
5655, 48unitsubm 15468 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  M )
)
57 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( Ms  ( CC  \  { 0 } ) )
5857subsubm 14450 . . . . . . . . . . . . 13  |-  ( ( CC  \  { 0 } )  e.  (SubMnd `  M )  ->  ( RR+  e.  (SubMnd `  ( Ms  ( CC  \  { 0 } ) ) )  <-> 
( RR+  e.  (SubMnd `  M )  /\  RR+  C_  ( CC  \  { 0 } ) ) ) )
592, 56, 58mp2b 9 . . . . . . . . . . . 12  |-  ( RR+  e.  (SubMnd `  ( Ms  ( CC  \  { 0 } ) ) )  <->  ( RR+  e.  (SubMnd `  M )  /\  RR+  C_  ( CC  \  { 0 } ) ) )
6052, 59mpbi 199 . . . . . . . . . . 11  |-  ( RR+  e.  (SubMnd `  M )  /\  RR+  C_  ( CC  \  { 0 } ) )
6160simpli 444 . . . . . . . . . 10  |-  RR+  e.  (SubMnd `  M )
62 eqid 2296 . . . . . . . . . . 11  |-  ( Ms  RR+ )  =  ( Ms  RR+ )
6362submbas 14448 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  -> 
RR+  =  ( Base `  ( Ms  RR+ ) ) )
6461, 63ax-mp 8 . . . . . . . . 9  |-  RR+  =  ( Base `  ( Ms  RR+ )
)
65 cnfld1 16415 . . . . . . . . . . . 12  |-  1  =  ( 1r ` fld )
6648, 65rngidval 15359 . . . . . . . . . . 11  |-  1  =  ( 0g `  M )
6762, 66subm0 14449 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  ->  1  =  ( 0g
`  ( Ms  RR+ )
) )
6861, 67ax-mp 8 . . . . . . . . 9  |-  1  =  ( 0g `  ( Ms  RR+ ) )
69 cncrng 16411 . . . . . . . . . . 11  |-fld  e.  CRing
7048crngmgp 15365 . . . . . . . . . . 11  |-  (fld  e.  CRing  ->  M  e. CMnd )
7169, 70mp1i 11 . . . . . . . . . 10  |-  ( ph  ->  M  e. CMnd )
7262submmnd 14447 . . . . . . . . . . 11  |-  ( RR+  e.  (SubMnd `  M )  ->  ( Ms  RR+ )  e.  Mnd )
7361, 72mp1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( Ms  RR+ )  e.  Mnd )
7462subcmn 15149 . . . . . . . . . 10  |-  ( ( M  e. CMnd  /\  ( Ms  RR+ )  e.  Mnd )  ->  ( Ms  RR+ )  e. CMnd )
7571, 73, 74syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( Ms  RR+ )  e. CMnd )
76 eqid 2296 . . . . . . . . . . . 12  |-  (flds  RR )  =  (flds  RR )
7776subrgrng 15564 . . . . . . . . . . 11  |-  ( RR  e.  (SubRing ` fld )  ->  (flds  RR )  e.  Ring )
787, 77ax-mp 8 . . . . . . . . . 10  |-  (flds  RR )  e.  Ring
79 rngmnd 15366 . . . . . . . . . 10  |-  ( (flds  RR )  e.  Ring  ->  (flds  RR )  e.  Mnd )
8078, 79mp1i 11 . . . . . . . . 9  |-  ( ph  ->  (flds  RR )  e.  Mnd )
8148oveq1i 5884 . . . . . . . . . . . 12  |-  ( Ms  RR+ )  =  ( (mulGrp ` fld )s  RR+ )
8276, 81reloggim 19968 . . . . . . . . . . 11  |-  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) GrpIso  (flds  RR ) )
83 gimghm 14744 . . . . . . . . . . 11  |-  ( ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) GrpIso  (flds  RR ) )  ->  ( log  |`  RR+ )  e.  (
( Ms  RR+ )  GrpHom  (flds  RR ) ) )
8482, 83ax-mp 8 . . . . . . . . . 10  |-  ( log  |`  RR+ )  e.  ( ( Ms  RR+ )  GrpHom  (flds  RR ) )
85 ghmmhm 14709 . . . . . . . . . 10  |-  ( ( log  |`  RR+ )  e.  ( ( Ms  RR+ )  GrpHom  (flds  RR ) )  ->  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) MndHom  (flds  RR ) ) )
8684, 85mp1i 11 . . . . . . . . 9  |-  ( ph  ->  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) MndHom  (flds  RR ) ) )
875, 10fisuppfi 14466 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {
1 } ) )  e.  Fin )
8864, 68, 75, 80, 5, 86, 10, 87gsummhm 15227 . . . . . . . 8  |-  ( ph  ->  ( (flds  RR )  gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
89 subgsubm 14655 . . . . . . . . . 10  |-  ( RR  e.  (SubGrp ` fld )  ->  RR  e.  (SubMnd ` fld ) )
909, 89syl 15 . . . . . . . . 9  |-  ( ph  ->  RR  e.  (SubMnd ` fld )
)
91 fco 5414 . . . . . . . . . 10  |-  ( ( ( log  |`  RR+ ) : RR+ --> RR  /\  F : A --> RR+ )  ->  (
( log  |`  RR+ )  o.  F ) : A --> RR )
9239, 10, 91syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( log  |`  RR+ )  o.  F ) : A --> RR )
935, 90, 92, 76gsumsubm 14471 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( (flds  RR )  gsumg  ( ( log  |`  RR+ )  o.  F ) ) )
9461a1i 10 . . . . . . . . . 10  |-  ( ph  -> 
RR+  e.  (SubMnd `  M
) )
955, 94, 10, 62gsumsubm 14471 . . . . . . . . 9  |-  ( ph  ->  ( M  gsumg  F )  =  ( ( Ms  RR+ )  gsumg  F ) )
9695fveq2d 5545 . . . . . . . 8  |-  ( ph  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
9788, 93, 963eqtr4d 2338 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) ) )
9866, 71, 5, 94, 10, 87gsumsubmcl 15217 . . . . . . . 8  |-  ( ph  ->  ( M  gsumg  F )  e.  RR+ )
99 fvres 5558 . . . . . . . 8  |-  ( ( M  gsumg  F )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( log `  ( M  gsumg  F ) ) )
10098, 99syl 15 . . . . . . 7  |-  ( ph  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( log `  ( M  gsumg  F ) ) )
10147, 97, 1003eqtrd 2332 . . . . . 6  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  =  ( log `  ( M  gsumg  F ) ) )
102101oveq1d 5889 . . . . 5  |-  ( ph  ->  ( -u (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( log `  ( M 
gsumg  F ) )  / 
( # `  A ) ) )
10398relogcld 19990 . . . . . . 7  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  RR )
104103recnd 8877 . . . . . 6  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  CC )
105104, 24, 25divrec2d 9556 . . . . 5  |-  ( ph  ->  ( ( log `  ( M  gsumg  F ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10626, 102, 1053eqtrd 2332 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10736oveq2d 5890 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  F )  =  (fld  gsumg  ( k  e.  A  |->  ( F `
 k ) ) ) )
10812rpcnd 10408 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  CC )
1095, 108gsumfsum 16455 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) )  = 
sum_ k  e.  A  ( F `  k ) )
110107, 109eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  F )  =  sum_ k  e.  A  ( F `  k )
)
1115, 20, 12fsumrpcl 12226 . . . . . . . 8  |-  ( ph  -> 
sum_ k  e.  A  ( F `  k )  e.  RR+ )
112110, 111eqeltrd 2370 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR+ )
11323nnrpd 10405 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  RR+ )
114112, 113rpdivcld 10423 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR+ )
115114relogcld 19990 . . . . 5  |-  ( ph  ->  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  e.  RR )
11618, 23nndivred 9810 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  RR )
117 rpssre 10380 . . . . . . . . 9  |-  RR+  C_  RR
118117a1i 10 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
119 relogcl 19948 . . . . . . . . . . 11  |-  ( w  e.  RR+  ->  ( log `  w )  e.  RR )
120119adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( log `  w )  e.  RR )
121120renegcld 9226 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  RR+ )  ->  -u ( log `  w )  e.  RR )
122 eqid 2296 . . . . . . . . 9  |-  ( w  e.  RR+  |->  -u ( log `  w ) )  =  ( w  e.  RR+  |->  -u ( log `  w
) )
123121, 122fmptd 5700 . . . . . . . 8  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) ) : RR+ --> RR )
124 ioorp 10743 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
125124eleq2i 2360 . . . . . . . . . . 11  |-  ( a  e.  ( 0 (,) 
+oo )  <->  a  e.  RR+ )
126124eleq2i 2360 . . . . . . . . . . 11  |-  ( b  e.  ( 0 (,) 
+oo )  <->  b  e.  RR+ )
127 iccssioo2 10738 . . . . . . . . . . 11  |-  ( ( a  e.  ( 0 (,)  +oo )  /\  b  e.  ( 0 (,)  +oo ) )  ->  (
a [,] b ) 
C_  ( 0 (,) 
+oo ) )
128125, 126, 127syl2anbr 466 . . . . . . . . . 10  |-  ( ( a  e.  RR+  /\  b  e.  RR+ )  ->  (
a [,] b ) 
C_  ( 0 (,) 
+oo ) )
129128, 124syl6sseq 3237 . . . . . . . . 9  |-  ( ( a  e.  RR+  /\  b  e.  RR+ )  ->  (
a [,] b ) 
C_  RR+ )
130129adantl 452 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  RR+  /\  b  e.  RR+ ) )  ->  (
a [,] b ) 
C_  RR+ )
13123nnrecred 9807 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR )
132113rpreccld 10416 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR+ )
133132rpge0d 10410 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( 1  /  ( # `  A
) ) )
134 elrege0 10762 . . . . . . . . . 10  |-  ( ( 1  /  ( # `  A ) )  e.  ( 0 [,)  +oo ) 
<->  ( ( 1  / 
( # `  A ) )  e.  RR  /\  0  <_  ( 1  / 
( # `  A ) ) ) )
135131, 133, 134sylanbrc 645 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  ( 0 [,) 
+oo ) )
136 fconst6g 5446 . . . . . . . . 9  |-  ( ( 1  /  ( # `  A ) )  e.  ( 0 [,)  +oo )  ->  ( A  X.  { ( 1  / 
( # `  A ) ) } ) : A --> ( 0 [,) 
+oo ) )
137135, 136syl 15 . . . . . . . 8  |-  ( ph  ->  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) : A --> ( 0 [,)  +oo ) )
138 0lt1 9312 . . . . . . . . 9  |-  0  <  1
139 fconstmpt 4748 . . . . . . . . . . 11  |-  ( A  X.  { ( 1  /  ( # `  A
) ) } )  =  ( k  e.  A  |->  ( 1  / 
( # `  A ) ) )
140139oveq2i 5885 . . . . . . . . . 10  |-  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )
141 rngmnd 15366 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  ->fld  e.  Mnd )
1422, 141mp1i 11 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Mnd )
143131recnd 8877 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  CC )
144 eqid 2296 . . . . . . . . . . . . 13  |-  (.g ` fld )  =  (.g ` fld )
14553, 144gsumconst 15225 . . . . . . . . . . . 12  |-  ( (fld  e. 
Mnd  /\  A  e.  Fin  /\  ( 1  / 
( # `  A ) )  e.  CC )  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  ( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) ) )
146142, 5, 143, 145syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  ( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) ) )
14723nnzd 10132 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
148 cnfldmulg 16422 . . . . . . . . . . . 12  |-  ( ( ( # `  A
)  e.  ZZ  /\  ( 1  /  ( # `
 A ) )  e.  CC )  -> 
( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) )  =  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) ) )
149147, 143, 148syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) )  =  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) ) )
15024, 25recidd 9547 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) )  =  1 )
151146, 149, 1503eqtrd 2332 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  1 )
152140, 151syl5eq 2340 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) )  =  1 )
153138, 152syl5breqr 4075 . . . . . . . 8  |-  ( ph  ->  0  <  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )
154 logccv 20026 . . . . . . . . . . . 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 ) ) ) )
1551543adant1 973 . . . . . . . . . . 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 ) ) ) )
156 ioossre 10728 . . . . . . . . . . . . . . 15  |-  ( 0 (,) 1 )  C_  RR
157 simp3 957 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  ( 0 (,) 1
) )
158156, 157sseldi 3191 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  RR )
159 simp21 988 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  x  e.  RR+ )
160159relogcld 19990 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  RR )
161158, 160remulcld 8879 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  RR )
162 1re 8853 . . . . . . . . . . . . . . 15  |-  1  e.  RR
163 resubcl 9127 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR  /\  t  e.  RR )  ->  ( 1  -  t
)  e.  RR )
164162, 158, 163sylancr 644 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
1  -  t )  e.  RR )
165 simp22 989 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  y  e.  RR+ )
166165relogcld 19990 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  RR )
167164, 166remulcld 8879 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  RR )
168161, 167readdcld 8878 . . . . . . . . . . . 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 )
169 simp1 955 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ph )
170 ioossicc 10751 . . . . . . . . . . . . . . 15  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
171170, 157sseldi 3191 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  ( 0 [,] 1
) )
172118, 130cvxcl 20295 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  t  e.  ( 0 [,] 1 ) ) )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  e.  RR+ )
173169, 159, 165, 171, 172syl13anc 1184 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  e.  RR+ )
174173relogcld 19990 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  e.  RR )
175168, 174ltnegd 9366 . . . . . . . . . . 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
) ) ) ) )
176155, 175mpbid 201 . . . . . . . . . 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 ) ) ) )
177 fveq2 5541 . . . . . . . . . . . . 13  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  ( log `  w )  =  ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
178177negeqd 9062 . . . . . . . . . . . 12  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  -u ( log `  w )  = 
-u ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
179 negex 9066 . . . . . . . . . . . 12  |-  -u ( log `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  e. 
_V
180178, 122, 179fvmpt 5618 . . . . . . . . . . 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 ) ) ) )
181173, 180syl 15 . . . . . . . . . 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
) ) ) )
182 fveq2 5541 . . . . . . . . . . . . . . . . 17  |-  ( w  =  x  ->  ( log `  w )  =  ( log `  x
) )
183182negeqd 9062 . . . . . . . . . . . . . . . 16  |-  ( w  =  x  ->  -u ( log `  w )  = 
-u ( log `  x
) )
184 negex 9066 . . . . . . . . . . . . . . . 16  |-  -u ( log `  x )  e. 
_V
185183, 122, 184fvmpt 5618 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  ( ( w  e.  RR+  |->  -u ( log `  w ) ) `
 x )  = 
-u ( log `  x
) )
186159, 185syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( w  e.  RR+  |->  -u ( log `  w
) ) `  x
)  =  -u ( log `  x ) )
187186oveq2d 5890 . . . . . . . . . . . . 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
) ) )
188158recnd 8877 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  CC )
189160recnd 8877 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  CC )
190188, 189mulneg2d 9249 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  -u ( log `  x ) )  =  -u ( t  x.  ( log `  x
) ) )
191187, 190eqtrd 2328 . . . . . . . . . . . 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
) ) )
192 fveq2 5541 . . . . . . . . . . . . . . . . 17  |-  ( w  =  y  ->  ( log `  w )  =  ( log `  y
) )
193192negeqd 9062 . . . . . . . . . . . . . . . 16  |-  ( w  =  y  ->  -u ( log `  w )  = 
-u ( log `  y
) )
194 negex 9066 . . . . . . . . . . . . . . . 16  |-  -u ( log `  y )  e. 
_V
195193, 122, 194fvmpt 5618 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR+  ->  ( ( w  e.  RR+  |->  -u ( log `  w ) ) `
 y )  = 
-u ( log `  y
) )
196165, 195syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( w  e.  RR+  |->  -u ( log `  w
) ) `  y
)  =  -u ( log `  y ) )
197196oveq2d 5890 . . . . . . . . . . . . 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
) ) )
198164recnd 8877 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
1  -  t )  e.  CC )
199166recnd 8877 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  CC )
200198, 199mulneg2d 9249 . . . . . . . . . . . . 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
) ) )
201197, 200eqtrd 2328 . . . . . . . . . . . 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
) ) )
202191, 201oveq12d 5892 . . . . . . . . . . 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
) ) ) )
203161recnd 8877 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  CC )
204167recnd 8877 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  CC )
205203, 204negdid 9186 . . . . . . . . . . 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 ) ) ) )
206202, 205eqtr4d 2331 . . . . . . . . . 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
) ) ) )
207176, 181, 2063brtr4d 4069 . . . . . . . . 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
) ) ) )
208118, 123, 130, 207scvxcvx 20296 . . . . . . . 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
) ) ) )
209118, 123, 130, 5, 137, 10, 153, 208jensen 20299 . . . . . . 7  |-  ( ph  ->  ( ( (fld  gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  e.  RR+  /\  (
( w  e.  RR+  |->  -u ( log `  w
) ) `  (
(fld  gsumg  ( ( A  X.  {
( 1  /  ( # `
 A ) ) } )  o F  x.  F ) )  /  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) ) ) )  <_  ( (fld  gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  ( ( w  e.  RR+  |->  -u ( log `  w ) )  o.  F ) ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) ) ) )
210209simprd 449 . . . . . 6  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) ) `  (
(fld  gsumg  ( ( A  X.  {
( 1  /  ( # `
 A ) ) } )  o F  x.  F ) )  /  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) ) ) )  <_  ( (fld  gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  ( ( w  e.  RR+  |->  -u ( log `  w ) )  o.  F ) ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) ) )
211131adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  (
1  /  ( # `  A ) )  e.  RR )
212139a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  X.  {
( 1  /  ( # `
 A ) ) } )  =  ( k  e.  A  |->  ( 1  /  ( # `  A ) ) ) )
2135, 211, 12, 212, 36offval2 6111 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F )  =  ( k  e.  A  |->  ( ( 1  /  ( # `  A
) )  x.  ( F `  k )
) ) )
214213oveq2d 5890 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  (fld  gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) ) )
215 cnfldadd 16400 . . . . . . . . . . . 12  |-  +  =  ( +g  ` fld )
216 cnfldmul 16401 . . . . . . . . . . . 12  |-  x.  =  ( .r ` fld )
2172a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Ring )
218 eqid 2296 . . . . . . . . . . . . . 14  |-  ( k  e.  A  |->  ( F `
 k ) )  =  ( k  e.  A  |->  ( F `  k ) )
219108, 218fmptd 5700 . . . . . . . . . . . . 13  |-  ( ph  ->  ( k  e.  A  |->  ( F `  k
) ) : A --> CC )
2205, 219fisuppfi 14466 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( k  e.  A  |->  ( F `
 k ) )
" ( _V  \  { 0 } ) )  e.  Fin )
22153, 1, 215, 216, 217, 5, 143, 108, 220gsummulc2 15407 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) )  =  ( ( 1  /  ( # `
 A ) )  x.  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
222 fss 5413 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> RR+  /\  RR+  C_  RR )  ->  F : A --> RR )
22310, 117, 222sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : A --> RR )
2245, 10fisuppfi 14466 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' F "
( _V  \  {
0 } ) )  e.  Fin )
2251, 4, 5, 9, 223, 224gsumsubgcl 15218 . . . . . . . . . . . . . 14  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR )
226225recnd 8877 . . . . . . . . . . . . 13  |-  ( ph  ->  (fld 
gsumg  F )  e.  CC )
227226, 24, 25divrec2d 9556 . . . . . . . . . . . 12  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( ( 1  /  ( # `  A
) )  x.  (fld  gsumg  F ) ) )
228107oveq2d 5890 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  F ) )  =  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
229227, 228eqtr2d 2329 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
230214, 221, 2293eqtrd 2332 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  ( (fld  gsumg  F )  /  ( # `  A
) ) )
231230, 152oveq12d 5892 . . . . . . . . 9  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( ( (fld  gsumg  F )  /  ( # `  A
) )  /  1
) )
232225, 23nndivred 9810 . . . . . . . . . . 11  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR )
233232recnd 8877 . . . . . . . . . 10  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  CC )
234233div1d 9544 . . . . . . . . 9  |-  ( ph  ->  ( ( (fld  gsumg  F )  /  ( # `
 A ) )  /  1 )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
235231, 234eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
236235fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) ) `  (
(fld  gsumg  ( ( A  X.  {
( 1  /  ( # `
 A ) ) } )  o F  x.  F ) )  /  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) ) ) )  =  ( ( w  e.  RR+  |->  -u ( log `  w ) ) `
 ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
237 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  ( log `  w
)  =  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
238237negeqd 9062 . . . . . . . . 9  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  -u ( log `  w
)  =  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
239 negex 9066 . . . . . . . . 9  |-  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) )  e.  _V
240238, 122, 239fvmpt 5618 . . . . . . . 8  |-  ( ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR+  ->  ( ( w  e.  RR+  |->  -u ( log `  w ) ) `
 ( (fld  gsumg  F )  /  ( # `
 A ) ) )  =  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
241114, 240syl 15 . . . . . . 7  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) ) `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  =  -u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
242236, 241eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) ) `  (
(fld  gsumg  ( ( A  X.  {
( 1  /  ( # `
 A ) ) } )  o F  x.  F ) )  /  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) ) ) )  =  -u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
24353, 1, 215, 216, 217, 5, 143, 31, 17gsummulc2 15407 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  -u ( log `  ( F `  k ) ) ) ) )  =  ( ( 1  /  ( # `
 A ) )  x.  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) ) ) )
244 negex 9066 . . . . . . . . . . . 12  |-  -u ( log `  ( F `  k ) )  e. 
_V
245244a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e. 
_V )
246 eqidd 2297 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) )  =  ( w  e.  RR+  |->  -u ( log `  w ) ) )
247 fveq2 5541 . . . . . . . . . . . . 13  |-  ( w  =  ( F `  k )  ->  ( log `  w )  =  ( log `  ( F `  k )
) )
248247negeqd 9062 . . . . . . . . . . . 12  |-  ( w  =  ( F `  k )  ->  -u ( log `  w )  = 
-u ( log `  ( F `  k )
) )
24912, 36, 246, 248fmptco 5707 . . . . . . . . . . 11  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) )  o.  F
)  =  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) ) )
2505, 211, 245, 212, 249offval2 6111 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  ( ( w  e.  RR+  |->  -u ( log `  w ) )  o.  F ) )  =  ( k  e.  A  |->  ( ( 1  /  ( # `  A
) )  x.  -u ( log `  ( F `  k ) ) ) ) )
251250oveq2d 5890 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  ( ( w  e.  RR+  |->  -u ( log `  w ) )  o.  F ) ) )  =  (fld  gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  -u ( log `  ( F `  k ) ) ) ) ) )
25219, 24, 25divrec2d 9556 . . . . . . . . 9  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) ) ) )
253243, 251, 2523eqtr4d 2338 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  ( ( w  e.  RR+  |->  -u ( log `  w ) )  o.  F ) ) )  =  ( (fld  gsumg  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) ) )  /  ( # `  A ) ) )
254253, 152oveq12d 5892 . . . . . . 7  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  ( ( w  e.  RR+  |->  -u ( log `  w ) )  o.  F ) ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( ( (fld  gsumg  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) ) )  /  ( # `  A ) )  / 
1 ) )
255116recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  CC )
256255div1d 9544 . . . . . . 7  |-  ( ph  ->  ( ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  /  1 )  =  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
257254, 256eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  ( ( w  e.  RR+  |->  -u ( log `  w ) )  o.  F ) ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
258210, 242, 2573brtr3d 4068 . . . . 5  |-  ( ph  -> 
-u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  <_  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
259115, 116, 258lenegcon1d 9370 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  <_  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
260106, 259eqbrtrrd 4061 . . 3  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  <_  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
261131, 103remulcld 8879 . . . 4  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  e.  RR )
262 efle 12414 . . . 4  |-  ( ( ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  e.  RR  /\  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) )  e.  RR )  ->  ( ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  <_  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  <-> 
( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )  <_  ( exp `  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) ) ) )
263261, 115, 262syl2anc 642 . . 3  |-  ( ph  ->  ( ( ( 1  /  ( # `  A
) )  x.  ( log `  ( M  gsumg  F ) ) )  <_  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) )  <->  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )  <_  ( exp `  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) ) ) )
264260, 263mpbid 201 . 2  |-  ( ph  ->  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )  <_  ( exp `  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) ) )
26598rpcnd 10408 . . 3  |-  ( ph  ->  ( M  gsumg  F )  e.  CC )
26698rpne0d 10411 . . 3  |-  ( ph  ->  ( M  gsumg  F )  =/=  0
)
267265, 266, 143cxpefd 20075 . 2  |-  ( ph  ->  ( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  =  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) ) )
268114reeflogd 19991 . . 3  |-  ( ph  ->  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
269268eqcomd 2301 . 2  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) ) )
270264, 267, 2693brtr4d 4069 1  |-  ( ph  ->  ( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  <_ 
( (fld 
gsumg  F )  /  ( # `
 A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703    |` cres 4707    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   ZZcz 10040   RR+crp 10370   (,)cioo 10672   [,)cico 10674   [,]cicc 10675   #chash 11353   sum_csu 12174   expce 12359   Basecbs 13164   ↾s cress 13165   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377  .gcmg 14382   MndHom cmhm 14429  SubMndcsubmnd 14430  SubGrpcsubg 14631    GrpHom cghm 14696   GrpIso cgim 14737  CMndccmn 15105   Abelcabel 15106  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354   DivRingcdr 15528  SubRingcsubrg 15557  ℂfldccnfld 16393   logclog 19928    ^ c ccxp 19929
This theorem is referenced by:  amgm  20301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-ghm 14697  df-gim 14739  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-subrg 15559  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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