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Theorem amgmlem 20284
Description: Lemma for amgm 20285. (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 16398 . . . . . . . 8  |-  0  =  ( 0g ` fld )
2 cnrng 16396 . . . . . . . . 9  |-fld  e.  Ring
3 rngabl 15370 . . . . . . . . 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 16423 . . . . . . . . . 10  |-  ( RR  e.  (SubRing ` fld )  /\  (flds  RR )  e.  DivRing )
76simpli 444 . . . . . . . . 9  |-  RR  e.  (SubRing ` fld )
8 subrgsubg 15551 . . . . . . . . 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 5663 . . . . . . . . . . . 12  |-  ( ( F : A --> RR+  /\  k  e.  A )  ->  ( F `  k )  e.  RR+ )
1210, 11sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  RR+ )
1312relogcld 19974 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  RR )
1413renegcld 9210 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  RR )
15 eqid 2283 . . . . . . . . 9  |-  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )  =  ( k  e.  A  |->  -u ( log `  ( F `  k )
) )
1614, 15fmptd 5684 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) : A --> RR )
175, 16fisuppfi 14450 . . . . . . . 8  |-  ( ph  ->  ( `' ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )
" ( _V  \  { 0 } ) )  e.  Fin )
181, 4, 5, 9, 16, 17gsumsubgcl 15202 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  RR )
1918recnd 8861 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  CC )
20 amgm.3 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
21 hashnncl 11354 . . . . . . . . 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 9762 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  CC )
2523nnne0d 9790 . . . . . 6  |-  ( ph  ->  ( # `  A
)  =/=  0 )
2619, 24, 25divnegd 9549 . . . . 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 8861 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  CC )
285, 27gsumfsum 16439 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  ( log `  ( F `
 k ) ) )
2927negnegd 9148 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u -u ( log `  ( F `  k ) )  =  ( log `  ( F `  k )
) )
3029sumeq2dv 12176 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  sum_ k  e.  A  ( log `  ( F `  k
) ) )
3114recnd 8861 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  CC )
325, 31fsumneg 12249 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  -u sum_ k  e.  A  -u ( log `  ( F `  k
) ) )
3328, 30, 323eqtr2rd 2322 . . . . . . . 8  |-  ( ph  -> 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
)  =  (fld  gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
345, 31gsumfsum 16439 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3534negeqd 9046 . . . . . . . 8  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3610feqmptd 5575 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
37 relogf1o 19924 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
38 f1of 5472 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
3937, 38mp1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( log  |`  RR+ ) : RR+ --> RR )
4039feqmptd 5575 . . . . . . . . . . 11  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x
) ) )
41 fvres 5542 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
4241mpteq2ia 4102 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
4340, 42syl6eq 2331 . . . . . . . . . 10  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x
) ) )
44 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  ( F `  k )  ->  ( log `  x )  =  ( log `  ( F `  k )
) )
4512, 36, 43, 44fmptco 5691 . . . . . . . . 9  |-  ( ph  ->  ( ( log  |`  RR+ )  o.  F )  =  ( k  e.  A  |->  ( log `  ( F `
 k ) ) ) )
4645oveq2d 5874 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
4733, 35, 463eqtr4d 2325 . . . . . . 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 5868 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
5049rpmsubg 16435 . . . . . . . . . . . . 13  |-  RR+  e.  (SubGrp `  ( Ms  ( CC 
\  { 0 } ) ) )
51 subgsubm 14639 . . . . . . . . . . . . 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 16383 . . . . . . . . . . . . . . 15  |-  CC  =  ( Base ` fld )
54 cndrng 16403 . . . . . . . . . . . . . . 15  |-fld  e.  DivRing
5553, 1, 54drngui 15518 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
5655, 48unitsubm 15452 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  M )
)
57 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( Ms  ( CC  \  { 0 } ) )
5857subsubm 14434 . . . . . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( Ms  RR+ )  =  ( Ms  RR+ )
6362submbas 14432 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  -> 
RR+  =  ( Base `  ( Ms  RR+ ) ) )
6461, 63ax-mp 8 . . . . . . . . 9  |-  RR+  =  ( Base `  ( Ms  RR+ )
)
65 cnfld1 16399 . . . . . . . . . . . 12  |-  1  =  ( 1r ` fld )
6648, 65rngidval 15343 . . . . . . . . . . 11  |-  1  =  ( 0g `  M )
6762, 66subm0 14433 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  ->  1  =  ( 0g
`  ( Ms  RR+ )
) )
6861, 67ax-mp 8 . . . . . . . . 9  |-  1  =  ( 0g `  ( Ms  RR+ ) )
69 cncrng 16395 . . . . . . . . . . 11  |-fld  e.  CRing
7048crngmgp 15349 . . . . . . . . . . 11  |-  (fld  e.  CRing  ->  M  e. CMnd )
7169, 70mp1i 11 . . . . . . . . . 10  |-  ( ph  ->  M  e. CMnd )
7262submmnd 14431 . . . . . . . . . . 11  |-  ( RR+  e.  (SubMnd `  M )  ->  ( Ms  RR+ )  e.  Mnd )
7361, 72mp1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( Ms  RR+ )  e.  Mnd )
7462subcmn 15133 . . . . . . . . . 10  |-  ( ( M  e. CMnd  /\  ( Ms  RR+ )  e.  Mnd )  ->  ( Ms  RR+ )  e. CMnd )
7571, 73, 74syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( Ms  RR+ )  e. CMnd )
76 eqid 2283 . . . . . . . . . . . 12  |-  (flds  RR )  =  (flds  RR )
7776subrgrng 15548 . . . . . . . . . . 11  |-  ( RR  e.  (SubRing ` fld )  ->  (flds  RR )  e.  Ring )
787, 77ax-mp 8 . . . . . . . . . 10  |-  (flds  RR )  e.  Ring
79 rngmnd 15350 . . . . . . . . . 10  |-  ( (flds  RR )  e.  Ring  ->  (flds  RR )  e.  Mnd )
8078, 79mp1i 11 . . . . . . . . 9  |-  ( ph  ->  (flds  RR )  e.  Mnd )
8148oveq1i 5868 . . . . . . . . . . . 12  |-  ( Ms  RR+ )  =  ( (mulGrp ` fld )s  RR+ )
8276, 81reloggim 19952 . . . . . . . . . . 11  |-  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) GrpIso  (flds  RR ) )
83 gimghm 14728 . . . . . . . . . . 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 14693 . . . . . . . . . 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 14450 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {
1 } ) )  e.  Fin )
8864, 68, 75, 80, 5, 86, 10, 87gsummhm 15211 . . . . . . . 8  |-  ( ph  ->  ( (flds  RR )  gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
89 subgsubm 14639 . . . . . . . . . 10  |-  ( RR  e.  (SubGrp ` fld )  ->  RR  e.  (SubMnd ` fld ) )
909, 89syl 15 . . . . . . . . 9  |-  ( ph  ->  RR  e.  (SubMnd ` fld )
)
91 fco 5398 . . . . . . . . . 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 14455 . . . . . . . 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 14455 . . . . . . . . 9  |-  ( ph  ->  ( M  gsumg  F )  =  ( ( Ms  RR+ )  gsumg  F ) )
9695fveq2d 5529 . . . . . . . 8  |-  ( ph  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
9788, 93, 963eqtr4d 2325 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) ) )
9866, 71, 5, 94, 10, 87gsumsubmcl 15201 . . . . . . . 8  |-  ( ph  ->  ( M  gsumg  F )  e.  RR+ )
99 fvres 5542 . . . . . . . 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 2319 . . . . . 6  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  =  ( log `  ( M  gsumg  F ) ) )
102101oveq1d 5873 . . . . 5  |-  ( ph  ->  ( -u (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( log `  ( M 
gsumg  F ) )  / 
( # `  A ) ) )
10398relogcld 19974 . . . . . . 7  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  RR )
104103recnd 8861 . . . . . 6  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  CC )
105104, 24, 25divrec2d 9540 . . . . 5  |-  ( ph  ->  ( ( log `  ( M  gsumg  F ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10626, 102, 1053eqtrd 2319 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10736oveq2d 5874 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  F )  =  (fld  gsumg  ( k  e.  A  |->  ( F `
 k ) ) ) )
10812rpcnd 10392 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  CC )
1095, 108gsumfsum 16439 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) )  = 
sum_ k  e.  A  ( F `  k ) )
110107, 109eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  F )  =  sum_ k  e.  A  ( F `  k )
)
1115, 20, 12fsumrpcl 12210 . . . . . . . 8  |-  ( ph  -> 
sum_ k  e.  A  ( F `  k )  e.  RR+ )
112110, 111eqeltrd 2357 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR+ )
11323nnrpd 10389 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  RR+ )
114112, 113rpdivcld 10407 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR+ )
115114relogcld 19974 . . . . 5  |-  ( ph  ->  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  e.  RR )
11618, 23nndivred 9794 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  RR )
117 rpssre 10364 . . . . . . . . 9  |-  RR+  C_  RR
118117a1i 10 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
119 relogcl 19932 . . . . . . . . . . 11  |-  ( w  e.  RR+  ->  ( log `  w )  e.  RR )
120119adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( log `  w )  e.  RR )
121120renegcld 9210 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  RR+ )  ->  -u ( log `  w )  e.  RR )
122 eqid 2283 . . . . . . . . 9  |-  ( w  e.  RR+  |->  -u ( log `  w ) )  =  ( w  e.  RR+  |->  -u ( log `  w
) )
123121, 122fmptd 5684 . . . . . . . 8  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) ) : RR+ --> RR )
124 ioorp 10727 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
125124eleq2i 2347 . . . . . . . . . . 11  |-  ( a  e.  ( 0 (,) 
+oo )  <->  a  e.  RR+ )
126124eleq2i 2347 . . . . . . . . . . 11  |-  ( b  e.  ( 0 (,) 
+oo )  <->  b  e.  RR+ )
127 iccssioo2 10722 . . . . . . . . . . 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 3224 . . . . . . . . 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 9791 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR )
132113rpreccld 10400 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR+ )
133132rpge0d 10394 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( 1  /  ( # `  A
) ) )
134 elrege0 10746 . . . . . . . . . 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 5430 . . . . . . . . 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 9296 . . . . . . . . 9  |-  0  <  1
139 fconstmpt 4732 . . . . . . . . . . 11  |-  ( A  X.  { ( 1  /  ( # `  A
) ) } )  =  ( k  e.  A  |->  ( 1  / 
( # `  A ) ) )
140139oveq2i 5869 . . . . . . . . . 10  |-  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )
141 rngmnd 15350 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  ->fld  e.  Mnd )
1422, 141mp1i 11 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Mnd )
143131recnd 8861 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  CC )
144 eqid 2283 . . . . . . . . . . . . 13  |-  (.g ` fld )  =  (.g ` fld )
14553, 144gsumconst 15209 . . . . . . . . . . . 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 10116 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
148 cnfldmulg 16406 . . . . . . . . . . . 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 9531 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) )  =  1 )
151146, 149, 1503eqtrd 2319 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  1 )
152140, 151syl5eq 2327 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) )  =  1 )
153138, 152syl5breqr 4059 . . . . . . . 8  |-  ( ph  ->  0  <  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )
154 logccv 20010 . . . . . . . . . . . 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 10712 . . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . . . 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 19974 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  RR )
161158, 160remulcld 8863 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  RR )
162 1re 8837 . . . . . . . . . . . . . . 15  |-  1  e.  RR
163 resubcl 9111 . . . . . . . . . . . . . . 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 19974 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  RR )
167164, 166remulcld 8863 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  RR )
168161, 167readdcld 8862 . . . . . . . . . . . 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 10735 . . . . . . . . . . . . . . 15  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
171170, 157sseldi 3178 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  ( 0 [,] 1
) )
172118, 130cvxcl 20279 . . . . . . . . . . . . . 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 19974 . . . . . . . . . . . 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 9350 . . . . . . . . . . 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 5525 . . . . . . . . . . . . 13  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  ( log `  w )  =  ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
178177negeqd 9046 . . . . . . . . . . . 12  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  -u ( log `  w )  = 
-u ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
179 negex 9050 . . . . . . . . . . . 12  |-  -u ( log `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  e. 
_V
180178, 122, 179fvmpt 5602 . . . . . . . . . . 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 5525 . . . . . . . . . . . . . . . . 17  |-  ( w  =  x  ->  ( log `  w )  =  ( log `  x
) )
183182negeqd 9046 . . . . . . . . . . . . . . . 16  |-  ( w  =  x  ->  -u ( log `  w )  = 
-u ( log `  x
) )
184 negex 9050 . . . . . . . . . . . . . . . 16  |-  -u ( log `  x )  e. 
_V
185183, 122, 184fvmpt 5602 . . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . 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 8861 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  CC )
189160recnd 8861 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  CC )
190188, 189mulneg2d 9233 . . . . . . . . . . . . 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 2315 . . . . . . . . . . . 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 5525 . . . . . . . . . . . . . . . . 17  |-  ( w  =  y  ->  ( log `  w )  =  ( log `  y
) )
193192negeqd 9046 . . . . . . . . . . . . . . . 16  |-  ( w  =  y  ->  -u ( log `  w )  = 
-u ( log `  y
) )
194 negex 9050 . . . . . . . . . . . . . . . 16  |-  -u ( log `  y )  e. 
_V
195193, 122, 194fvmpt 5602 . . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . 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 8861 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
1  -  t )  e.  CC )
199166recnd 8861 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  CC )
200198, 199mulneg2d 9233 . . . . . . . . . . . . 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 2315 . . . . . . . . . . . 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 5876 . . . . . . . . . . 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 8861 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  CC )
204167recnd 8861 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  CC )
205203, 204negdid 9170 . . . . . . . . . . 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 2318 . . . . . . . . . 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 4053 . . . . . . . . 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 20280 . . . . . . . 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 20283 . . . . . . 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 6095 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F )  =  ( k  e.  A  |->  ( ( 1  /  ( # `  A
) )  x.  ( F `  k )
) ) )
214213oveq2d 5874 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  (fld  gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) ) )
215 cnfldadd 16384 . . . . . . . . . . . 12  |-  +  =  ( +g  ` fld )
216 cnfldmul 16385 . . . . . . . . . . . 12  |-  x.  =  ( .r ` fld )
2172a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Ring )
218 eqid 2283 . . . . . . . . . . . . . 14  |-  ( k  e.  A  |->  ( F `
 k ) )  =  ( k  e.  A  |->  ( F `  k ) )
219108, 218fmptd 5684 . . . . . . . . . . . . 13  |-  ( ph  ->  ( k  e.  A  |->  ( F `  k
) ) : A --> CC )
2205, 219fisuppfi 14450 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( k  e.  A  |->  ( F `
 k ) )
" ( _V  \  { 0 } ) )  e.  Fin )
22153, 1, 215, 216, 217, 5, 143, 108, 220gsummulc2 15391 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) )  =  ( ( 1  /  ( # `
 A ) )  x.  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
222 fss 5397 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> RR+  /\  RR+  C_  RR )  ->  F : A --> RR )
22310, 117, 222sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : A --> RR )
2245, 10fisuppfi 14450 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' F "
( _V  \  {
0 } ) )  e.  Fin )
2251, 4, 5, 9, 223, 224gsumsubgcl 15202 . . . . . . . . . . . . . 14  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR )
226225recnd 8861 . . . . . . . . . . . . 13  |-  ( ph  ->  (fld 
gsumg  F )  e.  CC )
227226, 24, 25divrec2d 9540 . . . . . . . . . . . 12  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( ( 1  /  ( # `  A
) )  x.  (fld  gsumg  F ) ) )
228107oveq2d 5874 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  F ) )  =  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
229227, 228eqtr2d 2316 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
230214, 221, 2293eqtrd 2319 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  ( (fld  gsumg  F )  /  ( # `  A
) ) )
231230, 152oveq12d 5876 . . . . . . . . 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 9794 . . . . . . . . . . 11  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR )
233232recnd 8861 . . . . . . . . . 10  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  CC )
234233div1d 9528 . . . . . . . . 9  |-  ( ph  ->  ( ( (fld  gsumg  F )  /  ( # `
 A ) )  /  1 )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
235231, 234eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
236235fveq2d 5529 . . . . . . 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 5525 . . . . . . . . . 10  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  ( log `  w
)  =  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
238237negeqd 9046 . . . . . . . . 9  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  -u ( log `  w
)  =  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
239 negex 9050 . . . . . . . . 9  |-  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) )  e.  _V
240238, 122, 239fvmpt 5602 . . . . . . . 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 2315 . . . . . 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 15391 . . . . . . . . 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 9050 . . . . . . . . . . . 12  |-  -u ( log `  ( F `  k ) )  e. 
_V
245244a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e. 
_V )
246 eqidd 2284 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) )  =  ( w  e.  RR+  |->  -u ( log `  w ) ) )
247 fveq2 5525 . . . . . . . . . . . . 13  |-  ( w  =  ( F `  k )  ->  ( log `  w )  =  ( log `  ( F `  k )
) )
248247negeqd 9046 . . . . . . . . . . . 12  |-  ( w  =  ( F `  k )  ->  -u ( log `  w )  = 
-u ( log `  ( F `  k )
) )
24912, 36, 246, 248fmptco 5691 . . . . . . . . . . 11  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) )  o.  F
)  =  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) ) )
2505, 211, 245, 212, 249offval2 6095 . . . . . . . . . 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 5874 . . . . . . . . 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 9540 . . . . . . . . 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 2325 . . . . . . . 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 5876 . . . . . . 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 8861 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  CC )
256255div1d 9528 . . . . . . 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 2315 . . . . . 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 4052 . . . . 5  |-  ( ph  -> 
-u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  <_  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
259115, 116, 258lenegcon1d 9354 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  <_  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
260106, 259eqbrtrrd 4045 . . 3  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  <_  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
261131, 103remulcld 8863 . . . 4  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  e.  RR )
262 efle 12398 . . . 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 10392 . . 3  |-  ( ph  ->  ( M  gsumg  F )  e.  CC )
26698rpne0d 10395 . . 3  |-  ( ph  ->  ( M  gsumg  F )  =/=  0
)
267265, 266, 143cxpefd 20059 . 2  |-  ( ph  ->  ( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  =  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) ) )
268114reeflogd 19975 . . 3  |-  ( ph  ->  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
269268eqcomd 2288 . 2  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) ) )
270264, 267, 2693brtr4d 4053 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   ZZcz 10024   RR+crp 10354   (,)cioo 10656   [,)cico 10658   [,]cicc 10659   #chash 11337   sum_csu 12158   expce 12343   Basecbs 13148   ↾s cress 13149   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361  .gcmg 14366   MndHom cmhm 14413  SubMndcsubmnd 14414  SubGrpcsubg 14615    GrpHom cghm 14680   GrpIso cgim 14721  CMndccmn 15089   Abelcabel 15090  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   DivRingcdr 15512  SubRingcsubrg 15541  ℂfldccnfld 16377   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  amgm  20285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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