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Theorem amgmlem 20696
Description: Lemma for amgm 20697. (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 16649 . . . . . . . 8  |-  0  =  ( 0g ` fld )
2 cnrng 16647 . . . . . . . . 9  |-fld  e.  Ring
3 rngabl 15621 . . . . . . . . 9  |-  (fld  e.  Ring  ->fld  e.  Abel )
42, 3mp1i 12 . . . . . . . 8  |-  ( ph  ->fld  e. 
Abel )
5 amgm.2 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
6 resubdrg 16674 . . . . . . . . . 10  |-  ( RR  e.  (SubRing ` fld )  /\  (flds  RR )  e.  DivRing )
76simpli 445 . . . . . . . . 9  |-  RR  e.  (SubRing ` fld )
8 subrgsubg 15802 . . . . . . . . 9  |-  ( RR  e.  (SubRing ` fld )  ->  RR  e.  (SubGrp ` fld ) )
97, 8mp1i 12 . . . . . . . 8  |-  ( ph  ->  RR  e.  (SubGrp ` fld )
)
10 amgm.4 . . . . . . . . . . . 12  |-  ( ph  ->  F : A --> RR+ )
1110ffvelrnda 5810 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  RR+ )
1211relogcld 20386 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  RR )
1312renegcld 9397 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  RR )
14 eqid 2388 . . . . . . . . 9  |-  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )  =  ( k  e.  A  |->  -u ( log `  ( F `  k )
) )
1513, 14fmptd 5833 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) : A --> RR )
165, 15fisuppfi 14701 . . . . . . . 8  |-  ( ph  ->  ( `' ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )
" ( _V  \  { 0 } ) )  e.  Fin )
171, 4, 5, 9, 15, 16gsumsubgcl 15453 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  RR )
1817recnd 9048 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  CC )
19 amgm.3 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
20 hashnncl 11573 . . . . . . . . 9  |-  ( A  e.  Fin  ->  (
( # `  A )  e.  NN  <->  A  =/=  (/) ) )
215, 20syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( # `  A
)  e.  NN  <->  A  =/=  (/) ) )
2219, 21mpbird 224 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  NN )
2322nncnd 9949 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  CC )
2422nnne0d 9977 . . . . . 6  |-  ( ph  ->  ( # `  A
)  =/=  0 )
2518, 23, 24divnegd 9736 . . . . 5  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( -u (fld  gsumg  (
k  e.  A  |->  -u ( log `  ( F `
 k ) ) ) )  /  ( # `
 A ) ) )
2612recnd 9048 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  CC )
275, 26gsumfsum 16690 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  ( log `  ( F `
 k ) ) )
2826negnegd 9335 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u -u ( log `  ( F `  k ) )  =  ( log `  ( F `  k )
) )
2928sumeq2dv 12425 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  sum_ k  e.  A  ( log `  ( F `  k
) ) )
3013recnd 9048 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  CC )
315, 30fsumneg 12498 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  -u sum_ k  e.  A  -u ( log `  ( F `  k
) ) )
3227, 29, 313eqtr2rd 2427 . . . . . . . 8  |-  ( ph  -> 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
)  =  (fld  gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
335, 30gsumfsum 16690 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3433negeqd 9233 . . . . . . . 8  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3510feqmptd 5719 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
36 relogf1o 20332 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
37 f1of 5615 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
3836, 37mp1i 12 . . . . . . . . . . . 12  |-  ( ph  ->  ( log  |`  RR+ ) : RR+ --> RR )
3938feqmptd 5719 . . . . . . . . . . 11  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x
) ) )
40 fvres 5686 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
4140mpteq2ia 4233 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
4239, 41syl6eq 2436 . . . . . . . . . 10  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x
) ) )
43 fveq2 5669 . . . . . . . . . 10  |-  ( x  =  ( F `  k )  ->  ( log `  x )  =  ( log `  ( F `  k )
) )
4411, 35, 42, 43fmptco 5841 . . . . . . . . 9  |-  ( ph  ->  ( ( log  |`  RR+ )  o.  F )  =  ( k  e.  A  |->  ( log `  ( F `
 k ) ) ) )
4544oveq2d 6037 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
4632, 34, 453eqtr4d 2430 . . . . . . 7  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  =  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) ) )
47 amgm.1 . . . . . . . . . . . . . . 15  |-  M  =  (mulGrp ` fld )
4847oveq1i 6031 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
4948rpmsubg 16686 . . . . . . . . . . . . 13  |-  RR+  e.  (SubGrp `  ( Ms  ( CC 
\  { 0 } ) ) )
50 subgsubm 14890 . . . . . . . . . . . . 13  |-  ( RR+  e.  (SubGrp `  ( Ms  ( CC  \  { 0 } ) ) )  ->  RR+ 
e.  (SubMnd `  ( Ms  ( CC  \  { 0 } ) ) ) )
5149, 50ax-mp 8 . . . . . . . . . . . 12  |-  RR+  e.  (SubMnd `  ( Ms  ( CC 
\  { 0 } ) ) )
52 cnfldbas 16631 . . . . . . . . . . . . . . 15  |-  CC  =  ( Base ` fld )
53 cndrng 16654 . . . . . . . . . . . . . . 15  |-fld  e.  DivRing
5452, 1, 53drngui 15769 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
5554, 47unitsubm 15703 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  M )
)
56 eqid 2388 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( Ms  ( CC  \  { 0 } ) )
5756subsubm 14685 . . . . . . . . . . . . 13  |-  ( ( CC  \  { 0 } )  e.  (SubMnd `  M )  ->  ( RR+  e.  (SubMnd `  ( Ms  ( CC  \  { 0 } ) ) )  <-> 
( RR+  e.  (SubMnd `  M )  /\  RR+  C_  ( CC  \  { 0 } ) ) ) )
582, 55, 57mp2b 10 . . . . . . . . . . . 12  |-  ( RR+  e.  (SubMnd `  ( Ms  ( CC  \  { 0 } ) ) )  <->  ( RR+  e.  (SubMnd `  M )  /\  RR+  C_  ( CC  \  { 0 } ) ) )
5951, 58mpbi 200 . . . . . . . . . . 11  |-  ( RR+  e.  (SubMnd `  M )  /\  RR+  C_  ( CC  \  { 0 } ) )
6059simpli 445 . . . . . . . . . 10  |-  RR+  e.  (SubMnd `  M )
61 eqid 2388 . . . . . . . . . . 11  |-  ( Ms  RR+ )  =  ( Ms  RR+ )
6261submbas 14683 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  -> 
RR+  =  ( Base `  ( Ms  RR+ ) ) )
6360, 62ax-mp 8 . . . . . . . . 9  |-  RR+  =  ( Base `  ( Ms  RR+ )
)
64 cnfld1 16650 . . . . . . . . . . . 12  |-  1  =  ( 1r ` fld )
6547, 64rngidval 15594 . . . . . . . . . . 11  |-  1  =  ( 0g `  M )
6661, 65subm0 14684 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  ->  1  =  ( 0g
`  ( Ms  RR+ )
) )
6760, 66ax-mp 8 . . . . . . . . 9  |-  1  =  ( 0g `  ( Ms  RR+ ) )
68 cncrng 16646 . . . . . . . . . . 11  |-fld  e.  CRing
6947crngmgp 15600 . . . . . . . . . . 11  |-  (fld  e.  CRing  ->  M  e. CMnd )
7068, 69mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  M  e. CMnd )
7161submmnd 14682 . . . . . . . . . . 11  |-  ( RR+  e.  (SubMnd `  M )  ->  ( Ms  RR+ )  e.  Mnd )
7260, 71mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  ( Ms  RR+ )  e.  Mnd )
7361subcmn 15384 . . . . . . . . . 10  |-  ( ( M  e. CMnd  /\  ( Ms  RR+ )  e.  Mnd )  ->  ( Ms  RR+ )  e. CMnd )
7470, 72, 73syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( Ms  RR+ )  e. CMnd )
75 eqid 2388 . . . . . . . . . . . 12  |-  (flds  RR )  =  (flds  RR )
7675subrgrng 15799 . . . . . . . . . . 11  |-  ( RR  e.  (SubRing ` fld )  ->  (flds  RR )  e.  Ring )
777, 76ax-mp 8 . . . . . . . . . 10  |-  (flds  RR )  e.  Ring
78 rngmnd 15601 . . . . . . . . . 10  |-  ( (flds  RR )  e.  Ring  ->  (flds  RR )  e.  Mnd )
7977, 78mp1i 12 . . . . . . . . 9  |-  ( ph  ->  (flds  RR )  e.  Mnd )
8047oveq1i 6031 . . . . . . . . . . . 12  |-  ( Ms  RR+ )  =  ( (mulGrp ` fld )s  RR+ )
8175, 80reloggim 20361 . . . . . . . . . . 11  |-  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) GrpIso  (flds  RR ) )
82 gimghm 14979 . . . . . . . . . . 11  |-  ( ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) GrpIso  (flds  RR ) )  ->  ( log  |`  RR+ )  e.  (
( Ms  RR+ )  GrpHom  (flds  RR ) ) )
8381, 82ax-mp 8 . . . . . . . . . 10  |-  ( log  |`  RR+ )  e.  ( ( Ms  RR+ )  GrpHom  (flds  RR ) )
84 ghmmhm 14944 . . . . . . . . . 10  |-  ( ( log  |`  RR+ )  e.  ( ( Ms  RR+ )  GrpHom  (flds  RR ) )  ->  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) MndHom  (flds  RR ) ) )
8583, 84mp1i 12 . . . . . . . . 9  |-  ( ph  ->  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) MndHom  (flds  RR ) ) )
865, 10fisuppfi 14701 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {
1 } ) )  e.  Fin )
8763, 67, 74, 79, 5, 85, 10, 86gsummhm 15462 . . . . . . . 8  |-  ( ph  ->  ( (flds  RR )  gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
88 subgsubm 14890 . . . . . . . . . 10  |-  ( RR  e.  (SubGrp ` fld )  ->  RR  e.  (SubMnd ` fld ) )
899, 88syl 16 . . . . . . . . 9  |-  ( ph  ->  RR  e.  (SubMnd ` fld )
)
90 fco 5541 . . . . . . . . . 10  |-  ( ( ( log  |`  RR+ ) : RR+ --> RR  /\  F : A --> RR+ )  ->  (
( log  |`  RR+ )  o.  F ) : A --> RR )
9138, 10, 90syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( ( log  |`  RR+ )  o.  F ) : A --> RR )
925, 89, 91, 75gsumsubm 14706 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( (flds  RR )  gsumg  ( ( log  |`  RR+ )  o.  F ) ) )
9360a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
RR+  e.  (SubMnd `  M
) )
945, 93, 10, 61gsumsubm 14706 . . . . . . . . 9  |-  ( ph  ->  ( M  gsumg  F )  =  ( ( Ms  RR+ )  gsumg  F ) )
9594fveq2d 5673 . . . . . . . 8  |-  ( ph  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
9687, 92, 953eqtr4d 2430 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) ) )
9765, 70, 5, 93, 10, 86gsumsubmcl 15452 . . . . . . . 8  |-  ( ph  ->  ( M  gsumg  F )  e.  RR+ )
98 fvres 5686 . . . . . . . 8  |-  ( ( M  gsumg  F )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( log `  ( M  gsumg  F ) ) )
9997, 98syl 16 . . . . . . 7  |-  ( ph  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( log `  ( M  gsumg  F ) ) )
10046, 96, 993eqtrd 2424 . . . . . 6  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  =  ( log `  ( M  gsumg  F ) ) )
101100oveq1d 6036 . . . . 5  |-  ( ph  ->  ( -u (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( log `  ( M 
gsumg  F ) )  / 
( # `  A ) ) )
10297relogcld 20386 . . . . . . 7  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  RR )
103102recnd 9048 . . . . . 6  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  CC )
104103, 23, 24divrec2d 9727 . . . . 5  |-  ( ph  ->  ( ( log `  ( M  gsumg  F ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10525, 101, 1043eqtrd 2424 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10635oveq2d 6037 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  F )  =  (fld  gsumg  ( k  e.  A  |->  ( F `
 k ) ) ) )
10711rpcnd 10583 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  CC )
1085, 107gsumfsum 16690 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) )  = 
sum_ k  e.  A  ( F `  k ) )
109106, 108eqtrd 2420 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  F )  =  sum_ k  e.  A  ( F `  k )
)
1105, 19, 11fsumrpcl 12459 . . . . . . . 8  |-  ( ph  -> 
sum_ k  e.  A  ( F `  k )  e.  RR+ )
111109, 110eqeltrd 2462 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR+ )
11222nnrpd 10580 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  RR+ )
113111, 112rpdivcld 10598 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR+ )
114113relogcld 20386 . . . . 5  |-  ( ph  ->  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  e.  RR )
11517, 22nndivred 9981 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  RR )
116 rpssre 10555 . . . . . . . . 9  |-  RR+  C_  RR
117116a1i 11 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
118 relogcl 20341 . . . . . . . . . . 11  |-  ( w  e.  RR+  ->  ( log `  w )  e.  RR )
119118adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( log `  w )  e.  RR )
120119renegcld 9397 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  RR+ )  ->  -u ( log `  w )  e.  RR )
121 eqid 2388 . . . . . . . . 9  |-  ( w  e.  RR+  |->  -u ( log `  w ) )  =  ( w  e.  RR+  |->  -u ( log `  w
) )
122120, 121fmptd 5833 . . . . . . . 8  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) ) : RR+ --> RR )
123 ioorp 10921 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
124123eleq2i 2452 . . . . . . . . . . 11  |-  ( a  e.  ( 0 (,) 
+oo )  <->  a  e.  RR+ )
125123eleq2i 2452 . . . . . . . . . . 11  |-  ( b  e.  ( 0 (,) 
+oo )  <->  b  e.  RR+ )
126 iccssioo2 10916 . . . . . . . . . . 11  |-  ( ( a  e.  ( 0 (,)  +oo )  /\  b  e.  ( 0 (,)  +oo ) )  ->  (
a [,] b ) 
C_  ( 0 (,) 
+oo ) )
127124, 125, 126syl2anbr 467 . . . . . . . . . 10  |-  ( ( a  e.  RR+  /\  b  e.  RR+ )  ->  (
a [,] b ) 
C_  ( 0 (,) 
+oo ) )
128127, 123syl6sseq 3338 . . . . . . . . 9  |-  ( ( a  e.  RR+  /\  b  e.  RR+ )  ->  (
a [,] b ) 
C_  RR+ )
129128adantl 453 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  RR+  /\  b  e.  RR+ ) )  ->  (
a [,] b ) 
C_  RR+ )
13022nnrecred 9978 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR )
131112rpreccld 10591 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR+ )
132131rpge0d 10585 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( 1  /  ( # `  A
) ) )
133 elrege0 10940 . . . . . . . . . 10  |-  ( ( 1  /  ( # `  A ) )  e.  ( 0 [,)  +oo ) 
<->  ( ( 1  / 
( # `  A ) )  e.  RR  /\  0  <_  ( 1  / 
( # `  A ) ) ) )
134130, 132, 133sylanbrc 646 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  ( 0 [,) 
+oo ) )
135 fconst6g 5573 . . . . . . . . 9  |-  ( ( 1  /  ( # `  A ) )  e.  ( 0 [,)  +oo )  ->  ( A  X.  { ( 1  / 
( # `  A ) ) } ) : A --> ( 0 [,) 
+oo ) )
136134, 135syl 16 . . . . . . . 8  |-  ( ph  ->  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) : A --> ( 0 [,)  +oo ) )
137 0lt1 9483 . . . . . . . . 9  |-  0  <  1
138 fconstmpt 4862 . . . . . . . . . . 11  |-  ( A  X.  { ( 1  /  ( # `  A
) ) } )  =  ( k  e.  A  |->  ( 1  / 
( # `  A ) ) )
139138oveq2i 6032 . . . . . . . . . 10  |-  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )
140 rngmnd 15601 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  ->fld  e.  Mnd )
1412, 140mp1i 12 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Mnd )
142130recnd 9048 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  CC )
143 eqid 2388 . . . . . . . . . . . . 13  |-  (.g ` fld )  =  (.g ` fld )
14452, 143gsumconst 15460 . . . . . . . . . . . 12  |-  ( (fld  e. 
Mnd  /\  A  e.  Fin  /\  ( 1  / 
( # `  A ) )  e.  CC )  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  ( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) ) )
145141, 5, 142, 144syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  ( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) ) )
14622nnzd 10307 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
147 cnfldmulg 16657 . . . . . . . . . . . 12  |-  ( ( ( # `  A
)  e.  ZZ  /\  ( 1  /  ( # `
 A ) )  e.  CC )  -> 
( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) )  =  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) ) )
148146, 142, 147syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
) (.g ` fld ) ( 1  / 
( # `  A ) ) )  =  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) ) )
14923, 24recidd 9718 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) )  =  1 )
150145, 148, 1493eqtrd 2424 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  1 )
151139, 150syl5eq 2432 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) )  =  1 )
152137, 151syl5breqr 4190 . . . . . . . 8  |-  ( ph  ->  0  <  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )
153 logccv 20422 . . . . . . . . . . . 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 ) ) ) )
1541533adant1 975 . . . . . . . . . . 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 ) ) ) )
155 ioossre 10905 . . . . . . . . . . . . . . 15  |-  ( 0 (,) 1 )  C_  RR
156 simp3 959 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  ( 0 (,) 1
) )
157155, 156sseldi 3290 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  RR )
158 simp21 990 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  x  e.  RR+ )
159158relogcld 20386 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  RR )
160157, 159remulcld 9050 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  RR )
161 1re 9024 . . . . . . . . . . . . . . 15  |-  1  e.  RR
162 resubcl 9298 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR  /\  t  e.  RR )  ->  ( 1  -  t
)  e.  RR )
163161, 157, 162sylancr 645 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
1  -  t )  e.  RR )
164 simp22 991 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  y  e.  RR+ )
165164relogcld 20386 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  RR )
166163, 165remulcld 9050 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  RR )
167160, 166readdcld 9049 . . . . . . . . . . . 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 )
168 simp1 957 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ph )
169 ioossicc 10929 . . . . . . . . . . . . . . 15  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
170169, 156sseldi 3290 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  ( 0 [,] 1
) )
171117, 129cvxcl 20691 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  t  e.  ( 0 [,] 1 ) ) )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  e.  RR+ )
172168, 158, 164, 170, 171syl13anc 1186 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  e.  RR+ )
173172relogcld 20386 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  e.  RR )
174167, 173ltnegd 9537 . . . . . . . . . . 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
) ) ) ) )
175154, 174mpbid 202 . . . . . . . . . 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 ) ) ) )
176 fveq2 5669 . . . . . . . . . . . . 13  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  ( log `  w )  =  ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
177176negeqd 9233 . . . . . . . . . . . 12  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  -u ( log `  w )  = 
-u ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
178 negex 9237 . . . . . . . . . . . 12  |-  -u ( log `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  e. 
_V
179177, 121, 178fvmpt 5746 . . . . . . . . . . 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 ) ) ) )
180172, 179syl 16 . . . . . . . . . 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
) ) ) )
181 fveq2 5669 . . . . . . . . . . . . . . . . 17  |-  ( w  =  x  ->  ( log `  w )  =  ( log `  x
) )
182181negeqd 9233 . . . . . . . . . . . . . . . 16  |-  ( w  =  x  ->  -u ( log `  w )  = 
-u ( log `  x
) )
183 negex 9237 . . . . . . . . . . . . . . . 16  |-  -u ( log `  x )  e. 
_V
184182, 121, 183fvmpt 5746 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  ( ( w  e.  RR+  |->  -u ( log `  w ) ) `
 x )  = 
-u ( log `  x
) )
185158, 184syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( w  e.  RR+  |->  -u ( log `  w
) ) `  x
)  =  -u ( log `  x ) )
186185oveq2d 6037 . . . . . . . . . . . . 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
) ) )
187157recnd 9048 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  CC )
188159recnd 9048 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  CC )
189187, 188mulneg2d 9420 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  -u ( log `  x ) )  =  -u ( t  x.  ( log `  x
) ) )
190186, 189eqtrd 2420 . . . . . . . . . . . 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
) ) )
191 fveq2 5669 . . . . . . . . . . . . . . . . 17  |-  ( w  =  y  ->  ( log `  w )  =  ( log `  y
) )
192191negeqd 9233 . . . . . . . . . . . . . . . 16  |-  ( w  =  y  ->  -u ( log `  w )  = 
-u ( log `  y
) )
193 negex 9237 . . . . . . . . . . . . . . . 16  |-  -u ( log `  y )  e. 
_V
194192, 121, 193fvmpt 5746 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR+  ->  ( ( w  e.  RR+  |->  -u ( log `  w ) ) `
 y )  = 
-u ( log `  y
) )
195164, 194syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( w  e.  RR+  |->  -u ( log `  w
) ) `  y
)  =  -u ( log `  y ) )
196195oveq2d 6037 . . . . . . . . . . . . 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
) ) )
197163recnd 9048 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
1  -  t )  e.  CC )
198165recnd 9048 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  CC )
199197, 198mulneg2d 9420 . . . . . . . . . . . . 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
) ) )
200196, 199eqtrd 2420 . . . . . . . . . . . 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
) ) )
201190, 200oveq12d 6039 . . . . . . . . . . 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
) ) ) )
202160recnd 9048 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  CC )
203166recnd 9048 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  CC )
204202, 203negdid 9357 . . . . . . . . . . 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 ) ) ) )
205201, 204eqtr4d 2423 . . . . . . . . . 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
) ) ) )
206175, 180, 2053brtr4d 4184 . . . . . . . . 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
) ) ) )
207117, 122, 129, 206scvxcvx 20692 . . . . . . . 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
) ) ) )
208117, 122, 129, 5, 136, 10, 152, 207jensen 20695 . . . . . . 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 ) ) } ) ) ) ) )
209208simprd 450 . . . . . 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 ) ) } ) ) ) )
210130adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  (
1  /  ( # `  A ) )  e.  RR )
211138a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  X.  {
( 1  /  ( # `
 A ) ) } )  =  ( k  e.  A  |->  ( 1  /  ( # `  A ) ) ) )
2125, 210, 11, 211, 35offval2 6262 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F )  =  ( k  e.  A  |->  ( ( 1  /  ( # `  A
) )  x.  ( F `  k )
) ) )
213212oveq2d 6037 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  (fld  gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) ) )
214 cnfldadd 16632 . . . . . . . . . . . 12  |-  +  =  ( +g  ` fld )
215 cnfldmul 16633 . . . . . . . . . . . 12  |-  x.  =  ( .r ` fld )
2162a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Ring )
217 eqid 2388 . . . . . . . . . . . . . 14  |-  ( k  e.  A  |->  ( F `
 k ) )  =  ( k  e.  A  |->  ( F `  k ) )
218107, 217fmptd 5833 . . . . . . . . . . . . 13  |-  ( ph  ->  ( k  e.  A  |->  ( F `  k
) ) : A --> CC )
2195, 218fisuppfi 14701 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( k  e.  A  |->  ( F `
 k ) )
" ( _V  \  { 0 } ) )  e.  Fin )
22052, 1, 214, 215, 216, 5, 142, 107, 219gsummulc2 15642 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) )  =  ( ( 1  /  ( # `
 A ) )  x.  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
221 fss 5540 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> RR+  /\  RR+  C_  RR )  ->  F : A --> RR )
22210, 116, 221sylancl 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : A --> RR )
2235, 10fisuppfi 14701 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' F "
( _V  \  {
0 } ) )  e.  Fin )
2241, 4, 5, 9, 222, 223gsumsubgcl 15453 . . . . . . . . . . . . . 14  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR )
225224recnd 9048 . . . . . . . . . . . . 13  |-  ( ph  ->  (fld 
gsumg  F )  e.  CC )
226225, 23, 24divrec2d 9727 . . . . . . . . . . . 12  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( ( 1  /  ( # `  A
) )  x.  (fld  gsumg  F ) ) )
227106oveq2d 6037 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  F ) )  =  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
228226, 227eqtr2d 2421 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
229213, 220, 2283eqtrd 2424 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  ( (fld  gsumg  F )  /  ( # `  A
) ) )
230229, 151oveq12d 6039 . . . . . . . . 9  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( ( (fld  gsumg  F )  /  ( # `  A
) )  /  1
) )
231224, 22nndivred 9981 . . . . . . . . . . 11  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR )
232231recnd 9048 . . . . . . . . . 10  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  CC )
233232div1d 9715 . . . . . . . . 9  |-  ( ph  ->  ( ( (fld  gsumg  F )  /  ( # `
 A ) )  /  1 )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
234230, 233eqtrd 2420 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
235234fveq2d 5673 . . . . . . 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 ) ) ) )
236 fveq2 5669 . . . . . . . . . 10  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  ( log `  w
)  =  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
237236negeqd 9233 . . . . . . . . 9  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  -u ( log `  w
)  =  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
238 negex 9237 . . . . . . . . 9  |-  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) )  e.  _V
239237, 121, 238fvmpt 5746 . . . . . . . 8  |-  ( ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR+  ->  ( ( w  e.  RR+  |->  -u ( log `  w ) ) `
 ( (fld  gsumg  F )  /  ( # `
 A ) ) )  =  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
240113, 239syl 16 . . . . . . 7  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) ) `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  =  -u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
241235, 240eqtrd 2420 . . . . . 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 ) ) ) )
24252, 1, 214, 215, 216, 5, 142, 30, 16gsummulc2 15642 . . . . . . . . 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 )
) ) ) ) )
243 negex 9237 . . . . . . . . . . . 12  |-  -u ( log `  ( F `  k ) )  e. 
_V
244243a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e. 
_V )
245 eqidd 2389 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) )  =  ( w  e.  RR+  |->  -u ( log `  w ) ) )
246 fveq2 5669 . . . . . . . . . . . . 13  |-  ( w  =  ( F `  k )  ->  ( log `  w )  =  ( log `  ( F `  k )
) )
247246negeqd 9233 . . . . . . . . . . . 12  |-  ( w  =  ( F `  k )  ->  -u ( log `  w )  = 
-u ( log `  ( F `  k )
) )
24811, 35, 245, 247fmptco 5841 . . . . . . . . . . 11  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) )  o.  F
)  =  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) ) )
2495, 210, 244, 211, 248offval2 6262 . . . . . . . . . 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 ) ) ) ) )
250249oveq2d 6037 . . . . . . . . 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 ) ) ) ) ) )
25118, 23, 24divrec2d 9727 . . . . . . . . 9  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) ) ) )
252242, 250, 2513eqtr4d 2430 . . . . . . . 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 ) ) )
253252, 151oveq12d 6039 . . . . . . 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 ) )
254115recnd 9048 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  CC )
255254div1d 9715 . . . . . . 7  |-  ( ph  ->  ( ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  /  1 )  =  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
256253, 255eqtrd 2420 . . . . . 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 ) ) )
257209, 241, 2563brtr3d 4183 . . . . 5  |-  ( ph  -> 
-u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  <_  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
258114, 115, 257lenegcon1d 9541 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  <_  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
259105, 258eqbrtrrd 4176 . . 3  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  <_  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
260130, 102remulcld 9050 . . . 4  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  e.  RR )
261 efle 12647 . . . 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 ) ) ) ) ) )
262260, 114, 261syl2anc 643 . . 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 ) ) ) ) ) )
263259, 262mpbid 202 . 2  |-  ( ph  ->  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )  <_  ( exp `  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) ) )
26497rpcnd 10583 . . 3  |-  ( ph  ->  ( M  gsumg  F )  e.  CC )
26597rpne0d 10586 . . 3  |-  ( ph  ->  ( M  gsumg  F )  =/=  0
)
266264, 265, 142cxpefd 20471 . 2  |-  ( ph  ->  ( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  =  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) ) )
267113reeflogd 20387 . . 3  |-  ( ph  ->  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
268267eqcomd 2393 . 2  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) ) )
269263, 266, 2683brtr4d 4184 1  |-  ( ph  ->  ( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  <_ 
( (fld 
gsumg  F )  /  ( # `
 A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900    \ cdif 3261    C_ wss 3264   (/)c0 3572   {csn 3758   class class class wbr 4154    e. cmpt 4208    X. cxp 4817    |` cres 4821    o. ccom 4823   -->wf 5391   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021    o Fcof 6243   Fincfn 7046   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    +oocpnf 9051    < clt 9054    <_ cle 9055    - cmin 9224   -ucneg 9225    / cdiv 9610   NNcn 9933   ZZcz 10215   RR+crp 10545   (,)cioo 10849   [,)cico 10851   [,]cicc 10852   #chash 11546   sum_csu 12407   expce 12592   Basecbs 13397   ↾s cress 13398   0gc0g 13651    gsumg cgsu 13652   Mndcmnd 14612  .gcmg 14617   MndHom cmhm 14664  SubMndcsubmnd 14665  SubGrpcsubg 14866    GrpHom cghm 14931   GrpIso cgim 14972  CMndccmn 15340   Abelcabel 15341  mulGrpcmgp 15576   Ringcrg 15588   CRingccrg 15589   DivRingcdr 15763  SubRingcsubrg 15792  ℂfldccnfld 16627   logclog 20320    ^ c ccxp 20321
This theorem is referenced by:  amgm  20697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-mhm 14666  df-submnd 14667  df-grp 14740  df-minusg 14741  df-mulg 14743  df-subg 14869  df-ghm 14932  df-gim 14974  df-cntz 15044  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-cring 15592  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-dvr 15716  df-drng 15765  df-subrg 15794  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-cmp 17373  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322  df-cxp 20323
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