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Theorem amgmlem 20821
Description: Lemma for amgm 20822. (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 16718 . . . . . . . 8  |-  0  =  ( 0g ` fld )
2 cnrng 16716 . . . . . . . . 9  |-fld  e.  Ring
3 rngabl 15686 . . . . . . . . 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 16743 . . . . . . . . . 10  |-  ( RR  e.  (SubRing ` fld )  /\  (flds  RR )  e.  DivRing )
76simpli 445 . . . . . . . . 9  |-  RR  e.  (SubRing ` fld )
8 subrgsubg 15867 . . . . . . . . 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 5863 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  RR+ )
1211relogcld 20511 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  RR )
1312renegcld 9457 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  RR )
14 eqid 2436 . . . . . . . . 9  |-  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )  =  ( k  e.  A  |->  -u ( log `  ( F `  k )
) )
1513, 14fmptd 5886 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) : A --> RR )
165, 15fisuppfi 14766 . . . . . . . 8  |-  ( ph  ->  ( `' ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )
" ( _V  \  { 0 } ) )  e.  Fin )
171, 4, 5, 9, 15, 16gsumsubgcl 15518 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  RR )
1817recnd 9107 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  CC )
19 amgm.3 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
20 hashnncl 11638 . . . . . . . . 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 10009 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  CC )
2422nnne0d 10037 . . . . . 6  |-  ( ph  ->  ( # `  A
)  =/=  0 )
2518, 23, 24divnegd 9796 . . . . 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 9107 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  CC )
275, 26gsumfsum 16759 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  ( log `  ( F `
 k ) ) )
2826negnegd 9395 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u -u ( log `  ( F `  k ) )  =  ( log `  ( F `  k )
) )
2928sumeq2dv 12490 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  sum_ k  e.  A  ( log `  ( F `  k
) ) )
3013recnd 9107 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  CC )
315, 30fsumneg 12563 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  -u sum_ k  e.  A  -u ( log `  ( F `  k
) ) )
3227, 29, 313eqtr2rd 2475 . . . . . . . 8  |-  ( ph  -> 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
)  =  (fld  gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
335, 30gsumfsum 16759 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3433negeqd 9293 . . . . . . . 8  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3510feqmptd 5772 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
36 relogf1o 20457 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
37 f1of 5667 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
3836, 37mp1i 12 . . . . . . . . . . . 12  |-  ( ph  ->  ( log  |`  RR+ ) : RR+ --> RR )
3938feqmptd 5772 . . . . . . . . . . 11  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x
) ) )
40 fvres 5738 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
4140mpteq2ia 4284 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
4239, 41syl6eq 2484 . . . . . . . . . 10  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x
) ) )
43 fveq2 5721 . . . . . . . . . 10  |-  ( x  =  ( F `  k )  ->  ( log `  x )  =  ( log `  ( F `  k )
) )
4411, 35, 42, 43fmptco 5894 . . . . . . . . 9  |-  ( ph  ->  ( ( log  |`  RR+ )  o.  F )  =  ( k  e.  A  |->  ( log `  ( F `
 k ) ) ) )
4544oveq2d 6090 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
4632, 34, 453eqtr4d 2478 . . . . . . 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 6084 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
4948rpmsubg 16755 . . . . . . . . . . . . 13  |-  RR+  e.  (SubGrp `  ( Ms  ( CC 
\  { 0 } ) ) )
50 subgsubm 14955 . . . . . . . . . . . . 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 16700 . . . . . . . . . . . . . . 15  |-  CC  =  ( Base ` fld )
53 cndrng 16723 . . . . . . . . . . . . . . 15  |-fld  e.  DivRing
5452, 1, 53drngui 15834 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
5554, 47unitsubm 15768 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  M )
)
56 eqid 2436 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( Ms  ( CC  \  { 0 } ) )
5756subsubm 14750 . . . . . . . . . . . . 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 2436 . . . . . . . . . . 11  |-  ( Ms  RR+ )  =  ( Ms  RR+ )
6261submbas 14748 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  -> 
RR+  =  ( Base `  ( Ms  RR+ ) ) )
6360, 62ax-mp 8 . . . . . . . . 9  |-  RR+  =  ( Base `  ( Ms  RR+ )
)
64 cnfld1 16719 . . . . . . . . . . . 12  |-  1  =  ( 1r ` fld )
6547, 64rngidval 15659 . . . . . . . . . . 11  |-  1  =  ( 0g `  M )
6661, 65subm0 14749 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  ->  1  =  ( 0g
`  ( Ms  RR+ )
) )
6760, 66ax-mp 8 . . . . . . . . 9  |-  1  =  ( 0g `  ( Ms  RR+ ) )
68 cncrng 16715 . . . . . . . . . . 11  |-fld  e.  CRing
6947crngmgp 15665 . . . . . . . . . . 11  |-  (fld  e.  CRing  ->  M  e. CMnd )
7068, 69mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  M  e. CMnd )
7161submmnd 14747 . . . . . . . . . . 11  |-  ( RR+  e.  (SubMnd `  M )  ->  ( Ms  RR+ )  e.  Mnd )
7260, 71mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  ( Ms  RR+ )  e.  Mnd )
7361subcmn 15449 . . . . . . . . . 10  |-  ( ( M  e. CMnd  /\  ( Ms  RR+ )  e.  Mnd )  ->  ( Ms  RR+ )  e. CMnd )
7470, 72, 73syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( Ms  RR+ )  e. CMnd )
75 eqid 2436 . . . . . . . . . . . 12  |-  (flds  RR )  =  (flds  RR )
7675subrgrng 15864 . . . . . . . . . . 11  |-  ( RR  e.  (SubRing ` fld )  ->  (flds  RR )  e.  Ring )
777, 76ax-mp 8 . . . . . . . . . 10  |-  (flds  RR )  e.  Ring
78 rngmnd 15666 . . . . . . . . . 10  |-  ( (flds  RR )  e.  Ring  ->  (flds  RR )  e.  Mnd )
7977, 78mp1i 12 . . . . . . . . 9  |-  ( ph  ->  (flds  RR )  e.  Mnd )
8047oveq1i 6084 . . . . . . . . . . . 12  |-  ( Ms  RR+ )  =  ( (mulGrp ` fld )s  RR+ )
8175, 80reloggim 20486 . . . . . . . . . . 11  |-  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) GrpIso  (flds  RR ) )
82 gimghm 15044 . . . . . . . . . . 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 15009 . . . . . . . . . 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 14766 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {
1 } ) )  e.  Fin )
8763, 67, 74, 79, 5, 85, 10, 86gsummhm 15527 . . . . . . . 8  |-  ( ph  ->  ( (flds  RR )  gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
88 subgsubm 14955 . . . . . . . . . 10  |-  ( RR  e.  (SubGrp ` fld )  ->  RR  e.  (SubMnd ` fld ) )
899, 88syl 16 . . . . . . . . 9  |-  ( ph  ->  RR  e.  (SubMnd ` fld )
)
90 fco 5593 . . . . . . . . . 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 14771 . . . . . . . 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 14771 . . . . . . . . 9  |-  ( ph  ->  ( M  gsumg  F )  =  ( ( Ms  RR+ )  gsumg  F ) )
9594fveq2d 5725 . . . . . . . 8  |-  ( ph  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
9687, 92, 953eqtr4d 2478 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) ) )
9765, 70, 5, 93, 10, 86gsumsubmcl 15517 . . . . . . . 8  |-  ( ph  ->  ( M  gsumg  F )  e.  RR+ )
98 fvres 5738 . . . . . . . 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 2472 . . . . . 6  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  =  ( log `  ( M  gsumg  F ) ) )
101100oveq1d 6089 . . . . 5  |-  ( ph  ->  ( -u (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( log `  ( M 
gsumg  F ) )  / 
( # `  A ) ) )
10297relogcld 20511 . . . . . . 7  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  RR )
103102recnd 9107 . . . . . 6  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  CC )
104103, 23, 24divrec2d 9787 . . . . 5  |-  ( ph  ->  ( ( log `  ( M  gsumg  F ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10525, 101, 1043eqtrd 2472 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10635oveq2d 6090 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  F )  =  (fld  gsumg  ( k  e.  A  |->  ( F `
 k ) ) ) )
10711rpcnd 10643 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  CC )
1085, 107gsumfsum 16759 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) )  = 
sum_ k  e.  A  ( F `  k ) )
109106, 108eqtrd 2468 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  F )  =  sum_ k  e.  A  ( F `  k )
)
1105, 19, 11fsumrpcl 12524 . . . . . . . 8  |-  ( ph  -> 
sum_ k  e.  A  ( F `  k )  e.  RR+ )
111109, 110eqeltrd 2510 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR+ )
11222nnrpd 10640 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  RR+ )
113111, 112rpdivcld 10658 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR+ )
114113relogcld 20511 . . . . 5  |-  ( ph  ->  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  e.  RR )
11517, 22nndivred 10041 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  RR )
116 rpssre 10615 . . . . . . . . 9  |-  RR+  C_  RR
117116a1i 11 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
118 relogcl 20466 . . . . . . . . . . 11  |-  ( w  e.  RR+  ->  ( log `  w )  e.  RR )
119118adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( log `  w )  e.  RR )
120119renegcld 9457 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  RR+ )  ->  -u ( log `  w )  e.  RR )
121 eqid 2436 . . . . . . . . 9  |-  ( w  e.  RR+  |->  -u ( log `  w ) )  =  ( w  e.  RR+  |->  -u ( log `  w
) )
122120, 121fmptd 5886 . . . . . . . 8  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) ) : RR+ --> RR )
123 ioorp 10981 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
124123eleq2i 2500 . . . . . . . . . . 11  |-  ( a  e.  ( 0 (,) 
+oo )  <->  a  e.  RR+ )
125123eleq2i 2500 . . . . . . . . . . 11  |-  ( b  e.  ( 0 (,) 
+oo )  <->  b  e.  RR+ )
126 iccssioo2 10976 . . . . . . . . . . 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 3387 . . . . . . . . 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 10038 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR )
131112rpreccld 10651 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR+ )
132131rpge0d 10645 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( 1  /  ( # `  A
) ) )
133 elrege0 11000 . . . . . . . . . 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 5625 . . . . . . . . 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 9543 . . . . . . . . 9  |-  0  <  1
138 fconstmpt 4914 . . . . . . . . . . 11  |-  ( A  X.  { ( 1  /  ( # `  A
) ) } )  =  ( k  e.  A  |->  ( 1  / 
( # `  A ) ) )
139138oveq2i 6085 . . . . . . . . . 10  |-  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )
140 rngmnd 15666 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  ->fld  e.  Mnd )
1412, 140mp1i 12 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Mnd )
142130recnd 9107 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  CC )
143 eqid 2436 . . . . . . . . . . . . 13  |-  (.g ` fld )  =  (.g ` fld )
14452, 143gsumconst 15525 . . . . . . . . . . . 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 10367 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
147 cnfldmulg 16726 . . . . . . . . . . . 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 9778 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) )  =  1 )
150145, 148, 1493eqtrd 2472 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  1 )
151139, 150syl5eq 2480 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) )  =  1 )
152137, 151syl5breqr 4241 . . . . . . . 8  |-  ( ph  ->  0  <  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )
153 logccv 20547 . . . . . . . . . . . 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 10965 . . . . . . . . . . . . . . 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 3339 . . . . . . . . . . . . . 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 20511 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  RR )
160157, 159remulcld 9109 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  RR )
161 1re 9083 . . . . . . . . . . . . . . 15  |-  1  e.  RR
162 resubcl 9358 . . . . . . . . . . . . . . 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 20511 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  RR )
166163, 165remulcld 9109 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  RR )
167160, 166readdcld 9108 . . . . . . . . . . . 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 10989 . . . . . . . . . . . . . . 15  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
170169, 156sseldi 3339 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  ( 0 [,] 1
) )
171117, 129cvxcl 20816 . . . . . . . . . . . . . 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 20511 . . . . . . . . . . . 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 9597 . . . . . . . . . . 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 5721 . . . . . . . . . . . . 13  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  ( log `  w )  =  ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
177176negeqd 9293 . . . . . . . . . . . 12  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  -u ( log `  w )  = 
-u ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
178 negex 9297 . . . . . . . . . . . 12  |-  -u ( log `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  e. 
_V
179177, 121, 178fvmpt 5799 . . . . . . . . . . 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 5721 . . . . . . . . . . . . . . . . 17  |-  ( w  =  x  ->  ( log `  w )  =  ( log `  x
) )
182181negeqd 9293 . . . . . . . . . . . . . . . 16  |-  ( w  =  x  ->  -u ( log `  w )  = 
-u ( log `  x
) )
183 negex 9297 . . . . . . . . . . . . . . . 16  |-  -u ( log `  x )  e. 
_V
184182, 121, 183fvmpt 5799 . . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . 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 9107 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  CC )
188159recnd 9107 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  CC )
189187, 188mulneg2d 9480 . . . . . . . . . . . . 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 2468 . . . . . . . . . . . 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 5721 . . . . . . . . . . . . . . . . 17  |-  ( w  =  y  ->  ( log `  w )  =  ( log `  y
) )
192191negeqd 9293 . . . . . . . . . . . . . . . 16  |-  ( w  =  y  ->  -u ( log `  w )  = 
-u ( log `  y
) )
193 negex 9297 . . . . . . . . . . . . . . . 16  |-  -u ( log `  y )  e. 
_V
194192, 121, 193fvmpt 5799 . . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . 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 9107 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
1  -  t )  e.  CC )
198165recnd 9107 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  CC )
199197, 198mulneg2d 9480 . . . . . . . . . . . . 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 2468 . . . . . . . . . . . 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 6092 . . . . . . . . . . 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 9107 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  CC )
203166recnd 9107 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  CC )
204202, 203negdid 9417 . . . . . . . . . . 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 2471 . . . . . . . . . 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 4235 . . . . . . . . 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 20817 . . . . . . . 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 20820 . . . . . . 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 6315 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F )  =  ( k  e.  A  |->  ( ( 1  /  ( # `  A
) )  x.  ( F `  k )
) ) )
213212oveq2d 6090 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  (fld  gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) ) )
214 cnfldadd 16701 . . . . . . . . . . . 12  |-  +  =  ( +g  ` fld )
215 cnfldmul 16702 . . . . . . . . . . . 12  |-  x.  =  ( .r ` fld )
2162a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Ring )
217 eqid 2436 . . . . . . . . . . . . . 14  |-  ( k  e.  A  |->  ( F `
 k ) )  =  ( k  e.  A  |->  ( F `  k ) )
218107, 217fmptd 5886 . . . . . . . . . . . . 13  |-  ( ph  ->  ( k  e.  A  |->  ( F `  k
) ) : A --> CC )
2195, 218fisuppfi 14766 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( k  e.  A  |->  ( F `
 k ) )
" ( _V  \  { 0 } ) )  e.  Fin )
22052, 1, 214, 215, 216, 5, 142, 107, 219gsummulc2 15707 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) )  =  ( ( 1  /  ( # `
 A ) )  x.  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
221 fss 5592 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> RR+  /\  RR+  C_  RR )  ->  F : A --> RR )
22210, 116, 221sylancl 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : A --> RR )
2235, 10fisuppfi 14766 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' F "
( _V  \  {
0 } ) )  e.  Fin )
2241, 4, 5, 9, 222, 223gsumsubgcl 15518 . . . . . . . . . . . . . 14  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR )
225224recnd 9107 . . . . . . . . . . . . 13  |-  ( ph  ->  (fld 
gsumg  F )  e.  CC )
226225, 23, 24divrec2d 9787 . . . . . . . . . . . 12  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( ( 1  /  ( # `  A
) )  x.  (fld  gsumg  F ) ) )
227106oveq2d 6090 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  F ) )  =  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
228226, 227eqtr2d 2469 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
229213, 220, 2283eqtrd 2472 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  ( (fld  gsumg  F )  /  ( # `  A
) ) )
230229, 151oveq12d 6092 . . . . . . . . 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 10041 . . . . . . . . . . 11  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR )
232231recnd 9107 . . . . . . . . . 10  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  CC )
233232div1d 9775 . . . . . . . . 9  |-  ( ph  ->  ( ( (fld  gsumg  F )  /  ( # `
 A ) )  /  1 )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
234230, 233eqtrd 2468 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
235234fveq2d 5725 . . . . . . 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 5721 . . . . . . . . . 10  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  ( log `  w
)  =  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
237236negeqd 9293 . . . . . . . . 9  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  -u ( log `  w
)  =  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
238 negex 9297 . . . . . . . . 9  |-  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) )  e.  _V
239237, 121, 238fvmpt 5799 . . . . . . . 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 2468 . . . . . 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 15707 . . . . . . . . 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 9297 . . . . . . . . . . . 12  |-  -u ( log `  ( F `  k ) )  e. 
_V
244243a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e. 
_V )
245 eqidd 2437 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) )  =  ( w  e.  RR+  |->  -u ( log `  w ) ) )
246 fveq2 5721 . . . . . . . . . . . . 13  |-  ( w  =  ( F `  k )  ->  ( log `  w )  =  ( log `  ( F `  k )
) )
247246negeqd 9293 . . . . . . . . . . . 12  |-  ( w  =  ( F `  k )  ->  -u ( log `  w )  = 
-u ( log `  ( F `  k )
) )
24811, 35, 245, 247fmptco 5894 . . . . . . . . . . 11  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) )  o.  F
)  =  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) ) )
2495, 210, 244, 211, 248offval2 6315 . . . . . . . . . 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 6090 . . . . . . . . 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 9787 . . . . . . . . 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 2478 . . . . . . . 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 6092 . . . . . . 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 9107 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  CC )
255254div1d 9775 . . . . . . 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 2468 . . . . . 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 4234 . . . . 5  |-  ( ph  -> 
-u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  <_  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
258114, 115, 257lenegcon1d 9601 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  <_  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
259105, 258eqbrtrrd 4227 . . 3  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  <_  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
260130, 102remulcld 9109 . . . 4  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  e.  RR )
261 efle 12712 . . . 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 10643 . . 3  |-  ( ph  ->  ( M  gsumg  F )  e.  CC )
26597rpne0d 10646 . . 3  |-  ( ph  ->  ( M  gsumg  F )  =/=  0
)
266264, 265, 142cxpefd 20596 . 2  |-  ( ph  ->  ( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  =  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) ) )
267113reeflogd 20512 . . 3  |-  ( ph  ->  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
268267eqcomd 2441 . 2  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) ) )
269263, 266, 2683brtr4d 4235 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 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2949    \ cdif 3310    C_ wss 3313   (/)c0 3621   {csn 3807   class class class wbr 4205    e. cmpt 4259    X. cxp 4869    |` cres 4873    o. ccom 4875   -->wf 5443   -1-1-onto->wf1o 5446   ` cfv 5447  (class class class)co 6074    o Fcof 6296   Fincfn 7102   CCcc 8981   RRcr 8982   0cc0 8983   1c1 8984    + caddc 8986    x. cmul 8988    +oocpnf 9110    < clt 9113    <_ cle 9114    - cmin 9284   -ucneg 9285    / cdiv 9670   NNcn 9993   ZZcz 10275   RR+crp 10605   (,)cioo 10909   [,)cico 10911   [,]cicc 10912   #chash 11611   sum_csu 12472   expce 12657   Basecbs 13462   ↾s cress 13463   0gc0g 13716    gsumg cgsu 13717   Mndcmnd 14677  .gcmg 14682   MndHom cmhm 14729  SubMndcsubmnd 14730  SubGrpcsubg 14931    GrpHom cghm 14996   GrpIso cgim 15037  CMndccmn 15405   Abelcabel 15406  mulGrpcmgp 15641   Ringcrg 15653   CRingccrg 15654   DivRingcdr 15828  SubRingcsubrg 15857  ℂfldccnfld 16696   logclog 20445    ^ c ccxp 20446
This theorem is referenced by:  amgm  20822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061  ax-addf 9062  ax-mulf 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-1st 6342  df-2nd 6343  df-tpos 6472  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-er 6898  df-map 7013  df-pm 7014  df-ixp 7057  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-fi 7409  df-sup 7439  df-oi 7472  df-card 7819  df-cda 8041  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-q 10568  df-rp 10606  df-xneg 10703  df-xadd 10704  df-xmul 10705  df-ioo 10913  df-ioc 10914  df-ico 10915  df-icc 10916  df-fz 11037  df-fzo 11129  df-fl 11195  df-mod 11244  df-seq 11317  df-exp 11376  df-fac 11560  df-bc 11587  df-hash 11612  df-shft 11875  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-limsup 12258  df-clim 12275  df-rlim 12276  df-sum 12473  df-ef 12663  df-sin 12665  df-cos 12666  df-pi 12668  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-starv 13537  df-sca 13538  df-vsca 13539  df-tset 13541  df-ple 13542  df-ds 13544  df-unif 13545  df-hom 13546  df-cco 13547  df-rest 13643  df-topn 13644  df-topgen 13660  df-pt 13661  df-prds 13664  df-xrs 13719  df-0g 13720  df-gsum 13721  df-qtop 13726  df-imas 13727  df-xps 13729  df-mre 13804  df-mrc 13805  df-acs 13807  df-mnd 14683  df-mhm 14731  df-submnd 14732  df-grp 14805  df-minusg 14806  df-mulg 14808  df-subg 14934  df-ghm 14997  df-gim 15039  df-cntz 15109  df-cmn 15407  df-abl 15408  df-mgp 15642  df-rng 15656  df-cring 15657  df-ur 15658  df-oppr 15721  df-dvdsr 15739  df-unit 15740  df-invr 15770  df-dvr 15781  df-drng 15830  df-subrg 15859  df-psmet 16687  df-xmet 16688  df-met 16689  df-bl 16690  df-mopn 16691  df-fbas 16692  df-fg 16693  df-cnfld 16697  df-top 16956  df-bases 16958  df-topon 16959  df-topsp 16960  df-cld 17076  df-ntr 17077  df-cls 17078  df-nei 17155  df-lp 17193  df-perf 17194  df-cn 17284  df-cnp 17285  df-haus 17372  df-cmp 17443  df-tx 17587  df-hmeo 17780  df-fil 17871  df-fm 17963  df-flim 17964  df-flf 17965  df-xms 18343  df-ms 18344  df-tms 18345  df-cncf 18901  df-limc 19746  df-dv 19747  df-log 20447  df-cxp 20448
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