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Theorem amgmlem 20781
Description: Lemma for amgm 20782. (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 16680 . . . . . . . 8  |-  0  =  ( 0g ` fld )
2 cnrng 16678 . . . . . . . . 9  |-fld  e.  Ring
3 rngabl 15648 . . . . . . . . 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 16705 . . . . . . . . . 10  |-  ( RR  e.  (SubRing ` fld )  /\  (flds  RR )  e.  DivRing )
76simpli 445 . . . . . . . . 9  |-  RR  e.  (SubRing ` fld )
8 subrgsubg 15829 . . . . . . . . 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 5829 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  RR+ )
1211relogcld 20471 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  RR )
1312renegcld 9420 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  RR )
14 eqid 2404 . . . . . . . . 9  |-  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )  =  ( k  e.  A  |->  -u ( log `  ( F `  k )
) )
1513, 14fmptd 5852 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) : A --> RR )
165, 15fisuppfi 14728 . . . . . . . 8  |-  ( ph  ->  ( `' ( k  e.  A  |->  -u ( log `  ( F `  k ) ) )
" ( _V  \  { 0 } ) )  e.  Fin )
171, 4, 5, 9, 15, 16gsumsubgcl 15480 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  RR )
1817recnd 9070 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  e.  CC )
19 amgm.3 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
20 hashnncl 11600 . . . . . . . . 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 9972 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  CC )
2422nnne0d 10000 . . . . . 6  |-  ( ph  ->  ( # `  A
)  =/=  0 )
2518, 23, 24divnegd 9759 . . . . 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 9070 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( log `  ( F `  k ) )  e.  CC )
275, 26gsumfsum 16721 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  ( log `  ( F `
 k ) ) )
2826negnegd 9358 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u -u ( log `  ( F `  k ) )  =  ( log `  ( F `  k )
) )
2928sumeq2dv 12452 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  sum_ k  e.  A  ( log `  ( F `  k
) ) )
3013recnd 9070 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e.  CC )
315, 30fsumneg 12525 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  A  -u -u ( log `  ( F `  k )
)  =  -u sum_ k  e.  A  -u ( log `  ( F `  k
) ) )
3227, 29, 313eqtr2rd 2443 . . . . . . . 8  |-  ( ph  -> 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
)  =  (fld  gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
335, 30gsumfsum 16721 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3433negeqd 9256 . . . . . . . 8  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  = 
-u sum_ k  e.  A  -u ( log `  ( F `  k )
) )
3510feqmptd 5738 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
36 relogf1o 20417 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
37 f1of 5633 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
3836, 37mp1i 12 . . . . . . . . . . . 12  |-  ( ph  ->  ( log  |`  RR+ ) : RR+ --> RR )
3938feqmptd 5738 . . . . . . . . . . 11  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x
) ) )
40 fvres 5704 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
4140mpteq2ia 4251 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
4239, 41syl6eq 2452 . . . . . . . . . 10  |-  ( ph  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x
) ) )
43 fveq2 5687 . . . . . . . . . 10  |-  ( x  =  ( F `  k )  ->  ( log `  x )  =  ( log `  ( F `  k )
) )
4411, 35, 42, 43fmptco 5860 . . . . . . . . 9  |-  ( ph  ->  ( ( log  |`  RR+ )  o.  F )  =  ( k  e.  A  |->  ( log `  ( F `
 k ) ) ) )
4544oveq2d 6056 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( log `  ( F `  k )
) ) ) )
4632, 34, 453eqtr4d 2446 . . . . . . 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 6050 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
4948rpmsubg 16717 . . . . . . . . . . . . 13  |-  RR+  e.  (SubGrp `  ( Ms  ( CC 
\  { 0 } ) ) )
50 subgsubm 14917 . . . . . . . . . . . . 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 16662 . . . . . . . . . . . . . . 15  |-  CC  =  ( Base ` fld )
53 cndrng 16685 . . . . . . . . . . . . . . 15  |-fld  e.  DivRing
5452, 1, 53drngui 15796 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
5554, 47unitsubm 15730 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  M )
)
56 eqid 2404 . . . . . . . . . . . . . 14  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( Ms  ( CC  \  { 0 } ) )
5756subsubm 14712 . . . . . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  ( Ms  RR+ )  =  ( Ms  RR+ )
6261submbas 14710 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  -> 
RR+  =  ( Base `  ( Ms  RR+ ) ) )
6360, 62ax-mp 8 . . . . . . . . 9  |-  RR+  =  ( Base `  ( Ms  RR+ )
)
64 cnfld1 16681 . . . . . . . . . . . 12  |-  1  =  ( 1r ` fld )
6547, 64rngidval 15621 . . . . . . . . . . 11  |-  1  =  ( 0g `  M )
6661, 65subm0 14711 . . . . . . . . . 10  |-  ( RR+  e.  (SubMnd `  M )  ->  1  =  ( 0g
`  ( Ms  RR+ )
) )
6760, 66ax-mp 8 . . . . . . . . 9  |-  1  =  ( 0g `  ( Ms  RR+ ) )
68 cncrng 16677 . . . . . . . . . . 11  |-fld  e.  CRing
6947crngmgp 15627 . . . . . . . . . . 11  |-  (fld  e.  CRing  ->  M  e. CMnd )
7068, 69mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  M  e. CMnd )
7161submmnd 14709 . . . . . . . . . . 11  |-  ( RR+  e.  (SubMnd `  M )  ->  ( Ms  RR+ )  e.  Mnd )
7260, 71mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  ( Ms  RR+ )  e.  Mnd )
7361subcmn 15411 . . . . . . . . . 10  |-  ( ( M  e. CMnd  /\  ( Ms  RR+ )  e.  Mnd )  ->  ( Ms  RR+ )  e. CMnd )
7470, 72, 73syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( Ms  RR+ )  e. CMnd )
75 eqid 2404 . . . . . . . . . . . 12  |-  (flds  RR )  =  (flds  RR )
7675subrgrng 15826 . . . . . . . . . . 11  |-  ( RR  e.  (SubRing ` fld )  ->  (flds  RR )  e.  Ring )
777, 76ax-mp 8 . . . . . . . . . 10  |-  (flds  RR )  e.  Ring
78 rngmnd 15628 . . . . . . . . . 10  |-  ( (flds  RR )  e.  Ring  ->  (flds  RR )  e.  Mnd )
7977, 78mp1i 12 . . . . . . . . 9  |-  ( ph  ->  (flds  RR )  e.  Mnd )
8047oveq1i 6050 . . . . . . . . . . . 12  |-  ( Ms  RR+ )  =  ( (mulGrp ` fld )s  RR+ )
8175, 80reloggim 20446 . . . . . . . . . . 11  |-  ( log  |`  RR+ )  e.  ( ( Ms  RR+ ) GrpIso  (flds  RR ) )
82 gimghm 15006 . . . . . . . . . . 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 14971 . . . . . . . . . 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 14728 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {
1 } ) )  e.  Fin )
8763, 67, 74, 79, 5, 85, 10, 86gsummhm 15489 . . . . . . . 8  |-  ( ph  ->  ( (flds  RR )  gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
88 subgsubm 14917 . . . . . . . . . 10  |-  ( RR  e.  (SubGrp ` fld )  ->  RR  e.  (SubMnd ` fld ) )
899, 88syl 16 . . . . . . . . 9  |-  ( ph  ->  RR  e.  (SubMnd ` fld )
)
90 fco 5559 . . . . . . . . . 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 14733 . . . . . . . 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 14733 . . . . . . . . 9  |-  ( ph  ->  ( M  gsumg  F )  =  ( ( Ms  RR+ )  gsumg  F ) )
9594fveq2d 5691 . . . . . . . 8  |-  ( ph  ->  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) )  =  ( ( log  |`  RR+ ) `  ( ( Ms  RR+ )  gsumg  F ) ) )
9687, 92, 953eqtr4d 2446 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( log  |`  RR+ )  o.  F ) )  =  ( ( log  |`  RR+ ) `  ( M  gsumg  F ) ) )
9765, 70, 5, 93, 10, 86gsumsubmcl 15479 . . . . . . . 8  |-  ( ph  ->  ( M  gsumg  F )  e.  RR+ )
98 fvres 5704 . . . . . . . 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 2440 . . . . . 6  |-  ( ph  -> 
-u (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  =  ( log `  ( M  gsumg  F ) ) )
101100oveq1d 6055 . . . . 5  |-  ( ph  ->  ( -u (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( log `  ( M 
gsumg  F ) )  / 
( # `  A ) ) )
10297relogcld 20471 . . . . . . 7  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  RR )
103102recnd 9070 . . . . . 6  |-  ( ph  ->  ( log `  ( M  gsumg  F ) )  e.  CC )
104103, 23, 24divrec2d 9750 . . . . 5  |-  ( ph  ->  ( ( log `  ( M  gsumg  F ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10525, 101, 1043eqtrd 2440 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  =  ( ( 1  /  ( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) ) )
10635oveq2d 6056 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  F )  =  (fld  gsumg  ( k  e.  A  |->  ( F `
 k ) ) ) )
10711rpcnd 10606 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  CC )
1085, 107gsumfsum 16721 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) )  = 
sum_ k  e.  A  ( F `  k ) )
109106, 108eqtrd 2436 . . . . . . . 8  |-  ( ph  ->  (fld 
gsumg  F )  =  sum_ k  e.  A  ( F `  k )
)
1105, 19, 11fsumrpcl 12486 . . . . . . . 8  |-  ( ph  -> 
sum_ k  e.  A  ( F `  k )  e.  RR+ )
111109, 110eqeltrd 2478 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR+ )
11222nnrpd 10603 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  RR+ )
113111, 112rpdivcld 10621 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR+ )
114113relogcld 20471 . . . . 5  |-  ( ph  ->  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  e.  RR )
11517, 22nndivred 10004 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  RR )
116 rpssre 10578 . . . . . . . . 9  |-  RR+  C_  RR
117116a1i 11 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
118 relogcl 20426 . . . . . . . . . . 11  |-  ( w  e.  RR+  ->  ( log `  w )  e.  RR )
119118adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  RR+ )  ->  ( log `  w )  e.  RR )
120119renegcld 9420 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  RR+ )  ->  -u ( log `  w )  e.  RR )
121 eqid 2404 . . . . . . . . 9  |-  ( w  e.  RR+  |->  -u ( log `  w ) )  =  ( w  e.  RR+  |->  -u ( log `  w
) )
122120, 121fmptd 5852 . . . . . . . 8  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) ) : RR+ --> RR )
123 ioorp 10944 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
124123eleq2i 2468 . . . . . . . . . . 11  |-  ( a  e.  ( 0 (,) 
+oo )  <->  a  e.  RR+ )
125123eleq2i 2468 . . . . . . . . . . 11  |-  ( b  e.  ( 0 (,) 
+oo )  <->  b  e.  RR+ )
126 iccssioo2 10939 . . . . . . . . . . 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 3354 . . . . . . . . 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 10001 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR )
131112rpreccld 10614 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  RR+ )
132131rpge0d 10608 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( 1  /  ( # `  A
) ) )
133 elrege0 10963 . . . . . . . . . 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 5591 . . . . . . . . 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 9506 . . . . . . . . 9  |-  0  <  1
138 fconstmpt 4880 . . . . . . . . . . 11  |-  ( A  X.  { ( 1  /  ( # `  A
) ) } )  =  ( k  e.  A  |->  ( 1  / 
( # `  A ) ) )
139138oveq2i 6051 . . . . . . . . . 10  |-  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) )  =  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )
140 rngmnd 15628 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  ->fld  e.  Mnd )
1412, 140mp1i 12 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Mnd )
142130recnd 9070 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  ( # `
 A ) )  e.  CC )
143 eqid 2404 . . . . . . . . . . . . 13  |-  (.g ` fld )  =  (.g ` fld )
14452, 143gsumconst 15487 . . . . . . . . . . . 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 10330 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
147 cnfldmulg 16688 . . . . . . . . . . . 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 9741 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  x.  ( 1  /  ( # `  A
) ) )  =  1 )
150145, 148, 1493eqtrd 2440 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( 1  /  ( # `
 A ) ) ) )  =  1 )
151139, 150syl5eq 2448 . . . . . . . . 9  |-  ( ph  ->  (fld 
gsumg  ( A  X.  { ( 1  /  ( # `  A ) ) } ) )  =  1 )
152137, 151syl5breqr 4208 . . . . . . . 8  |-  ( ph  ->  0  <  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )
153 logccv 20507 . . . . . . . . . . . 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 10928 . . . . . . . . . . . . . . 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 3306 . . . . . . . . . . . . . 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 20471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  RR )
160157, 159remulcld 9072 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  RR )
161 1re 9046 . . . . . . . . . . . . . . 15  |-  1  e.  RR
162 resubcl 9321 . . . . . . . . . . . . . . 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 20471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  RR )
166163, 165remulcld 9072 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  RR )
167160, 166readdcld 9071 . . . . . . . . . . . 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 10952 . . . . . . . . . . . . . . 15  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
170169, 156sseldi 3306 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  ( 0 [,] 1
) )
171117, 129cvxcl 20776 . . . . . . . . . . . . . 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 20471 . . . . . . . . . . . 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 9560 . . . . . . . . . . 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 5687 . . . . . . . . . . . . 13  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  ( log `  w )  =  ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
177176negeqd 9256 . . . . . . . . . . . 12  |-  ( w  =  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  ->  -u ( log `  w )  = 
-u ( log `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) ) )
178 negex 9260 . . . . . . . . . . . 12  |-  -u ( log `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  e. 
_V
179177, 121, 178fvmpt 5765 . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . 17  |-  ( w  =  x  ->  ( log `  w )  =  ( log `  x
) )
182181negeqd 9256 . . . . . . . . . . . . . . . 16  |-  ( w  =  x  ->  -u ( log `  w )  = 
-u ( log `  x
) )
183 negex 9260 . . . . . . . . . . . . . . . 16  |-  -u ( log `  x )  e. 
_V
184182, 121, 183fvmpt 5765 . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . 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 9070 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  t  e.  CC )
188159recnd 9070 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  x )  e.  CC )
189187, 188mulneg2d 9443 . . . . . . . . . . . . 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 2436 . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . 17  |-  ( w  =  y  ->  ( log `  w )  =  ( log `  y
) )
192191negeqd 9256 . . . . . . . . . . . . . . . 16  |-  ( w  =  y  ->  -u ( log `  w )  = 
-u ( log `  y
) )
193 negex 9260 . . . . . . . . . . . . . . . 16  |-  -u ( log `  y )  e. 
_V
194192, 121, 193fvmpt 5765 . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . 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 9070 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
1  -  t )  e.  CC )
198165recnd 9070 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  ( log `  y )  e.  CC )
199197, 198mulneg2d 9443 . . . . . . . . . . . . 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 2436 . . . . . . . . . . . 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 6058 . . . . . . . . . . 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 9070 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
t  x.  ( log `  x ) )  e.  CC )
203166recnd 9070 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  RR+  /\  y  e.  RR+  /\  x  <  y
)  /\  t  e.  ( 0 (,) 1
) )  ->  (
( 1  -  t
)  x.  ( log `  y ) )  e.  CC )
204202, 203negdid 9380 . . . . . . . . . . 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 2439 . . . . . . . . . 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 4202 . . . . . . . . 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 20777 . . . . . . . 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 20780 . . . . . . 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 6281 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F )  =  ( k  e.  A  |->  ( ( 1  /  ( # `  A
) )  x.  ( F `  k )
) ) )
213212oveq2d 6056 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  (fld  gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) ) )
214 cnfldadd 16663 . . . . . . . . . . . 12  |-  +  =  ( +g  ` fld )
215 cnfldmul 16664 . . . . . . . . . . . 12  |-  x.  =  ( .r ` fld )
2162a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->fld  e. 
Ring )
217 eqid 2404 . . . . . . . . . . . . . 14  |-  ( k  e.  A  |->  ( F `
 k ) )  =  ( k  e.  A  |->  ( F `  k ) )
218107, 217fmptd 5852 . . . . . . . . . . . . 13  |-  ( ph  ->  ( k  e.  A  |->  ( F `  k
) ) : A --> CC )
2195, 218fisuppfi 14728 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( k  e.  A  |->  ( F `
 k ) )
" ( _V  \  { 0 } ) )  e.  Fin )
22052, 1, 214, 215, 216, 5, 142, 107, 219gsummulc2 15669 . . . . . . . . . . 11  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  ( ( 1  / 
( # `  A ) )  x.  ( F `
 k ) ) ) )  =  ( ( 1  /  ( # `
 A ) )  x.  (fld 
gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
221 fss 5558 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> RR+  /\  RR+  C_  RR )  ->  F : A --> RR )
22210, 116, 221sylancl 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : A --> RR )
2235, 10fisuppfi 14728 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' F "
( _V  \  {
0 } ) )  e.  Fin )
2241, 4, 5, 9, 222, 223gsumsubgcl 15480 . . . . . . . . . . . . . 14  |-  ( ph  ->  (fld 
gsumg  F )  e.  RR )
225224recnd 9070 . . . . . . . . . . . . 13  |-  ( ph  ->  (fld 
gsumg  F )  e.  CC )
226225, 23, 24divrec2d 9750 . . . . . . . . . . . 12  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( ( 1  /  ( # `  A
) )  x.  (fld  gsumg  F ) ) )
227106oveq2d 6056 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  F ) )  =  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) ) )
228226, 227eqtr2d 2437 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  (fld  gsumg  ( k  e.  A  |->  ( F `  k
) ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
229213, 220, 2283eqtrd 2440 . . . . . . . . . 10  |-  ( ph  ->  (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  =  ( (fld  gsumg  F )  /  ( # `  A
) ) )
230229, 151oveq12d 6058 . . . . . . . . 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 10004 . . . . . . . . . . 11  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  RR )
232231recnd 9070 . . . . . . . . . 10  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  e.  CC )
233232div1d 9738 . . . . . . . . 9  |-  ( ph  ->  ( ( (fld  gsumg  F )  /  ( # `
 A ) )  /  1 )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
234230, 233eqtrd 2436 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( ( A  X.  { ( 1  / 
( # `  A ) ) } )  o F  x.  F ) )  /  (fld  gsumg  ( A  X.  {
( 1  /  ( # `
 A ) ) } ) ) )  =  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
235234fveq2d 5691 . . . . . . 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 5687 . . . . . . . . . 10  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  ( log `  w
)  =  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
237236negeqd 9256 . . . . . . . . 9  |-  ( w  =  ( (fld  gsumg  F )  /  ( # `
 A ) )  ->  -u ( log `  w
)  =  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
238 negex 9260 . . . . . . . . 9  |-  -u ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) )  e.  _V
239237, 121, 238fvmpt 5765 . . . . . . . 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 2436 . . . . . 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 15669 . . . . . . . . 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 9260 . . . . . . . . . . . 12  |-  -u ( log `  ( F `  k ) )  e. 
_V
244243a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  -u ( log `  ( F `  k ) )  e. 
_V )
245 eqidd 2405 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  RR+  |->  -u ( log `  w
) )  =  ( w  e.  RR+  |->  -u ( log `  w ) ) )
246 fveq2 5687 . . . . . . . . . . . . 13  |-  ( w  =  ( F `  k )  ->  ( log `  w )  =  ( log `  ( F `  k )
) )
247246negeqd 9256 . . . . . . . . . . . 12  |-  ( w  =  ( F `  k )  ->  -u ( log `  w )  = 
-u ( log `  ( F `  k )
) )
24811, 35, 245, 247fmptco 5860 . . . . . . . . . . 11  |-  ( ph  ->  ( ( w  e.  RR+  |->  -u ( log `  w
) )  o.  F
)  =  ( k  e.  A  |->  -u ( log `  ( F `  k ) ) ) )
2495, 210, 244, 211, 248offval2 6281 . . . . . . . . . 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 6056 . . . . . . . . 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 9750 . . . . . . . . 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 2446 . . . . . . . 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 6058 . . . . . . 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 9070 . . . . . . . 8  |-  ( ph  ->  ( (fld 
gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  e.  CC )
255254div1d 9738 . . . . . . 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 2436 . . . . . 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 4201 . . . . 5  |-  ( ph  -> 
-u ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) )  <_  ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) ) )
258114, 115, 257lenegcon1d 9564 . . . 4  |-  ( ph  -> 
-u ( (fld  gsumg  ( k  e.  A  |-> 
-u ( log `  ( F `  k )
) ) )  / 
( # `  A ) )  <_  ( log `  ( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
259105, 258eqbrtrrd 4194 . . 3  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  <_  ( log `  (
(fld  gsumg  F )  /  ( # `  A ) ) ) )
260130, 102remulcld 9072 . . . 4  |-  ( ph  ->  ( ( 1  / 
( # `  A ) )  x.  ( log `  ( M  gsumg  F ) ) )  e.  RR )
261 efle 12674 . . . 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 10606 . . 3  |-  ( ph  ->  ( M  gsumg  F )  e.  CC )
26597rpne0d 10609 . . 3  |-  ( ph  ->  ( M  gsumg  F )  =/=  0
)
266264, 265, 142cxpefd 20556 . 2  |-  ( ph  ->  ( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  =  ( exp `  (
( 1  /  ( # `
 A ) )  x.  ( log `  ( M  gsumg  F ) ) ) ) )
267113reeflogd 20472 . . 3  |-  ( ph  ->  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )  =  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
268267eqcomd 2409 . 2  |-  ( ph  ->  ( (fld 
gsumg  F )  /  ( # `
 A ) )  =  ( exp `  ( log `  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) ) )
269263, 266, 2683brtr4d 4202 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 1721    =/= wne 2567   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172    e. cmpt 4226    X. cxp 4835    |` cres 4839    o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    o Fcof 6262   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    +oocpnf 9073    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   ZZcz 10238   RR+crp 10568   (,)cioo 10872   [,)cico 10874   [,]cicc 10875   #chash 11573   sum_csu 12434   expce 12619   Basecbs 13424   ↾s cress 13425   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639  .gcmg 14644   MndHom cmhm 14691  SubMndcsubmnd 14692  SubGrpcsubg 14893    GrpHom cghm 14958   GrpIso cgim 14999  CMndccmn 15367   Abelcabel 15368  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616   DivRingcdr 15790  SubRingcsubrg 15819  ℂfldccnfld 16658   logclog 20405    ^ c ccxp 20406
This theorem is referenced by:  amgm  20782
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-mulg 14770  df-subg 14896  df-ghm 14959  df-gim 15001  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-subrg 15821  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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