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Theorem amgm 20117
Description: Inequality of arithmetic and geometric means. Here  ( M  gsumg  F ) calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements  F ( x ) ,  x  e.  A together), and  (fld 
gsumg  F ) calculates the group sum in the additive group (i.e. the sum of the elements). (Contributed by Mario Carneiro, 20-Jun-2015.)
Hypothesis
Ref Expression
amgm.1  |-  M  =  (mulGrp ` fld )
Assertion
Ref Expression
amgm  |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A
--> ( 0 [,)  +oo ) )  ->  (
( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) )

Proof of Theorem amgm
StepHypRef Expression
1 amgm.1 . . . . . . . . . 10  |-  M  =  (mulGrp ` fld )
2 cnfldbas 16215 . . . . . . . . . 10  |-  CC  =  ( Base ` fld )
31, 2mgpbas 15166 . . . . . . . . 9  |-  CC  =  ( Base `  M )
4 cnfld1 16231 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
51, 4rngidval 15178 . . . . . . . . 9  |-  1  =  ( 0g `  M )
6 cnfldmul 16217 . . . . . . . . . 10  |-  x.  =  ( .r ` fld )
71, 6mgpplusg 15164 . . . . . . . . 9  |-  x.  =  ( +g  `  M )
8 cncrng 16227 . . . . . . . . . 10  |-fld  e.  CRing
91crngmgp 15184 . . . . . . . . . 10  |-  (fld  e.  CRing  ->  M  e. CMnd )
108, 9mp1i 13 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  M  e. CMnd )
11 simpl1 963 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  A  e.  Fin )
12 simpl3 965 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  F : A
--> ( 0 [,)  +oo ) )
13 0re 8718 . . . . . . . . . . . 12  |-  0  e.  RR
14 pnfxr 10334 . . . . . . . . . . . 12  |-  +oo  e.  RR*
15 icossre 10608 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
1613, 14, 15mp2an 656 . . . . . . . . . . 11  |-  ( 0 [,)  +oo )  C_  RR
17 ax-resscn 8674 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17sstri 3109 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  CC
19 fss 5254 . . . . . . . . . 10  |-  ( ( F : A --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  CC )  ->  F : A
--> CC )
2012, 18, 19sylancl 646 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  F : A
--> CC )
2111, 12fisuppfi 14285 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' F " ( _V  \  { 1 } ) )  e.  Fin )
22 disjdif 3432 . . . . . . . . . 10  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
2322a1i 12 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( {
x }  i^i  ( A  \  { x }
) )  =  (/) )
24 undif2 3436 . . . . . . . . . 10  |-  ( { x }  u.  ( A  \  { x }
) )  =  ( { x }  u.  A )
25 simprl 735 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  x  e.  A )
2625snssd 3660 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  { x }  C_  A )
27 ssequn1 3255 . . . . . . . . . . 11  |-  ( { x }  C_  A  <->  ( { x }  u.  A )  =  A )
2826, 27sylib 190 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( {
x }  u.  A
)  =  A )
2924, 28syl5req 2298 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  A  =  ( { x }  u.  ( A  \  { x } ) ) )
303, 5, 7, 10, 11, 20, 21, 23, 29gsumsplit 15042 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  F )  =  ( ( M  gsumg  ( F  |`  { x } ) )  x.  ( M  gsumg  ( F  |`  ( A  \  { x }
) ) ) ) )
3112, 26feqresmpt 5428 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |` 
{ x } )  =  ( y  e. 
{ x }  |->  ( F `  y ) ) )
3231oveq2d 5726 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  { x } ) )  =  ( M  gsumg  ( y  e.  {
x }  |->  ( F `
 y ) ) ) )
33 cnrng 16228 . . . . . . . . . . . 12  |-fld  e.  Ring
341rngmgp 15182 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->  M  e.  Mnd )
3533, 34mp1i 13 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  M  e.  Mnd )
36 ffvelrn 5515 . . . . . . . . . . . 12  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
3720, 25, 36syl2anc 645 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F `  x )  e.  CC )
38 fveq2 5377 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
393, 38gsumsn 15055 . . . . . . . . . . 11  |-  ( ( M  e.  Mnd  /\  x  e.  A  /\  ( F `  x )  e.  CC )  -> 
( M  gsumg  ( y  e.  {
x }  |->  ( F `
 y ) ) )  =  ( F `
 x ) )
4035, 25, 37, 39syl3anc 1187 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( y  e.  { x }  |->  ( F `  y ) ) )  =  ( F `  x ) )
41 simprr 736 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F `  x )  =  0 )
4232, 40, 413eqtrd 2289 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  { x } ) )  =  0 )
4342oveq1d 5725 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  ( F  |`  { x } ) )  x.  ( M  gsumg  ( F  |`  ( A  \  { x }
) ) ) )  =  ( 0  x.  ( M  gsumg  ( F  |`  ( A  \  { x }
) ) ) ) )
44 diffi 6974 . . . . . . . . . . 11  |-  ( A  e.  Fin  ->  ( A  \  { x }
)  e.  Fin )
4511, 44syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( A  \  { x } )  e.  Fin )
46 difss 3220 . . . . . . . . . . 11  |-  ( A 
\  { x }
)  C_  A
47 fssres 5265 . . . . . . . . . . 11  |-  ( ( F : A --> CC  /\  ( A  \  { x } )  C_  A
)  ->  ( F  |`  ( A  \  {
x } ) ) : ( A  \  { x } ) --> CC )
4820, 46, 47sylancl 646 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |`  ( A  \  {
x } ) ) : ( A  \  { x } ) --> CC )
4945, 48fisuppfi 14285 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' ( F  |`  ( A 
\  { x }
) ) " ( _V  \  { 1 } ) )  e.  Fin )
503, 5, 10, 45, 48, 49gsumcl 15033 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  ( A  \  { x } ) ) )  e.  CC )
5150mul02d 8890 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0  x.  ( M  gsumg  ( F  |`  ( A  \  {
x } ) ) ) )  =  0 )
5230, 43, 513eqtrd 2289 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  F )  =  0 )
5352oveq1d 5725 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  =  ( 0  ^ c  ( 1  /  ( # `  A
) ) ) )
54 simpl2 964 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  A  =/=  (/) )
55 hashnncl 11232 . . . . . . . . . . 11  |-  ( A  e.  Fin  ->  (
( # `  A )  e.  NN  <->  A  =/=  (/) ) )
5611, 55syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( # `
 A )  e.  NN  <->  A  =/=  (/) ) )
5754, 56mpbird 225 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  NN )
5857nncnd 9642 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  CC )
5957nnne0d 9670 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  =/=  0 )
6058, 59reccld 9409 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  e.  CC )
6158, 59recne0d 9410 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  =/=  0
)
6260, 610cxpd 19925 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0  ^ c  ( 1  /  ( # `  A
) ) )  =  0 )
6353, 62eqtrd 2285 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  =  0 )
64 cnfld0 16230 . . . . . . . 8  |-  0  =  ( 0g ` fld )
65 rngcmn 15206 . . . . . . . . 9  |-  (fld  e.  Ring  ->fld  e. CMnd )
6633, 65mp1i 13 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->fld  e. CMnd )
67 rege0subm 16260 . . . . . . . . 9  |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
6867a1i 12 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0 [,)  +oo )  e.  (SubMnd ` fld ) )
6911, 12fisuppfi 14285 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' F " ( _V  \  { 0 } ) )  e.  Fin )
7064, 66, 11, 68, 12, 69gsumsubmcl 15036 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  (fld  gsumg  F )  e.  ( 0 [,)  +oo )
)
71 elrege0 10624 . . . . . . 7  |-  ( (fld  gsumg  F )  e.  ( 0 [,) 
+oo )  <->  ( (fld  gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) ) )
7270, 71sylib 190 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( (fld  gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) ) )
7357nnred 9641 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  RR )
7457nngt0d 9669 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <  (
# `  A )
)
75 divge0 9505 . . . . . 6  |-  ( ( ( (fld 
gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) )  /\  ( ( # `  A
)  e.  RR  /\  0  <  ( # `  A
) ) )  -> 
0  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
7672, 73, 74, 75syl12anc 1185 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <_  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
7763, 76eqbrtrd 3940 . . . 4  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
7877expr 601 . . 3  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( ( F `  x )  =  0  ->  ( ( M 
gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
7978rexlimdva 2629 . 2  |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A
--> ( 0 [,)  +oo ) )  ->  ( E. x  e.  A  ( F `  x )  =  0  ->  (
( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
80 ralnex 2517 . . 3  |-  ( A. x  e.  A  -.  ( F `  x )  =  0  <->  -.  E. x  e.  A  ( F `  x )  =  0 )
81 simpl1 963 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A  e.  Fin )
82 simpl2 964 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A  =/=  (/) )
83 simpl3 965 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F : A --> ( 0 [,)  +oo ) )
84 ffn 5246 . . . . . . 7  |-  ( F : A --> ( 0 [,)  +oo )  ->  F  Fn  A )
8583, 84syl 17 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F  Fn  A )
86 ffvelrn 5515 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> ( 0 [,)  +oo )  /\  x  e.  A )  ->  ( F `  x )  e.  ( 0 [,)  +oo ) )
87863ad2antl3 1124 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  ( 0 [,)  +oo ) )
88 elrege0 10624 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
8987, 88sylib 190 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( ( F `  x )  e.  RR  /\  0  <_  ( F `  x ) ) )
9089simprd 451 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  0  <_  ( F `  x ) )
9189simpld 447 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
92 leloe 8788 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  ( F `  x )  e.  RR )  -> 
( 0  <_  ( F `  x )  <->  ( 0  <  ( F `
 x )  \/  0  =  ( F `
 x ) ) ) )
9313, 91, 92sylancr 647 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( 0  <_  ( F `  x )  <->  ( 0  <  ( F `
 x )  \/  0  =  ( F `
 x ) ) ) )
9490, 93mpbid 203 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( 0  <  ( F `  x )  \/  0  =  ( F `  x )
) )
9594ord 368 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  0  < 
( F `  x
)  ->  0  =  ( F `  x ) ) )
96 eqcom 2255 . . . . . . . . . . 11  |-  ( 0  =  ( F `  x )  <->  ( F `  x )  =  0 )
9795, 96syl6ib 219 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  0  < 
( F `  x
)  ->  ( F `  x )  =  0 ) )
9897con1d 118 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =  0  ->  0  <  ( F `  x ) ) )
99 elrp 10235 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  RR+  <->  ( ( F `
 x )  e.  RR  /\  0  < 
( F `  x
) ) )
10099baib 876 . . . . . . . . . 10  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  RR+  <->  0  <  ( F `  x ) ) )
10191, 100syl 17 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( ( F `  x )  e.  RR+  <->  0  <  ( F `  x ) ) )
10298, 101sylibrd 227 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =  0  ->  ( F `  x )  e.  RR+ ) )
103102ralimdva 2583 . . . . . . 7  |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A
--> ( 0 [,)  +oo ) )  ->  ( A. x  e.  A  -.  ( F `  x
)  =  0  ->  A. x  e.  A  ( F `  x )  e.  RR+ ) )
104103imp 420 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A. x  e.  A  ( F `  x )  e.  RR+ )
105 ffnfv 5537 . . . . . 6  |-  ( F : A --> RR+  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  RR+ ) )
10685, 104, 105sylanbrc 648 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F : A --> RR+ )
1071, 81, 82, 106amgmlem 20116 . . . 4  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  -> 
( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  <_ 
( (fld 
gsumg  F )  /  ( # `
 A ) ) )
108107ex 425 . . 3  |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A
--> ( 0 [,)  +oo ) )  ->  ( A. x  e.  A  -.  ( F `  x
)  =  0  -> 
( ( M  gsumg  F )  ^ c  ( 1  /  ( # `  A
) ) )  <_ 
( (fld 
gsumg  F )  /  ( # `
 A ) ) ) )
10980, 108syl5bir 211 . 2  |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A
--> ( 0 [,)  +oo ) )  ->  ( -.  E. x  e.  A  ( F `  x )  =  0  ->  (
( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) ) )
11079, 109pm2.61d 152 1  |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A
--> ( 0 [,)  +oo ) )  ->  (
( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510   _Vcvv 2727    \ cdif 3075    u. cun 3076    i^i cin 3077    C_ wss 3078   (/)c0 3362   {csn 3544   class class class wbr 3920    e. cmpt 3974    |` cres 4582    Fn wfn 4587   -->wf 4588   ` cfv 4592  (class class class)co 5710   Fincfn 6749   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    x. cmul 8622    +oocpnf 8744   RR*cxr 8746    < clt 8747    <_ cle 8748    / cdiv 9303   NNcn 9626   RR+crp 10233   [,)cico 10536   #chash 11215    gsumg cgsu 13275   Mndcmnd 14196  SubMndcsubmnd 14249  CMndccmn 14924  mulGrpcmgp 15160   Ringcrg 15172   CRingccrg 15173  ℂfldccnfld 16209    ^ c ccxp 19745
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-fi 7049  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-q 10196  df-rp 10234  df-xneg 10331  df-xadd 10332  df-xmul 10333  df-ioo 10538  df-ioc 10539  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-shft 11439  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-limsup 11822  df-clim 11839  df-rlim 11840  df-sum 12036  df-ef 12223  df-sin 12225  df-cos 12226  df-pi 12228  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-hom 13106  df-cco 13107  df-rest 13201  df-topn 13202  df-topgen 13218  df-pt 13219  df-prds 13222  df-xrs 13277  df-0g 13278  df-gsum 13279  df-qtop 13284  df-imas 13285  df-xps 13287  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-mhm 14250  df-submnd 14251  df-grp 14324  df-minusg 14325  df-mulg 14327  df-subg 14453  df-ghm 14516  df-gim 14558  df-cntz 14628  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-cring 15176  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-subrg 15378  df-xmet 16205  df-met 16206  df-bl 16207  df-mopn 16208  df-cnfld 16210  df-top 16468  df-bases 16470  df-topon 16471  df-topsp 16472  df-cld 16588  df-ntr 16589  df-cls 16590  df-nei 16667  df-lp 16700  df-perf 16701  df-cn 16789  df-cnp 16790  df-haus 16875  df-cmp 16946  df-tx 17089  df-hmeo 17278  df-fbas 17352  df-fg 17353  df-fil 17373  df-fm 17465  df-flim 17466  df-flf 17467  df-xms 17717  df-ms 17718  df-tms 17719  df-cncf 18214  df-limc 19048  df-dv 19049  df-log 19746  df-cxp 19747
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