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Theorem amgm 20280
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 amgm.1 . . . . . . . . . 10  |-  M  =  (mulGrp ` fld )
2 cnfldbas 16378 . . . . . . . . . 10  |-  CC  =  ( Base ` fld )
31, 2mgpbas 15326 . . . . . . . . 9  |-  CC  =  ( Base `  M )
4 cnfld1 16394 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
51, 4rngidval 15338 . . . . . . . . 9  |-  1  =  ( 0g `  M )
6 cnfldmul 16380 . . . . . . . . . 10  |-  x.  =  ( .r ` fld )
71, 6mgpplusg 15324 . . . . . . . . 9  |-  x.  =  ( +g  `  M )
8 cncrng 16390 . . . . . . . . . 10  |-fld  e.  CRing
91crngmgp 15344 . . . . . . . . . 10  |-  (fld  e.  CRing  ->  M  e. CMnd )
108, 9mp1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  M  e. CMnd )
11 simpl1 958 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  A  e.  Fin )
12 simpl3 960 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  F : A
--> ( 0 [,)  +oo ) )
13 0re 8833 . . . . . . . . . . . 12  |-  0  e.  RR
14 pnfxr 10450 . . . . . . . . . . . 12  |-  +oo  e.  RR*
15 icossre 10725 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
1613, 14, 15mp2an 653 . . . . . . . . . . 11  |-  ( 0 [,)  +oo )  C_  RR
17 ax-resscn 8789 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17sstri 3188 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  CC
19 fss 5362 . . . . . . . . . 10  |-  ( ( F : A --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  CC )  ->  F : A
--> CC )
2012, 18, 19sylancl 643 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  F : A
--> CC )
2111, 12fisuppfi 14445 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' F " ( _V  \  { 1 } ) )  e.  Fin )
22 disjdif 3526 . . . . . . . . . 10  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
2322a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( {
x }  i^i  ( A  \  { x }
) )  =  (/) )
24 undif2 3530 . . . . . . . . . 10  |-  ( { x }  u.  ( A  \  { x }
) )  =  ( { x }  u.  A )
25 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  x  e.  A )
2625snssd 3760 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  { x }  C_  A )
27 ssequn1 3345 . . . . . . . . . . 11  |-  ( { x }  C_  A  <->  ( { x }  u.  A )  =  A )
2826, 27sylib 188 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( {
x }  u.  A
)  =  A )
2924, 28syl5req 2328 . . . . . . . . 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 15202 . . . . . . . 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 5537 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |` 
{ x } )  =  ( y  e. 
{ x }  |->  ( F `  y ) ) )
3231oveq2d 5835 . . . . . . . . . 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 16391 . . . . . . . . . . . 12  |-fld  e.  Ring
341rngmgp 15342 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->  M  e.  Mnd )
3533, 34mp1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  M  e.  Mnd )
36 ffvelrn 5624 . . . . . . . . . . . 12  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
3720, 25, 36syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F `  x )  e.  CC )
38 fveq2 5485 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
393, 38gsumsn 15215 . . . . . . . . . . 11  |-  ( ( M  e.  Mnd  /\  x  e.  A  /\  ( F `  x )  e.  CC )  -> 
( M  gsumg  ( y  e.  {
x }  |->  ( F `
 y ) ) )  =  ( F `
 x ) )
4035, 25, 37, 39syl3anc 1182 . . . . . . . . . 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 733 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F `  x )  =  0 )
4232, 40, 413eqtrd 2319 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  { x } ) )  =  0 )
4342oveq1d 5834 . . . . . . . 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 7084 . . . . . . . . . . 11  |-  ( A  e.  Fin  ->  ( A  \  { x }
)  e.  Fin )
4511, 44syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( A  \  { x } )  e.  Fin )
46 difss 3303 . . . . . . . . . . 11  |-  ( A 
\  { x }
)  C_  A
47 fssres 5373 . . . . . . . . . . 11  |-  ( ( F : A --> CC  /\  ( A  \  { x } )  C_  A
)  ->  ( F  |`  ( A  \  {
x } ) ) : ( A  \  { x } ) --> CC )
4820, 46, 47sylancl 643 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |`  ( A  \  {
x } ) ) : ( A  \  { x } ) --> CC )
4945, 48fisuppfi 14445 . . . . . . . . . 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 15193 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  ( A  \  { x } ) ) )  e.  CC )
5150mul02d 9005 . . . . . . . 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 2319 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  F )  =  0 )
5352oveq1d 5834 . . . . . 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 959 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  A  =/=  (/) )
55 hashnncl 11349 . . . . . . . . . . 11  |-  ( A  e.  Fin  ->  (
( # `  A )  e.  NN  <->  A  =/=  (/) ) )
5611, 55syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( # `
 A )  e.  NN  <->  A  =/=  (/) ) )
5754, 56mpbird 223 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  NN )
5857nncnd 9757 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  CC )
5957nnne0d 9785 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  =/=  0 )
6058, 59reccld 9524 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  e.  CC )
6158, 59recne0d 9525 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  =/=  0
)
6260, 610cxpd 20052 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0  ^ c  ( 1  /  ( # `  A
) ) )  =  0 )
6353, 62eqtrd 2315 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  =  0 )
64 cnfld0 16393 . . . . . . . 8  |-  0  =  ( 0g ` fld )
65 rngcmn 15366 . . . . . . . . 9  |-  (fld  e.  Ring  ->fld  e. CMnd )
6633, 65mp1i 11 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->fld  e. CMnd )
67 rege0subm 16423 . . . . . . . . 9  |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
6867a1i 10 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0 [,)  +oo )  e.  (SubMnd ` fld ) )
6911, 12fisuppfi 14445 . . . . . . . 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 15196 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  (fld  gsumg  F )  e.  ( 0 [,)  +oo )
)
71 elrege0 10741 . . . . . . 7  |-  ( (fld  gsumg  F )  e.  ( 0 [,) 
+oo )  <->  ( (fld  gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) ) )
7270, 71sylib 188 . . . . . 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 9756 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  RR )
7457nngt0d 9784 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <  (
# `  A )
)
75 divge0 9620 . . . . . 6  |-  ( ( ( (fld 
gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) )  /\  ( ( # `  A
)  e.  RR  /\  0  <  ( # `  A
) ) )  -> 
0  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
7672, 73, 74, 75syl12anc 1180 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <_  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
7763, 76eqbrtrd 4043 . . . 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 598 . . 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 2667 . 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 2553 . . 3  |-  ( A. x  e.  A  -.  ( F `  x )  =  0  <->  -.  E. x  e.  A  ( F `  x )  =  0 )
81 simpl1 958 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A  e.  Fin )
82 simpl2 959 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A  =/=  (/) )
83 simpl3 960 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F : A --> ( 0 [,)  +oo ) )
84 ffn 5354 . . . . . . 7  |-  ( F : A --> ( 0 [,)  +oo )  ->  F  Fn  A )
8583, 84syl 15 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F  Fn  A )
86 ffvelrn 5624 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> ( 0 [,)  +oo )  /\  x  e.  A )  ->  ( F `  x )  e.  ( 0 [,)  +oo ) )
87863ad2antl3 1119 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  ( 0 [,)  +oo ) )
88 elrege0 10741 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
8987, 88sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( ( F `  x )  e.  RR  /\  0  <_  ( F `  x ) ) )
9089simprd 449 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  0  <_  ( F `  x ) )
9189simpld 445 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
92 leloe 8903 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  ( F `  x )  e.  RR )  -> 
( 0  <_  ( F `  x )  <->  ( 0  <  ( F `
 x )  \/  0  =  ( F `
 x ) ) ) )
9313, 91, 92sylancr 644 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( 0  <_  ( F `  x )  <->  ( 0  <  ( F `
 x )  \/  0  =  ( F `
 x ) ) ) )
9490, 93mpbid 201 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( 0  <  ( F `  x )  \/  0  =  ( F `  x )
) )
9594ord 366 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  0  < 
( F `  x
)  ->  0  =  ( F `  x ) ) )
96 eqcom 2285 . . . . . . . . . . 11  |-  ( 0  =  ( F `  x )  <->  ( F `  x )  =  0 )
9795, 96syl6ib 217 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  0  < 
( F `  x
)  ->  ( F `  x )  =  0 ) )
9897con1d 116 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =  0  ->  0  <  ( F `  x ) ) )
99 elrp 10351 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  RR+  <->  ( ( F `
 x )  e.  RR  /\  0  < 
( F `  x
) ) )
10099baib 871 . . . . . . . . . 10  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  RR+  <->  0  <  ( F `  x ) ) )
10191, 100syl 15 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( ( F `  x )  e.  RR+  <->  0  <  ( F `  x ) ) )
10298, 101sylibrd 225 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =  0  ->  ( F `  x )  e.  RR+ ) )
103102ralimdva 2621 . . . . . . 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 418 . . . . . 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 5646 . . . . . 6  |-  ( F : A --> RR+  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  RR+ ) )
10685, 104, 105sylanbrc 645 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F : A --> RR+ )
1071, 81, 82, 106amgmlem 20279 . . . 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 423 . . 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 209 . 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 150 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 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    e. cmpt 4077    |` cres 4689    Fn wfn 5215   -->wf 5216   ` cfv 5220  (class class class)co 5819   Fincfn 6858   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    x. cmul 8737    +oocpnf 8859   RR*cxr 8861    < clt 8862    <_ cle 8863    / cdiv 9418   NNcn 9741   RR+crp 10349   [,)cico 10653   #chash 11332    gsumg cgsu 13396   Mndcmnd 14356  SubMndcsubmnd 14409  CMndccmn 15084  mulGrpcmgp 15320   Ringcrg 15332   CRingccrg 15333  ℂfldccnfld 16372    ^ c ccxp 19908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4186  ax-pr 4212  ax-un 4510  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4303  df-id 4307  df-po 4312  df-so 4313  df-fr 4350  df-se 4351  df-we 4352  df-ord 4393  df-on 4394  df-lim 4395  df-suc 4396  df-om 4655  df-xp 4693  df-rel 4694  df-cnv 4695  df-co 4696  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fun 5222  df-fn 5223  df-f 5224  df-f1 5225  df-fo 5226  df-f1o 5227  df-fv 5228  df-isom 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10655  df-ioc 10656  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-fl 10920  df-mod 10969  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-shft 11557  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-limsup 11940  df-clim 11957  df-rlim 11958  df-sum 12154  df-ef 12344  df-sin 12346  df-cos 12347  df-pi 12349  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-starv 13218  df-sca 13219  df-vsca 13220  df-tset 13222  df-ple 13223  df-ds 13225  df-hom 13227  df-cco 13228  df-rest 13322  df-topn 13323  df-topgen 13339  df-pt 13340  df-prds 13343  df-xrs 13398  df-0g 13399  df-gsum 13400  df-qtop 13405  df-imas 13406  df-xps 13408  df-mre 13483  df-mrc 13484  df-acs 13486  df-mnd 14362  df-mhm 14410  df-submnd 14411  df-grp 14484  df-minusg 14485  df-mulg 14487  df-subg 14613  df-ghm 14676  df-gim 14718  df-cntz 14788  df-cmn 15086  df-abl 15087  df-mgp 15321  df-rng 15335  df-cring 15336  df-ur 15337  df-oppr 15400  df-dvdsr 15418  df-unit 15419  df-invr 15449  df-dvr 15460  df-drng 15509  df-subrg 15538  df-xmet 16368  df-met 16369  df-bl 16370  df-mopn 16371  df-cnfld 16373  df-top 16631  df-bases 16633  df-topon 16634  df-topsp 16635  df-cld 16751  df-ntr 16752  df-cls 16753  df-nei 16830  df-lp 16863  df-perf 16864  df-cn 16952  df-cnp 16953  df-haus 17038  df-cmp 17109  df-tx 17252  df-hmeo 17441  df-fbas 17515  df-fg 17516  df-fil 17536  df-fm 17628  df-flim 17629  df-flf 17630  df-xms 17880  df-ms 17881  df-tms 17882  df-cncf 18377  df-limc 19211  df-dv 19212  df-log 19909  df-cxp 19910
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