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Theorem amgm 20287
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 16385 . . . . . . . . . 10  |-  CC  =  ( Base ` fld )
31, 2mgpbas 15333 . . . . . . . . 9  |-  CC  =  ( Base `  M )
4 cnfld1 16401 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
51, 4rngidval 15345 . . . . . . . . 9  |-  1  =  ( 0g `  M )
6 cnfldmul 16387 . . . . . . . . . 10  |-  x.  =  ( .r ` fld )
71, 6mgpplusg 15331 . . . . . . . . 9  |-  x.  =  ( +g  `  M )
8 cncrng 16397 . . . . . . . . . 10  |-fld  e.  CRing
91crngmgp 15351 . . . . . . . . . 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 8840 . . . . . . . . . . . 12  |-  0  e.  RR
14 pnfxr 10457 . . . . . . . . . . . 12  |-  +oo  e.  RR*
15 icossre 10732 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
1613, 14, 15mp2an 653 . . . . . . . . . . 11  |-  ( 0 [,)  +oo )  C_  RR
17 ax-resscn 8796 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17sstri 3190 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  CC
19 fss 5399 . . . . . . . . . 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 14452 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' F " ( _V  \  { 1 } ) )  e.  Fin )
22 disjdif 3528 . . . . . . . . . 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 3532 . . . . . . . . . 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 3762 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  { x }  C_  A )
27 ssequn1 3347 . . . . . . . . . . 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 2330 . . . . . . . . 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 15209 . . . . . . . 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 5578 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |` 
{ x } )  =  ( y  e. 
{ x }  |->  ( F `  y ) ) )
3231oveq2d 5876 . . . . . . . . . 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 16398 . . . . . . . . . . . 12  |-fld  e.  Ring
341rngmgp 15349 . . . . . . . . . . . 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 5665 . . . . . . . . . . . 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 5527 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
393, 38gsumsn 15222 . . . . . . . . . . 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 2321 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  { x } ) )  =  0 )
4342oveq1d 5875 . . . . . . . 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 7091 . . . . . . . . . . 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 3305 . . . . . . . . . . 11  |-  ( A 
\  { x }
)  C_  A
47 fssres 5410 . . . . . . . . . . 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 14452 . . . . . . . . . 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 15200 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  ( A  \  { x } ) ) )  e.  CC )
5150mul02d 9012 . . . . . . . 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 2321 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  F )  =  0 )
5352oveq1d 5875 . . . . . 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 11356 . . . . . . . . . . 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 9764 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  CC )
5957nnne0d 9792 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  =/=  0 )
6058, 59reccld 9531 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  e.  CC )
6158, 59recne0d 9532 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  =/=  0
)
6260, 610cxpd 20059 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0  ^ c  ( 1  /  ( # `  A
) ) )  =  0 )
6353, 62eqtrd 2317 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  =  0 )
64 cnfld0 16400 . . . . . . . 8  |-  0  =  ( 0g ` fld )
65 rngcmn 15373 . . . . . . . . 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 16430 . . . . . . . . 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 14452 . . . . . . . 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 15203 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  (fld  gsumg  F )  e.  ( 0 [,)  +oo )
)
71 elrege0 10748 . . . . . . 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 9763 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  RR )
7457nngt0d 9791 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <  (
# `  A )
)
75 divge0 9627 . . . . . 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 4045 . . . 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 2669 . 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 2555 . . 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 5391 . . . . . . 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 5665 . . . . . . . . . . . . . . . 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 10748 . . . . . . . . . . . . . . 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 8910 . . . . . . . . . . . . . 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 2287 . . . . . . . . . . 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 10358 . . . . . . . . . . 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 2623 . . . . . . 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 5687 . . . . . 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 20286 . . . 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 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   _Vcvv 2790    \ cdif 3151    u. cun 3152    i^i cin 3153    C_ wss 3154   (/)c0 3457   {csn 3642   class class class wbr 4025    e. cmpt 4079    |` cres 4693    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860   Fincfn 6865   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744    +oocpnf 8866   RR*cxr 8868    < clt 8869    <_ cle 8870    / cdiv 9425   NNcn 9748   RR+crp 10356   [,)cico 10660   #chash 11339    gsumg cgsu 13403   Mndcmnd 14363  SubMndcsubmnd 14416  CMndccmn 15091  mulGrpcmgp 15327   Ringcrg 15339   CRingccrg 15340  ℂfldccnfld 16379    ^ c ccxp 19915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-mhm 14417  df-submnd 14418  df-grp 14491  df-minusg 14492  df-mulg 14494  df-subg 14620  df-ghm 14683  df-gim 14725  df-cntz 14795  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-cring 15343  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-subrg 15545  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-cmp 17116  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-cxp 19917
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