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Theorem amgm 20697
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 . . . . . . . . 9  |-  M  =  (mulGrp ` fld )
2 cnfldbas 16631 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
31, 2mgpbas 15582 . . . . . . . 8  |-  CC  =  ( Base `  M )
4 cnfld1 16650 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
51, 4rngidval 15594 . . . . . . . 8  |-  1  =  ( 0g `  M )
6 cnfldmul 16633 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
71, 6mgpplusg 15580 . . . . . . . 8  |-  x.  =  ( +g  `  M )
8 cncrng 16646 . . . . . . . . 9  |-fld  e.  CRing
91crngmgp 15600 . . . . . . . . 9  |-  (fld  e.  CRing  ->  M  e. CMnd )
108, 9mp1i 12 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  M  e. CMnd )
11 simpl1 960 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  A  e.  Fin )
12 simpl3 962 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  F : A
--> ( 0 [,)  +oo ) )
13 0re 9025 . . . . . . . . . . 11  |-  0  e.  RR
14 pnfxr 10646 . . . . . . . . . . 11  |-  +oo  e.  RR*
15 icossre 10924 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
1613, 14, 15mp2an 654 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  RR
17 ax-resscn 8981 . . . . . . . . . 10  |-  RR  C_  CC
1816, 17sstri 3301 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  CC
19 fss 5540 . . . . . . . . 9  |-  ( ( F : A --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  CC )  ->  F : A
--> CC )
2012, 18, 19sylancl 644 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  F : A
--> CC )
2111, 12fisuppfi 14701 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' F " ( _V  \  { 1 } ) )  e.  Fin )
22 disjdif 3644 . . . . . . . . 9  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
2322a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( {
x }  i^i  ( A  \  { x }
) )  =  (/) )
24 undif2 3648 . . . . . . . . 9  |-  ( { x }  u.  ( A  \  { x }
) )  =  ( { x }  u.  A )
25 simprl 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  x  e.  A )
2625snssd 3887 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  { x }  C_  A )
27 ssequn1 3461 . . . . . . . . . 10  |-  ( { x }  C_  A  <->  ( { x }  u.  A )  =  A )
2826, 27sylib 189 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( {
x }  u.  A
)  =  A )
2924, 28syl5req 2433 . . . . . . . 8  |-  ( ( ( 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 15458 . . . . . . 7  |-  ( ( ( 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 5720 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |` 
{ x } )  =  ( y  e. 
{ x }  |->  ( F `  y ) ) )
3231oveq2d 6037 . . . . . . . . 9  |-  ( ( ( 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 16647 . . . . . . . . . . 11  |-fld  e.  Ring
341rngmgp 15598 . . . . . . . . . . 11  |-  (fld  e.  Ring  ->  M  e.  Mnd )
3533, 34mp1i 12 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  M  e.  Mnd )
3620, 25ffvelrnd 5811 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F `  x )  e.  CC )
37 fveq2 5669 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
383, 37gsumsn 15471 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  x  e.  A  /\  ( F `  x )  e.  CC )  -> 
( M  gsumg  ( y  e.  {
x }  |->  ( F `
 y ) ) )  =  ( F `
 x ) )
3935, 25, 36, 38syl3anc 1184 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( y  e.  { x }  |->  ( F `  y ) ) )  =  ( F `  x ) )
40 simprr 734 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F `  x )  =  0 )
4132, 39, 403eqtrd 2424 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  { x } ) )  =  0 )
4241oveq1d 6036 . . . . . . 7  |-  ( ( ( 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 }
) ) ) ) )
43 diffi 7276 . . . . . . . . . 10  |-  ( A  e.  Fin  ->  ( A  \  { x }
)  e.  Fin )
4411, 43syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( A  \  { x } )  e.  Fin )
45 difss 3418 . . . . . . . . . 10  |-  ( A 
\  { x }
)  C_  A
46 fssres 5551 . . . . . . . . . 10  |-  ( ( F : A --> CC  /\  ( A  \  { x } )  C_  A
)  ->  ( F  |`  ( A  \  {
x } ) ) : ( A  \  { x } ) --> CC )
4720, 45, 46sylancl 644 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |`  ( A  \  {
x } ) ) : ( A  \  { x } ) --> CC )
4844, 47fisuppfi 14701 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' ( F  |`  ( A 
\  { x }
) ) " ( _V  \  { 1 } ) )  e.  Fin )
493, 5, 10, 44, 47, 48gsumcl 15449 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  ( A  \  { x } ) ) )  e.  CC )
5049mul02d 9197 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0  x.  ( M  gsumg  ( F  |`  ( A  \  {
x } ) ) ) )  =  0 )
5130, 42, 503eqtrd 2424 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  F )  =  0 )
5251oveq1d 6036 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  =  ( 0  ^ c  ( 1  /  ( # `  A
) ) ) )
53 simpl2 961 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  A  =/=  (/) )
54 hashnncl 11573 . . . . . . . . . 10  |-  ( A  e.  Fin  ->  (
( # `  A )  e.  NN  <->  A  =/=  (/) ) )
5511, 54syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( # `
 A )  e.  NN  <->  A  =/=  (/) ) )
5653, 55mpbird 224 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  NN )
5756nncnd 9949 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  CC )
5856nnne0d 9977 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  =/=  0 )
5957, 58reccld 9716 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  e.  CC )
6057, 58recne0d 9717 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  =/=  0
)
6159, 600cxpd 20469 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0  ^ c  ( 1  /  ( # `  A
) ) )  =  0 )
6252, 61eqtrd 2420 . . . 4  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  =  0 )
63 cnfld0 16649 . . . . . . 7  |-  0  =  ( 0g ` fld )
64 rngcmn 15622 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
6533, 64mp1i 12 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->fld  e. CMnd )
66 rege0subm 16679 . . . . . . . 8  |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
6766a1i 11 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0 [,)  +oo )  e.  (SubMnd ` fld ) )
6811, 12fisuppfi 14701 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' F " ( _V  \  { 0 } ) )  e.  Fin )
6963, 65, 11, 67, 12, 68gsumsubmcl 15452 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  (fld  gsumg  F )  e.  ( 0 [,)  +oo )
)
70 elrege0 10940 . . . . . 6  |-  ( (fld  gsumg  F )  e.  ( 0 [,) 
+oo )  <->  ( (fld  gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) ) )
7169, 70sylib 189 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( (fld  gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) ) )
7256nnred 9948 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  RR )
7356nngt0d 9976 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <  (
# `  A )
)
74 divge0 9812 . . . . 5  |-  ( ( ( (fld 
gsumg  F )  e.  RR  /\  0  <_  (fld  gsumg  F ) )  /\  ( ( # `  A
)  e.  RR  /\  0  <  ( # `  A
) ) )  -> 
0  <_  ( (fld  gsumg  F )  /  ( # `
 A ) ) )
7571, 72, 73, 74syl12anc 1182 . . . 4  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <_  ( (fld 
gsumg  F )  /  ( # `
 A ) ) )
7662, 75eqbrtrd 4174 . . 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 ) ) )
7776rexlimdvaa 2775 . 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 ) ) ) )
78 ralnex 2660 . . 3  |-  ( A. x  e.  A  -.  ( F `  x )  =  0  <->  -.  E. x  e.  A  ( F `  x )  =  0 )
79 simpl1 960 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A  e.  Fin )
80 simpl2 961 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A  =/=  (/) )
81 simpl3 962 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F : A --> ( 0 [,)  +oo ) )
82 ffn 5532 . . . . . . 7  |-  ( F : A --> ( 0 [,)  +oo )  ->  F  Fn  A )
8381, 82syl 16 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F  Fn  A )
84 ffvelrn 5808 . . . . . . . . . . . . . . . 16  |-  ( ( F : A --> ( 0 [,)  +oo )  /\  x  e.  A )  ->  ( F `  x )  e.  ( 0 [,)  +oo ) )
85843ad2antl3 1121 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  ( 0 [,)  +oo ) )
86 elrege0 10940 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  x )  e.  RR  /\  0  <_ 
( F `  x
) ) )
8785, 86sylib 189 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( ( F `  x )  e.  RR  /\  0  <_  ( F `  x ) ) )
8887simprd 450 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  0  <_  ( F `  x ) )
8987simpld 446 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
90 leloe 9095 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  ( F `  x )  e.  RR )  -> 
( 0  <_  ( F `  x )  <->  ( 0  <  ( F `
 x )  \/  0  =  ( F `
 x ) ) ) )
9113, 89, 90sylancr 645 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( 0  <_  ( F `  x )  <->  ( 0  <  ( F `
 x )  \/  0  =  ( F `
 x ) ) ) )
9288, 91mpbid 202 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( 0  <  ( F `  x )  \/  0  =  ( F `  x )
) )
9392ord 367 . . . . . . . . . . 11  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  0  < 
( F `  x
)  ->  0  =  ( F `  x ) ) )
94 eqcom 2390 . . . . . . . . . . 11  |-  ( 0  =  ( F `  x )  <->  ( F `  x )  =  0 )
9593, 94syl6ib 218 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  0  < 
( F `  x
)  ->  ( F `  x )  =  0 ) )
9695con1d 118 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =  0  ->  0  <  ( F `  x ) ) )
97 elrp 10547 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  RR+  <->  ( ( F `
 x )  e.  RR  /\  0  < 
( F `  x
) ) )
9897baib 872 . . . . . . . . . 10  |-  ( ( F `  x )  e.  RR  ->  (
( F `  x
)  e.  RR+  <->  0  <  ( F `  x ) ) )
9989, 98syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( ( F `  x )  e.  RR+  <->  0  <  ( F `  x ) ) )
10096, 99sylibrd 226 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =  0  ->  ( F `  x )  e.  RR+ ) )
101100ralimdva 2728 . . . . . . 7  |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A
--> ( 0 [,)  +oo ) )  ->  ( A. x  e.  A  -.  ( F `  x
)  =  0  ->  A. x  e.  A  ( F `  x )  e.  RR+ ) )
102101imp 419 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  A. x  e.  A  ( F `  x )  e.  RR+ )
103 ffnfv 5834 . . . . . 6  |-  ( F : A --> RR+  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  RR+ ) )
10483, 102, 103sylanbrc 646 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  A. x  e.  A  -.  ( F `  x )  =  0 )  ->  F : A --> RR+ )
1051, 79, 80, 104amgmlem 20696 . . . 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 ) ) )
106105ex 424 . . 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 ) ) ) )
10778, 106syl5bir 210 . 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 ) ) ) )
10877, 107pm2.61d 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 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   _Vcvv 2900    \ cdif 3261    u. cun 3262    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   class class class wbr 4154    e. cmpt 4208    |` cres 4821    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   Fincfn 7046   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    x. cmul 8929    +oocpnf 9051   RR*cxr 9053    < clt 9054    <_ cle 9055    / cdiv 9610   NNcn 9933   RR+crp 10545   [,)cico 10851   #chash 11546    gsumg cgsu 13652   Mndcmnd 14612  SubMndcsubmnd 14665  CMndccmn 15340  mulGrpcmgp 15576   Ringcrg 15588   CRingccrg 15589  ℂfldccnfld 16627    ^ c ccxp 20321
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-mhm 14666  df-submnd 14667  df-grp 14740  df-minusg 14741  df-mulg 14743  df-subg 14869  df-ghm 14932  df-gim 14974  df-cntz 15044  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-cring 15592  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-dvr 15716  df-drng 15765  df-subrg 15794  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-cmp 17373  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322  df-cxp 20323
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