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Theorem amgm 20821
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 16699 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
31, 2mgpbas 15646 . . . . . . . 8  |-  CC  =  ( Base `  M )
4 cnfld1 16718 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
51, 4rngidval 15658 . . . . . . . 8  |-  1  =  ( 0g `  M )
6 cnfldmul 16701 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
71, 6mgpplusg 15644 . . . . . . . 8  |-  x.  =  ( +g  `  M )
8 cncrng 16714 . . . . . . . . 9  |-fld  e.  CRing
91crngmgp 15664 . . . . . . . . 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 9083 . . . . . . . . . . 11  |-  0  e.  RR
14 pnfxr 10705 . . . . . . . . . . 11  |-  +oo  e.  RR*
15 icossre 10983 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
1613, 14, 15mp2an 654 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  RR
17 ax-resscn 9039 . . . . . . . . . 10  |-  RR  C_  CC
1816, 17sstri 3349 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  CC
19 fss 5591 . . . . . . . . 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 14765 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( `' F " ( _V  \  { 1 } ) )  e.  Fin )
22 disjdif 3692 . . . . . . . . 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 3696 . . . . . . . . 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 3935 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  { x }  C_  A )
27 ssequn1 3509 . . . . . . . . . 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 2480 . . . . . . . 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 15522 . . . . . . 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 5772 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F  |` 
{ x } )  =  ( y  e. 
{ x }  |->  ( F `  y ) ) )
3231oveq2d 6089 . . . . . . . . 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 16715 . . . . . . . . . . 11  |-fld  e.  Ring
341rngmgp 15662 . . . . . . . . . . 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 5863 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( F `  x )  e.  CC )
37 fveq2 5720 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
383, 37gsumsn 15535 . . . . . . . . . 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 2471 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  { x } ) )  =  0 )
4241oveq1d 6088 . . . . . . 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 7331 . . . . . . . . . 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 3466 . . . . . . . . . 10  |-  ( A 
\  { x }
)  C_  A
46 fssres 5602 . . . . . . . . . 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 14765 . . . . . . . . 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 15513 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  ( F  |`  ( A  \  { x } ) ) )  e.  CC )
5049mul02d 9256 . . . . . . 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 2471 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( M  gsumg  F )  =  0 )
5251oveq1d 6088 . . . . 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 11637 . . . . . . . . . 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 10008 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  CC )
5856nnne0d 10036 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  =/=  0 )
5957, 58reccld 9775 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  e.  CC )
6057, 58recne0d 9776 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 1  /  ( # `  A
) )  =/=  0
)
6159, 600cxpd 20593 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( 0  ^ c  ( 1  /  ( # `  A
) ) )  =  0 )
6252, 61eqtrd 2467 . . . 4  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( ( M  gsumg  F )  ^ c 
( 1  /  ( # `
 A ) ) )  =  0 )
63 cnfld0 16717 . . . . . . 7  |-  0  =  ( 0g ` fld )
64 rngcmn 15686 . . . . . . . 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 16747 . . . . . . . 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 14765 . . . . . . 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 15516 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  (fld  gsumg  F )  e.  ( 0 [,)  +oo )
)
70 elrege0 10999 . . . . . 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 10007 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  ( # `  A
)  e.  RR )
7356nngt0d 10035 . . . . 5  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) 
+oo ) )  /\  ( x  e.  A  /\  ( F `  x
)  =  0 ) )  ->  0  <  (
# `  A )
)
74 divge0 9871 . . . . 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 4224 . . 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 2823 . 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 2707 . . 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 5583 . . . . . . 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 5860 . . . . . . . . . . . . . . . 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 10999 . . . . . . . . . . . . . . 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 9153 . . . . . . . . . . . . . 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 2437 . . . . . . . . . . 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 10606 . . . . . . . . . . 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 2776 . . . . . . 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 5886 . . . . . 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 20820 . . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204    e. cmpt 4258    |` cres 4872    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113    / cdiv 9669   NNcn 9992   RR+crp 10604   [,)cico 10910   #chash 11610    gsumg cgsu 13716   Mndcmnd 14676  SubMndcsubmnd 14729  CMndccmn 15404  mulGrpcmgp 15640   Ringcrg 15652   CRingccrg 15653  ℂfldccnfld 16695    ^ c ccxp 20445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-mulg 14807  df-subg 14933  df-ghm 14996  df-gim 15038  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-subrg 15858  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-cmp 17442  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-cxp 20447
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