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Theorem sge0tsms 38010
Description: Σ^ applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0tsms.g  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
sge0tsms.x  |-  ( ph  ->  X  e.  V )
sge0tsms.f  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
Assertion
Ref Expression
sge0tsms  |-  ( ph  ->  (Σ^ `  F )  e.  ( G tsums  F ) )

Proof of Theorem sge0tsms
Dummy variables  s 
t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2422 . . . 4  |-  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  =  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )
21a1i 11 . . 3  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  =  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
3 xrltso 11441 . . . . . 6  |-  <  Or  RR*
43supex 7980 . . . . 5  |-  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  e.  _V
54a1i 11 . . . 4  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  e. 
_V )
6 elsncg 4019 . . . 4  |-  ( sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  e.  _V  ->  ( sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  e.  { sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  ) } 
<->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  =  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  ) ) )
75, 6syl 17 . . 3  |-  ( ph  ->  ( sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  e.  { sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  ) } 
<->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  =  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  ) ) )
82, 7mpbird 235 . 2  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  e. 
{ sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  ) } )
9 sge0tsms.x . . . . . . 7  |-  ( ph  ->  X  e.  V )
109adantr 466 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  X  e.  V )
11 sge0tsms.f . . . . . . 7  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
1211adantr 466 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F : X
--> ( 0 [,] +oo ) )
13 simpr 462 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
1410, 12, 13sge0pnfval 38003 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  = +oo )
15 ffn 5743 . . . . . . . . . 10  |-  ( F : X --> ( 0 [,] +oo )  ->  F  Fn  X )
1611, 15syl 17 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
1716adantr 466 . . . . . . . 8  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F  Fn  X )
18 fvelrnb 5925 . . . . . . . 8  |-  ( F  Fn  X  ->  ( +oo  e.  ran  F  <->  E. y  e.  X  ( F `  y )  = +oo ) )
1917, 18syl 17 . . . . . . 7  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( +oo  e.  ran  F  <->  E. y  e.  X  ( F `  y )  = +oo ) )
2013, 19mpbid 213 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  E. y  e.  X  ( F `  y )  = +oo )
21 iccssxr 11718 . . . . . . . . . . . . . 14  |-  ( 0 [,] +oo )  C_  RR*
22 sge0tsms.g . . . . . . . . . . . . . . 15  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
23 simpr 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  ( ~P X  i^i  Fin ) )
2411adantr 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
25 elinel1 3651 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  ~P X )
26 elpwi 3988 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~P X  ->  x  C_  X )
2725, 26syl 17 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  C_  X )
2827adantl 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  C_  X )
29 fssres 5763 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> ( 0 [,] +oo )  /\  x  C_  X )  -> 
( F  |`  x
) : x --> ( 0 [,] +oo ) )
3024, 28, 29syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
31 elinel2 3652 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  Fin )
3231adantl 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
33 0red 9645 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  0  e.  RR )
3430, 32, 33fdmfifsupp 7896 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) finSupp  0
)
3522, 23, 30, 34gsumge0cl 38001 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  x
) )  e.  ( 0 [,] +oo )
)
3621, 35sseldi 3462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  x
) )  e.  RR* )
3736ralrimiva 2839 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  ( ~P X  i^i  Fin ) ( G  gsumg  ( F  |`  x ) )  e. 
RR* )
38373ad2ant1 1026 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  A. x  e.  ( ~P X  i^i  Fin ) ( G  gsumg  ( F  |`  x ) )  e. 
RR* )
39 eqid 2422 . . . . . . . . . . . 12  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  x ) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) )
4039rnmptss 6064 . . . . . . . . . . 11  |-  ( A. x  e.  ( ~P X  i^i  Fin ) ( G  gsumg  ( F  |`  x
) )  e.  RR*  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) 
C_  RR* )
4138, 40syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) 
C_  RR* )
42 snelpwi 4663 . . . . . . . . . . . . . 14  |-  ( y  e.  X  ->  { y }  e.  ~P X
)
43 snfi 7654 . . . . . . . . . . . . . . 15  |-  { y }  e.  Fin
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  X  ->  { y }  e.  Fin )
4542, 44elind 3650 . . . . . . . . . . . . 13  |-  ( y  e.  X  ->  { y }  e.  ( ~P X  i^i  Fin )
)
46453ad2ant2 1027 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  { y }  e.  ( ~P X  i^i  Fin ) )
4711adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  F : X --> ( 0 [,] +oo ) )
48 snssi 4141 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  X  ->  { y }  C_  X )
4948adantl 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  { y }  C_  X )
5047, 49fssresd 5764 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } ) : { y } --> ( 0 [,] +oo ) )
5150feqmptd 5931 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } )  =  ( x  e.  { y } 
|->  ( ( F  |`  { y } ) `
 x ) ) )
52 fvres 5892 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { y }  ->  ( ( F  |`  { y } ) `
 x )  =  ( F `  x
) )
5352mpteq2ia 4503 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  { y } 
|->  ( ( F  |`  { y } ) `
 x ) )  =  ( x  e. 
{ y }  |->  ( F `  x ) )
5453a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  X )  ->  (
x  e.  { y }  |->  ( ( F  |`  { y } ) `
 x ) )  =  ( x  e. 
{ y }  |->  ( F `  x ) ) )
5551, 54eqtrd 2463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } )  =  ( x  e.  { y } 
|->  ( F `  x
) ) )
5655oveq2d 6318 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X )  ->  ( G  gsumg  ( F  |`  { y } ) )  =  ( G  gsumg  ( x  e.  {
y }  |->  ( F `
 x ) ) ) )
57563adant3 1025 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( G  gsumg  ( F  |`  { y } ) )  =  ( G 
gsumg  ( x  e.  { y }  |->  ( F `  x ) ) ) )
58 xrge0cmn 18998 . . . . . . . . . . . . . . . . 17  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
5922, 58eqeltri 2506 . . . . . . . . . . . . . . . 16  |-  G  e. CMnd
60 cmnmnd 17433 . . . . . . . . . . . . . . . 16  |-  ( G  e. CMnd  ->  G  e.  Mnd )
6159, 60ax-mp 5 . . . . . . . . . . . . . . 15  |-  G  e. 
Mnd
6261a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  G  e.  Mnd )
63 simp2 1006 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  y  e.  X
)
6411ffvelrnda 6034 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ( 0 [,] +oo ) )
65643adant3 1025 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  e.  ( 0 [,] +oo )
)
66 df-ss 3450 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] +oo )  C_ 
RR* 
<->  ( ( 0 [,] +oo )  i^i  RR* )  =  ( 0 [,] +oo ) )
6721, 66mpbi 211 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 [,] +oo )  i^i  RR* )  =  ( 0 [,] +oo )
6867eqcomi 2435 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] +oo )  =  ( ( 0 [,] +oo )  i^i  RR* )
69 ovex 6330 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,] +oo )  e. 
_V
70 xrsbas 18972 . . . . . . . . . . . . . . . . . 18  |-  RR*  =  ( Base `  RR*s )
7122, 70ressbas 15167 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 [,] +oo )  e.  _V  ->  ( (
0 [,] +oo )  i^i  RR* )  =  (
Base `  G )
)
7269, 71ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] +oo )  i^i  RR* )  =  (
Base `  G )
7368, 72eqtri 2451 . . . . . . . . . . . . . . 15  |-  ( 0 [,] +oo )  =  ( Base `  G
)
74 fveq2 5878 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
7573, 74gsumsn 17575 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  y  e.  X  /\  ( F `  y )  e.  ( 0 [,] +oo ) )  ->  ( G  gsumg  ( x  e.  {
y }  |->  ( F `
 x ) ) )  =  ( F `
 y ) )
7662, 63, 65, 75syl3anc 1264 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( G  gsumg  ( x  e.  { y } 
|->  ( F `  x
) ) )  =  ( F `  y
) )
77 simp3 1007 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  = +oo )
7857, 76, 773eqtrrd 2468 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  =  ( G  gsumg  ( F  |`  { y } ) ) )
79 reseq2 5116 . . . . . . . . . . . . . . 15  |-  ( x  =  { y }  ->  ( F  |`  x )  =  ( F  |`  { y } ) )
8079oveq2d 6318 . . . . . . . . . . . . . 14  |-  ( x  =  { y }  ->  ( G  gsumg  ( F  |`  x ) )  =  ( G  gsumg  ( F  |`  { y } ) ) )
8180eqeq2d 2436 . . . . . . . . . . . . 13  |-  ( x  =  { y }  ->  ( +oo  =  ( G  gsumg  ( F  |`  x
) )  <-> +oo  =  ( G  gsumg  ( F  |`  { y } ) ) ) )
8281rspcev 3182 . . . . . . . . . . . 12  |-  ( ( { y }  e.  ( ~P X  i^i  Fin )  /\ +oo  =  ( G  gsumg  ( F  |`  { y } ) ) )  ->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  ( G 
gsumg  ( F  |`  x ) ) )
8346, 78, 82syl2anc 665 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  ( G 
gsumg  ( F  |`  x ) ) )
84 pnfxr 11413 . . . . . . . . . . . . 13  |- +oo  e.  RR*
8584a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  RR* )
8639elrnmpt 5097 . . . . . . . . . . . 12  |-  ( +oo  e.  RR*  ->  ( +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) )  <->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  ( G  gsumg  ( F  |`  x )
) ) )
8785, 86syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) )  <->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  ( G  gsumg  ( F  |`  x )
) ) )
8883, 87mpbird 235 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) )
89 supxrpnf 11605 . . . . . . . . . 10  |-  ( ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) 
C_  RR*  /\ +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) )  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  = +oo )
9041, 88, 89syl2anc 665 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
91903exp 1204 . . . . . . . 8  |-  ( ph  ->  ( y  e.  X  ->  ( ( F `  y )  = +oo  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo ) ) )
9291adantr 466 . . . . . . 7  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( y  e.  X  ->  ( ( F `  y )  = +oo  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  = +oo ) ) )
9392rexlimdv 2915 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( E. y  e.  X  ( F `  y )  = +oo  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  = +oo ) )
9420, 93mpd 15 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  = +oo )
9514, 94eqtr4d 2466 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
969adantr 466 . . . . . 6  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  X  e.  V )
9711adantr 466 . . . . . . 7  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,] +oo ) )
98 simpr 462 . . . . . . 7  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  -. +oo  e.  ran  F )
9997, 98fge0iccico 38000 . . . . . 6  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,) +oo ) )
10096, 99sge0reval 38002 . . . . 5  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  (Σ^ `  F
)  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
10124, 28feqresmpt 5932 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x )  =  ( y  e.  x  |->  ( F `  y
) ) )
102101adantlr 719 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x )  =  ( y  e.  x  |->  ( F `  y
) ) )
103102oveq2d 6318 . . . . . . . . 9  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  x
) )  =  ( G  gsumg  ( y  e.  x  |->  ( F `  y
) ) ) )
10422fveq2i 5881 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
105 eqid 2422 . . . . . . . . . . . . . 14  |-  ( RR*ss  ( 0 [,] +oo ) )  =  (
RR*ss  ( 0 [,] +oo ) )
106 xrsadd 18973 . . . . . . . . . . . . . 14  |-  +e 
=  ( +g  `  RR*s
)
107105, 106ressplusg 15227 . . . . . . . . . . . . 13  |-  ( ( 0 [,] +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) ) )
10869, 107ax-mp 5 . . . . . . . . . . . 12  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
109108eqcomi 2435 . . . . . . . . . . 11  |-  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )  =  +e
110104, 109eqtr2i 2452 . . . . . . . . . 10  |-  +e 
=  ( +g  `  G
)
11122oveq1i 6312 . . . . . . . . . . 11  |-  ( Gs  ( 0 [,) +oo )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )
112 icossicc 11722 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
11369, 112pm3.2i 456 . . . . . . . . . . . 12  |-  ( ( 0 [,] +oo )  e.  _V  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )
114 ressabs 15176 . . . . . . . . . . . 12  |-  ( ( ( 0 [,] +oo )  e.  _V  /\  (
0 [,) +oo )  C_  ( 0 [,] +oo ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) ) )
115113, 114ax-mp 5 . . . . . . . . . . 11  |-  ( (
RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
116111, 115eqtr2i 2452 . . . . . . . . . 10  |-  ( RR*ss  ( 0 [,) +oo ) )  =  ( Gs  ( 0 [,) +oo ) )
11759elexi 3091 . . . . . . . . . . 11  |-  G  e. 
_V
118117a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  G  e.  _V )
119 simpr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  ( ~P X  i^i  Fin ) )
120112a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
0 [,) +oo )  C_  ( 0 [,] +oo ) )
121 0xr 9688 . . . . . . . . . . . . 13  |-  0  e.  RR*
122121a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  0  e.  RR* )
12384a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  -> +oo  e.  RR* )
12497ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  F : X --> ( 0 [,] +oo ) )
12527sselda 3464 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( ~P X  i^i  Fin )  /\  y  e.  x
)  ->  y  e.  X )
126125adantll 718 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  X )
127124, 126ffvelrnd 6035 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( 0 [,] +oo ) )
12821, 127sseldi 3462 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  RR* )
129 iccgelb 11692 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `
 y )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( F `  y
) )
130122, 123, 127, 129syl3anc 1264 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  0  <_  ( F `  y
) )
131 id 23 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F `  y )  = +oo  ->  ( F `  y )  = +oo )
132131eqcomd 2430 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  y )  = +oo  -> +oo  =  ( F `  y ) )
133132adantl 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  -> +oo  =  ( F `  y )
)
134 ffun 5745 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : X --> ( 0 [,] +oo )  ->  Fun  F )
13511, 134syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  Fun  F )
136135ad2antrr 730 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  Fun  F )
13723, 125sylan 473 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  X )
138 fdm 5747 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( F : X --> ( 0 [,] +oo )  ->  dom  F  =  X )
13911, 138syl 17 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  dom  F  =  X )
140139eqcomd 2430 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  X  =  dom  F
)
141140ad2antrr 730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  X  =  dom  F )
142137, 141eleqtrd 2512 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  dom  F )
143 fvelrn 6027 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  ran  F
)
144136, 142, 143syl2anc 665 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ran  F )
145144adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  ->  ( F `  y )  e.  ran  F )
146133, 145eqeltrd 2510 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  F )
147146adantlllr 37222 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\ 
-. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  F )
14898ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\ 
-. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  /\  ( F `  y )  = +oo )  ->  -. +oo  e.  ran  F )
149147, 148pm2.65da 578 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  -.  ( F `  y )  = +oo )
150149neqned 2627 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  =/= +oo )
151 ge0xrre 37464 . . . . . . . . . . . . . 14  |-  ( ( ( F `  y
)  e.  ( 0 [,] +oo )  /\  ( F `  y )  =/= +oo )  -> 
( F `  y
)  e.  RR )
152127, 150, 151syl2anc 665 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  RR )
153152ltpnfd 11424 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  < +oo )
154122, 123, 128, 130, 153elicod 11686 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( 0 [,) +oo ) )
155 eqid 2422 . . . . . . . . . . 11  |-  ( y  e.  x  |->  ( F `
 y ) )  =  ( y  e.  x  |->  ( F `  y ) )
156154, 155fmptd 6058 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
y  e.  x  |->  ( F `  y ) ) : x --> ( 0 [,) +oo ) )
157 0e0icopnf 11743 . . . . . . . . . . 11  |-  0  e.  ( 0 [,) +oo )
158157a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  0  e.  ( 0 [,) +oo ) )
15921sseli 3460 . . . . . . . . . . . 12  |-  ( y  e.  ( 0 [,] +oo )  ->  y  e. 
RR* )
160 xaddid2 11534 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( 0 +e y )  =  y )
161 xaddid1 11533 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( y +e 0 )  =  y )
162160, 161jca 534 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( 0 +e y )  =  y  /\  ( y +e 0 )  =  y ) )
163159, 162syl 17 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,] +oo )  ->  ( ( 0 +e y )  =  y  /\  ( y +e 0 )  =  y ) )
164163adantl 467 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  ( 0 [,] +oo ) )  ->  (
( 0 +e
y )  =  y  /\  ( y +e 0 )  =  y ) )
16573, 110, 116, 118, 119, 120, 156, 158, 164gsumress 16507 . . . . . . . . 9  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( y  e.  x  |->  ( F `  y
) ) )  =  ( ( RR*ss  (
0 [,) +oo )
)  gsumg  ( y  e.  x  |->  ( F `  y
) ) ) )
166 rege0subm 19012 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  e.  (SubMnd ` fld )
167166a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
0 [,) +oo )  e.  (SubMnd ` fld ) )
168 eqid 2422 . . . . . . . . . . . 12  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
169119, 167, 156, 168gsumsubm 16608 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (fld  gsumg  ( y  e.  x  |->  ( F `  y
) ) )  =  ( (flds  ( 0 [,) +oo )
)  gsumg  ( y  e.  x  |->  ( F `  y
) ) ) )
170 eqidd 2423 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
(flds  (
0 [,) +oo )
)  gsumg  ( y  e.  x  |->  ( F `  y
) ) )  =  ( (flds  ( 0 [,) +oo )
)  gsumg  ( y  e.  x  |->  ( F `  y
) ) ) )
171 vex 3084 . . . . . . . . . . . . . 14  |-  x  e. 
_V
172171mptex 6148 . . . . . . . . . . . . 13  |-  ( y  e.  x  |->  ( F `
 y ) )  e.  _V
173172a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
y  e.  x  |->  ( F `  y ) )  e.  _V )
174 ovex 6330 . . . . . . . . . . . . 13  |-  (flds  ( 0 [,) +oo ) )  e.  _V
175174a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (flds  ( 0 [,) +oo ) )  e.  _V )
176 ovex 6330 . . . . . . . . . . . . 13  |-  ( RR*ss  ( 0 [,) +oo ) )  e.  _V
177176a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( RR*ss  ( 0 [,) +oo ) )  e.  _V )
178 rge0ssre 11741 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,) +oo )  C_  RR
179 ax-resscn 9597 . . . . . . . . . . . . . . . . 17  |-  RR  C_  CC
180178, 179sstri 3473 . . . . . . . . . . . . . . . 16  |-  ( 0 [,) +oo )  C_  CC
181 cnfldbas 18962 . . . . . . . . . . . . . . . . 17  |-  CC  =  ( Base ` fld )
182168, 181ressbas2 15168 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
183180, 182ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
184183eqcomi 2435 . . . . . . . . . . . . . 14  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  ( 0 [,) +oo )
185112, 21sstri 3473 . . . . . . . . . . . . . . 15  |-  ( 0 [,) +oo )  C_  RR*
186 eqid 2422 . . . . . . . . . . . . . . . 16  |-  ( RR*ss  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
187186, 70ressbas2 15168 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,) +oo )  C_ 
RR*  ->  ( 0 [,) +oo )  =  ( Base `  ( RR*ss  (
0 [,) +oo )
) ) )
188185, 187ax-mp 5 . . . . . . . . . . . . . 14  |-  ( 0 [,) +oo )  =  ( Base `  ( RR*ss  ( 0 [,) +oo ) ) )
189184, 188eqtri 2451 . . . . . . . . . . . . 13  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  (
Base `  ( RR*ss  ( 0 [,) +oo ) ) )
190189a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( Base `  (flds  ( 0 [,) +oo )
) )  =  (
Base `  ( RR*ss  ( 0 [,) +oo ) ) ) )
191 rge0srg 19025 . . . . . . . . . . . . . . 15  |-  (flds  ( 0 [,) +oo ) )  e. SRing
192191a1i 11 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  (flds  ( 0 [,) +oo ) )  e. SRing )
193 simpl 458 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
194 simpr 462 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
195 eqid 2422 . . . . . . . . . . . . . . 15  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  (
Base `  (flds  ( 0 [,) +oo ) ) )
196 eqid 2422 . . . . . . . . . . . . . . 15  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
197195, 196srgacl 17745 . . . . . . . . . . . . . 14  |-  ( ( (flds  ( 0 [,) +oo )
)  e. SRing  /\  s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  (
s ( +g  `  (flds  ( 0 [,) +oo ) ) ) t )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
198192, 193, 194, 197syl3anc 1264 . . . . . . . . . . . . 13  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  (
s ( +g  `  (flds  ( 0 [,) +oo ) ) ) t )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
199198adantl 467 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  (
s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  -> 
( s ( +g  `  (flds  ( 0 [,) +oo )
) ) t )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
200178a1i 11 . . . . . . . . . . . . . . . 16  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  ( 0 [,) +oo )  C_  RR )
201 id 23 . . . . . . . . . . . . . . . . 17  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
202201, 184syl6eleq 2520 . . . . . . . . . . . . . . . 16  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  ( 0 [,) +oo ) )
203200, 202sseldd 3465 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  RR )
204203adantr 466 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  s  e.  RR )
205178a1i 11 . . . . . . . . . . . . . . . 16  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  ( 0 [,) +oo )  C_  RR )
206 id 23 . . . . . . . . . . . . . . . . 17  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
207206, 184syl6eleq 2520 . . . . . . . . . . . . . . . 16  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  ( 0 [,) +oo ) )
208205, 207sseldd 3465 . . . . . . . . . . . . . . 15  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  RR )
209208adantl 467 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  t  e.  RR )
210 rexadd 11526 . . . . . . . . . . . . . . . 16  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s +e
t )  =  ( s  +  t ) )
211210eqcomd 2430 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s  +  t )  =  ( s +e t ) )
212166elexi 3091 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,) +oo )  e. 
_V
213 cnfldadd 18963 . . . . . . . . . . . . . . . . . . . . 21  |-  +  =  ( +g  ` fld )
214168, 213ressplusg 15227 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
215212, 214ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
216215, 213eqtr3i 2453 . . . . . . . . . . . . . . . . . 18  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  ( +g  ` fld )
217216, 213eqtr4i 2454 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  +
218217oveqi 6315 . . . . . . . . . . . . . . . 16  |-  ( s ( +g  `  (flds  ( 0 [,) +oo ) ) ) t )  =  ( s  +  t )
219218a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s ( +g  `  (flds  ( 0 [,) +oo )
) ) t )  =  ( s  +  t ) )
220186, 106ressplusg 15227 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) )
221212, 220ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )
222221eqcomi 2435 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )  =  +e
223222oveqi 6315 . . . . . . . . . . . . . . . 16  |-  ( s ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) t )  =  ( s +e t )
224223a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) t )  =  ( s +e
t ) )
225211, 219, 2243eqtr4d 2473 . . . . . . . . . . . . . 14  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s ( +g  `  (flds  ( 0 [,) +oo )
) ) t )  =  ( s ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) t ) )
226204, 209, 225syl2anc 665 . . . . . . . . . . . . 13  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  (
s ( +g  `  (flds  ( 0 [,) +oo ) ) ) t )  =  ( s ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) t ) )
227226adantl 467 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  (
s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  -> 
( s ( +g  `  (flds  ( 0 [,) +oo )
) ) t )  =  ( s ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) t ) )
228 funmpt 5634 . . . . . . . . . . . . 13  |-  Fun  (
y  e.  x  |->  ( F `  y ) )
229228a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  Fun  ( y  e.  x  |->  ( F `  y
) ) )
230154, 183syl6eleq 2520 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
231230ralrimiva 2839 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  A. y  e.  x  ( F `  y )  e.  (
Base `  (flds  ( 0 [,) +oo ) ) ) )
232155rnmptss 6064 . . . . . . . . . . . . 13  |-  ( A. y  e.  x  ( F `  y )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  ran  ( y  e.  x  |->  ( F `
 y ) ) 
C_  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
233231, 232syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ran  ( y  e.  x  |->  ( F `  y
) )  C_  ( Base `  (flds  ( 0 [,) +oo )
) ) )
234173, 175, 177, 190, 199, 227, 229, 233gsumpropd2 16505 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
(flds  (
0 [,) +oo )
)  gsumg  ( y  e.  x  |->  ( F `  y
) ) )  =  ( ( RR*ss  (
0 [,) +oo )
)  gsumg  ( y  e.  x  |->  ( F `  y
) ) ) )
235169, 170, 2343eqtrd 2467 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (fld  gsumg  ( y  e.  x  |->  ( F `  y
) ) )  =  ( ( RR*ss  (
0 [,) +oo )
)  gsumg  ( y  e.  x  |->  ( F `  y
) ) ) )
23631adantl 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
237152recnd 9670 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  CC )
238236, 237gsumfsum 19022 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (fld  gsumg  ( y  e.  x  |->  ( F `  y
) ) )  = 
sum_ y  e.  x  ( F `  y ) )
239235, 238eqtr3d 2465 . . . . . . . . 9  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,) +oo ) )  gsumg  ( y  e.  x  |->  ( F `  y
) ) )  = 
sum_ y  e.  x  ( F `  y ) )
240103, 165, 2393eqtrrd 2468 . . . . . . . 8  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sum_ y  e.  x  ( F `  y )  =  ( G  gsumg  ( F  |`  x
) ) )
241240mpteq2dva 4507 . . . . . . 7  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  x ) ) ) )
242241rneqd 5078 . . . . . 6  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) )
243242supeq1d 7963 . . . . 5  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
244100, 243eqtrd 2463 . . . 4  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  (Σ^ `  F
)  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
24595, 244pm2.61dan 798 . . 3  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
24622, 9, 11, 1xrge0tsms 21839 . . 3  |-  ( ph  ->  ( G tsums  F )  =  { sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  ) } )
247245, 246eleq12d 2504 . 2  |-  ( ph  ->  ( (Σ^ `  F )  e.  ( G tsums  F )  <->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )  e.  { sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  ) } ) )
2488, 247mpbird 235 1  |-  ( ph  ->  (Σ^ `  F )  e.  ( G tsums  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081    i^i cin 3435    C_ wss 3436   ~Pcpw 3979   {csn 3996   class class class wbr 4420    |-> cmpt 4479   dom cdm 4850   ran crn 4851    |` cres 4852   Fun wfun 5592    Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302   Fincfn 7574   supcsup 7957   CCcc 9538   RRcr 9539   0cc0 9540    + caddc 9543   +oocpnf 9673   RR*cxr 9675    < clt 9676    <_ cle 9677   +ecxad 11408   [,)cico 11638   [,]cicc 11639   sum_csu 13740   Basecbs 15109   ↾s cress 15110   +g cplusg 15178    gsumg cgsu 15327   RR*scxrs 15386   Mndcmnd 16523  SubMndcsubmnd 16569  CMndccmn 17418  SRingcsrg 17727  ℂfldccnfld 18958   tsums ctsu 21127  Σ^csumge0 37992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xadd 11411  df-ioo 11640  df-ioc 11641  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-ordt 15387  df-xrs 15388  df-mre 15480  df-mrc 15481  df-acs 15483  df-ps 16434  df-tsr 16435  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-grp 16661  df-minusg 16662  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-srg 17728  df-ring 17770  df-cring 17771  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-ntr 20022  df-nei 20101  df-cn 20230  df-haus 20318  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-tsms 21128  df-sumge0 37993
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator