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Theorem sge0tsms 38222
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 2451 . . . 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 11440 . . . . . 6  |-  <  Or  RR*
43supex 7977 . . . . 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 3991 . . . 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 236 . 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 467 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  X  e.  V )
11 sge0tsms.f . . . . . . 7  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
1211adantr 467 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F : X
--> ( 0 [,] +oo ) )
13 simpr 463 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
1410, 12, 13sge0pnfval 38215 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  = +oo )
15 ffn 5728 . . . . . . . . . 10  |-  ( F : X --> ( 0 [,] +oo )  ->  F  Fn  X )
1611, 15syl 17 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
1716adantr 467 . . . . . . . 8  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F  Fn  X )
18 fvelrnb 5912 . . . . . . . 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 214 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  E. y  e.  X  ( F `  y )  = +oo )
21 iccssxr 11717 . . . . . . . . . . . . . 14  |-  ( 0 [,] +oo )  C_  RR*
22 sge0tsms.g . . . . . . . . . . . . . . 15  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
23 simpr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  ( ~P X  i^i  Fin ) )
2411adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
25 elinel1 3619 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  ~P X )
26 elpwi 3960 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~P X  ->  x  C_  X )
2725, 26syl 17 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  C_  X )
2827adantl 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  C_  X )
29 fssres 5749 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> ( 0 [,] +oo )  /\  x  C_  X )  -> 
( F  |`  x
) : x --> ( 0 [,] +oo ) )
3024, 28, 29syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
31 elinel2 3620 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  Fin )
3231adantl 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
33 0red 9644 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  0  e.  RR )
3430, 32, 33fdmfifsupp 7893 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) finSupp  0
)
3522, 23, 30, 34gsumge0cl 38213 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  x
) )  e.  ( 0 [,] +oo )
)
3621, 35sseldi 3430 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  x
) )  e.  RR* )
3736ralrimiva 2802 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  ( ~P X  i^i  Fin ) ( G  gsumg  ( F  |`  x ) )  e. 
RR* )
38373ad2ant1 1029 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  A. x  e.  ( ~P X  i^i  Fin ) ( G  gsumg  ( F  |`  x ) )  e. 
RR* )
39 eqid 2451 . . . . . . . . . . . 12  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  x ) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) )
4039rnmptss 6052 . . . . . . . . . . 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 4645 . . . . . . . . . . . . . 14  |-  ( y  e.  X  ->  { y }  e.  ~P X
)
43 snfi 7650 . . . . . . . . . . . . . . 15  |-  { y }  e.  Fin
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  X  ->  { y }  e.  Fin )
4542, 44elind 3618 . . . . . . . . . . . . 13  |-  ( y  e.  X  ->  { y }  e.  ( ~P X  i^i  Fin )
)
46453ad2ant2 1030 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  { y }  e.  ( ~P X  i^i  Fin ) )
4711adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  F : X --> ( 0 [,] +oo ) )
48 snssi 4116 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  X  ->  { y }  C_  X )
4948adantl 468 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  { y }  C_  X )
5047, 49fssresd 5750 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } ) : { y } --> ( 0 [,] +oo ) )
5150feqmptd 5918 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } )  =  ( x  e.  { y } 
|->  ( ( F  |`  { y } ) `
 x ) ) )
52 fvres 5879 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { y }  ->  ( ( F  |`  { y } ) `
 x )  =  ( F `  x
) )
5352mpteq2ia 4485 . . . . . . . . . . . . . . . . 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 2485 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } )  =  ( x  e.  { y } 
|->  ( F `  x
) ) )
5655oveq2d 6306 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X )  ->  ( G  gsumg  ( F  |`  { y } ) )  =  ( G  gsumg  ( x  e.  {
y }  |->  ( F `
 x ) ) ) )
57563adant3 1028 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( G  gsumg  ( F  |`  { y } ) )  =  ( G 
gsumg  ( x  e.  { y }  |->  ( F `  x ) ) ) )
58 xrge0cmn 19010 . . . . . . . . . . . . . . . . 17  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
5922, 58eqeltri 2525 . . . . . . . . . . . . . . . 16  |-  G  e. CMnd
60 cmnmnd 17445 . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  y  e.  X
)
6411ffvelrnda 6022 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ( 0 [,] +oo ) )
65643adant3 1028 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  e.  ( 0 [,] +oo )
)
66 df-ss 3418 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] +oo )  C_ 
RR* 
<->  ( ( 0 [,] +oo )  i^i  RR* )  =  ( 0 [,] +oo ) )
6721, 66mpbi 212 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 [,] +oo )  i^i  RR* )  =  ( 0 [,] +oo )
6867eqcomi 2460 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] +oo )  =  ( ( 0 [,] +oo )  i^i  RR* )
69 ovex 6318 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,] +oo )  e. 
_V
70 xrsbas 18984 . . . . . . . . . . . . . . . . . 18  |-  RR*  =  ( Base `  RR*s )
7122, 70ressbas 15179 . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . 15  |-  ( 0 [,] +oo )  =  ( Base `  G
)
74 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
7573, 74gsumsn 17587 . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( G  gsumg  ( x  e.  { y } 
|->  ( F `  x
) ) )  =  ( F `  y
) )
77 simp3 1010 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  = +oo )
7857, 76, 773eqtrrd 2490 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  =  ( G  gsumg  ( F  |`  { y } ) ) )
79 reseq2 5100 . . . . . . . . . . . . . . 15  |-  ( x  =  { y }  ->  ( F  |`  x )  =  ( F  |`  { y } ) )
8079oveq2d 6306 . . . . . . . . . . . . . 14  |-  ( x  =  { y }  ->  ( G  gsumg  ( F  |`  x ) )  =  ( G  gsumg  ( F  |`  { y } ) ) )
8180eqeq2d 2461 . . . . . . . . . . . . 13  |-  ( x  =  { y }  ->  ( +oo  =  ( G  gsumg  ( F  |`  x
) )  <-> +oo  =  ( G  gsumg  ( F  |`  { y } ) ) ) )
8281rspcev 3150 . . . . . . . . . . . 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 667 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  ( G 
gsumg  ( F  |`  x ) ) )
84 pnfxr 11412 . . . . . . . . . . . . 13  |- +oo  e.  RR*
8584a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  RR* )
8639elrnmpt 5081 . . . . . . . . . . . 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 236 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) )
89 supxrpnf 11604 . . . . . . . . . 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 667 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
91903exp 1207 . . . . . . . 8  |-  ( ph  ->  ( y  e.  X  ->  ( ( F `  y )  = +oo  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo ) ) )
9291adantr 467 . . . . . . 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 2877 . . . . . 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 2488 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
969adantr 467 . . . . . 6  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  X  e.  V )
9711adantr 467 . . . . . . 7  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,] +oo ) )
98 simpr 463 . . . . . . 7  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  -. +oo  e.  ran  F )
9997, 98fge0iccico 38212 . . . . . 6  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,) +oo ) )
10096, 99sge0reval 38214 . . . . 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 5919 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x )  =  ( y  e.  x  |->  ( F `  y
) ) )
102101adantlr 721 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x )  =  ( y  e.  x  |->  ( F `  y
) ) )
103102oveq2d 6306 . . . . . . . . 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 5868 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
105 eqid 2451 . . . . . . . . . . . . . 14  |-  ( RR*ss  ( 0 [,] +oo ) )  =  (
RR*ss  ( 0 [,] +oo ) )
106 xrsadd 18985 . . . . . . . . . . . . . 14  |-  +e 
=  ( +g  `  RR*s
)
107105, 106ressplusg 15239 . . . . . . . . . . . . 13  |-  ( ( 0 [,] +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) ) )
10869, 107ax-mp 5 . . . . . . . . . . . 12  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
109108eqcomi 2460 . . . . . . . . . . 11  |-  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )  =  +e
110104, 109eqtr2i 2474 . . . . . . . . . 10  |-  +e 
=  ( +g  `  G
)
11122oveq1i 6300 . . . . . . . . . . 11  |-  ( Gs  ( 0 [,) +oo )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )
112 icossicc 11721 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
11369, 112pm3.2i 457 . . . . . . . . . . . 12  |-  ( ( 0 [,] +oo )  e.  _V  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )
114 ressabs 15188 . . . . . . . . . . . 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 2474 . . . . . . . . . 10  |-  ( RR*ss  ( 0 [,) +oo ) )  =  ( Gs  ( 0 [,) +oo ) )
11759elexi 3055 . . . . . . . . . . 11  |-  G  e. 
_V
118117a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  G  e.  _V )
119 simpr 463 . . . . . . . . . 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 9687 . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  F : X --> ( 0 [,] +oo ) )
12527sselda 3432 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( ~P X  i^i  Fin )  /\  y  e.  x
)  ->  y  e.  X )
126125adantll 720 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  X )
127124, 126ffvelrnd 6023 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( 0 [,] +oo ) )
12821, 127sseldi 3430 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  RR* )
129 iccgelb 11691 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `
 y )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( F `  y
) )
130122, 123, 127, 129syl3anc 1268 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  0  <_  ( F `  y
) )
131 id 22 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F `  y )  = +oo  ->  ( F `  y )  = +oo )
132131eqcomd 2457 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  y )  = +oo  -> +oo  =  ( F `  y ) )
133132adantl 468 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  -> +oo  =  ( F `  y )
)
134 ffun 5731 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : X --> ( 0 [,] +oo )  ->  Fun  F )
13511, 134syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  Fun  F )
136135ad2antrr 732 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  Fun  F )
13723, 125sylan 474 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  X )
138 fdm 5733 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( F : X --> ( 0 [,] +oo )  ->  dom  F  =  X )
13911, 138syl 17 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  dom  F  =  X )
140139eqcomd 2457 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  X  =  dom  F
)
141140ad2antrr 732 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  X  =  dom  F )
142137, 141eleqtrd 2531 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  dom  F )
143 fvelrn 6015 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  ran  F
)
144136, 142, 143syl2anc 667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ran  F )
145144adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  ->  ( F `  y )  e.  ran  F )
146133, 145eqeltrd 2529 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  F )
147146adantlllr 37361 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\ 
-. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  F )
14898ad3antrrr 736 . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  -.  ( F `  y )  = +oo )
150149neqned 2631 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  =/= +oo )
151 ge0xrre 37633 . . . . . . . . . . . . . 14  |-  ( ( ( F `  y
)  e.  ( 0 [,] +oo )  /\  ( F `  y )  =/= +oo )  -> 
( F `  y
)  e.  RR )
152127, 150, 151syl2anc 667 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  RR )
153152ltpnfd 11423 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  < +oo )
154122, 123, 128, 130, 153elicod 11685 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( 0 [,) +oo ) )
155 eqid 2451 . . . . . . . . . . 11  |-  ( y  e.  x  |->  ( F `
 y ) )  =  ( y  e.  x  |->  ( F `  y ) )
156154, 155fmptd 6046 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
y  e.  x  |->  ( F `  y ) ) : x --> ( 0 [,) +oo ) )
157 0e0icopnf 11742 . . . . . . . . . . 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 3428 . . . . . . . . . . . 12  |-  ( y  e.  ( 0 [,] +oo )  ->  y  e. 
RR* )
160 xaddid2 11533 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( 0 +e y )  =  y )
161 xaddid1 11532 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( y +e 0 )  =  y )
162160, 161jca 535 . . . . . . . . . . . 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 468 . . . . . . . . . 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 16519 . . . . . . . . 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 19024 . . . . . . . . . . . . 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 2451 . . . . . . . . . . . 12  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
169119, 167, 156, 168gsumsubm 16620 . . . . . . . . . . 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 2452 . . . . . . . . . . 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 3048 . . . . . . . . . . . . . 14  |-  x  e. 
_V
172171mptex 6136 . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . 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 11740 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,) +oo )  C_  RR
179 ax-resscn 9596 . . . . . . . . . . . . . . . . 17  |-  RR  C_  CC
180178, 179sstri 3441 . . . . . . . . . . . . . . . 16  |-  ( 0 [,) +oo )  C_  CC
181 cnfldbas 18974 . . . . . . . . . . . . . . . . 17  |-  CC  =  ( Base ` fld )
182168, 181ressbas2 15180 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
183180, 182ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
184183eqcomi 2460 . . . . . . . . . . . . . 14  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  ( 0 [,) +oo )
185112, 21sstri 3441 . . . . . . . . . . . . . . 15  |-  ( 0 [,) +oo )  C_  RR*
186 eqid 2451 . . . . . . . . . . . . . . . 16  |-  ( RR*ss  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
187186, 70ressbas2 15180 . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . 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 19038 . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
194 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
195 eqid 2451 . . . . . . . . . . . . . . 15  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  (
Base `  (flds  ( 0 [,) +oo ) ) )
196 eqid 2451 . . . . . . . . . . . . . . 15  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
197195, 196srgacl 17757 . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 22 . . . . . . . . . . . . . . . . 17  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
202201, 184syl6eleq 2539 . . . . . . . . . . . . . . . 16  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  ( 0 [,) +oo ) )
203200, 202sseldd 3433 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  RR )
204203adantr 467 . . . . . . . . . . . . . 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 22 . . . . . . . . . . . . . . . . 17  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
207206, 184syl6eleq 2539 . . . . . . . . . . . . . . . 16  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  ( 0 [,) +oo ) )
208205, 207sseldd 3433 . . . . . . . . . . . . . . 15  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  RR )
209208adantl 468 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  t  e.  RR )
210 rexadd 11525 . . . . . . . . . . . . . . . 16  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s +e
t )  =  ( s  +  t ) )
211210eqcomd 2457 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s  +  t )  =  ( s +e t ) )
212166elexi 3055 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,) +oo )  e. 
_V
213 cnfldadd 18975 . . . . . . . . . . . . . . . . . . . . 21  |-  +  =  ( +g  ` fld )
214168, 213ressplusg 15239 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
215212, 214ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
216215, 213eqtr3i 2475 . . . . . . . . . . . . . . . . . 18  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  ( +g  ` fld )
217216, 213eqtr4i 2476 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  +
218217oveqi 6303 . . . . . . . . . . . . . . . 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 15239 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) )
221212, 220ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )
222221eqcomi 2460 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )  =  +e
223222oveqi 6303 . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . 14  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s ( +g  `  (flds  ( 0 [,) +oo )
) ) t )  =  ( s ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) t ) )
226204, 209, 225syl2anc 667 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 5618 . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
231230ralrimiva 2802 . . . . . . . . . . . . 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 6052 . . . . . . . . . . . . 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 16517 . . . . . . . . . . 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 2489 . . . . . . . . . 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 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
237152recnd 9669 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  CC )
238236, 237gsumfsum 19034 . . . . . . . . . 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 2487 . . . . . . . . 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 2490 . . . . . . . 8  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sum_ y  e.  x  ( F `  y )  =  ( G  gsumg  ( F  |`  x
) ) )
241240mpteq2dva 4489 . . . . . . 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 5062 . . . . . 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 7960 . . . . 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 2485 . . . 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 800 . . 3  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
24622, 9, 11, 1xrge0tsms 21852 . . 3  |-  ( ph  ->  ( G tsums  F )  =  { sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  ) } )
247245, 246eleq12d 2523 . 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 236 1  |-  ( ph  ->  (Σ^ `  F )  e.  ( G tsums  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   {csn 3968   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   ran crn 4835    |` cres 4836   Fun wfun 5576    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   Fincfn 7569   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539    + caddc 9542   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676   +ecxad 11407   [,)cico 11637   [,]cicc 11638   sum_csu 13752   Basecbs 15121   ↾s cress 15122   +g cplusg 15190    gsumg cgsu 15339   RR*scxrs 15398   Mndcmnd 16535  SubMndcsubmnd 16581  CMndccmn 17430  SRingcsrg 17739  ℂfldccnfld 18970   tsums ctsu 21140  Σ^csumge0 38204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-ordt 15399  df-xrs 15400  df-mre 15492  df-mrc 15493  df-acs 15495  df-ps 16446  df-tsr 16447  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-srg 17740  df-ring 17782  df-cring 17783  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-ntr 20035  df-nei 20114  df-cn 20243  df-haus 20331  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-tsms 21141  df-sumge0 38205
This theorem is referenced by: (None)
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