Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0tsms Structured version   Visualization version   Unicode version

Theorem sge0tsms 38336
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 2471 . . . 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 11463 . . . . . 6  |-  <  Or  RR*
43supex 7995 . . . . 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 3983 . . . 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 240 . 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 472 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  X  e.  V )
11 sge0tsms.f . . . . . . 7  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
1211adantr 472 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F : X
--> ( 0 [,] +oo ) )
13 simpr 468 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
1410, 12, 13sge0pnfval 38329 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  = +oo )
15 ffn 5739 . . . . . . . . . 10  |-  ( F : X --> ( 0 [,] +oo )  ->  F  Fn  X )
1611, 15syl 17 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
1716adantr 472 . . . . . . . 8  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F  Fn  X )
18 fvelrnb 5926 . . . . . . . 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 215 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  E. y  e.  X  ( F `  y )  = +oo )
21 iccssxr 11742 . . . . . . . . . . . . . 14  |-  ( 0 [,] +oo )  C_  RR*
22 sge0tsms.g . . . . . . . . . . . . . . 15  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
23 simpr 468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  ( ~P X  i^i  Fin ) )
2411adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
25 elinel1 3610 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  ~P X )
26 elpwi 3951 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~P X  ->  x  C_  X )
2725, 26syl 17 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  C_  X )
2827adantl 473 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  C_  X )
29 fssres 5761 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> ( 0 [,] +oo )  /\  x  C_  X )  -> 
( F  |`  x
) : x --> ( 0 [,] +oo ) )
3024, 28, 29syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
31 elinel2 3611 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  Fin )
3231adantl 473 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
33 0red 9662 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  0  e.  RR )
3430, 32, 33fdmfifsupp 7911 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) finSupp  0
)
3522, 23, 30, 34gsumge0cl 38327 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  x
) )  e.  ( 0 [,] +oo )
)
3621, 35sseldi 3416 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  x
) )  e.  RR* )
3736ralrimiva 2809 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  ( ~P X  i^i  Fin ) ( G  gsumg  ( F  |`  x ) )  e. 
RR* )
38373ad2ant1 1051 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  A. x  e.  ( ~P X  i^i  Fin ) ( G  gsumg  ( F  |`  x ) )  e. 
RR* )
39 eqid 2471 . . . . . . . . . . . 12  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  x ) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) )
4039rnmptss 6068 . . . . . . . . . . 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 7668 . . . . . . . . . . . . . . 15  |-  { y }  e.  Fin
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  X  ->  { y }  e.  Fin )
4542, 44elind 3609 . . . . . . . . . . . . 13  |-  ( y  e.  X  ->  { y }  e.  ( ~P X  i^i  Fin )
)
46453ad2ant2 1052 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  { y }  e.  ( ~P X  i^i  Fin ) )
4711adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  F : X --> ( 0 [,] +oo ) )
48 snssi 4107 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  X  ->  { y }  C_  X )
4948adantl 473 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  X )  ->  { y }  C_  X )
5047, 49fssresd 5762 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } ) : { y } --> ( 0 [,] +oo ) )
5150feqmptd 5932 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } )  =  ( x  e.  { y } 
|->  ( ( F  |`  { y } ) `
 x ) ) )
52 fvres 5893 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { y }  ->  ( ( F  |`  { y } ) `
 x )  =  ( F `  x
) )
5352mpteq2ia 4478 . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  X )  ->  ( F  |`  { y } )  =  ( x  e.  { y } 
|->  ( F `  x
) ) )
5655oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X )  ->  ( G  gsumg  ( F  |`  { y } ) )  =  ( G  gsumg  ( x  e.  {
y }  |->  ( F `
 x ) ) ) )
57563adant3 1050 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( G  gsumg  ( F  |`  { y } ) )  =  ( G 
gsumg  ( x  e.  { y }  |->  ( F `  x ) ) ) )
58 xrge0cmn 19087 . . . . . . . . . . . . . . . . 17  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
5922, 58eqeltri 2545 . . . . . . . . . . . . . . . 16  |-  G  e. CMnd
60 cmnmnd 17523 . . . . . . . . . . . . . . . 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 1031 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  y  e.  X
)
6411ffvelrnda 6037 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  X )  ->  ( F `  y )  e.  ( 0 [,] +oo ) )
65643adant3 1050 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  e.  ( 0 [,] +oo )
)
66 df-ss 3404 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] +oo )  C_ 
RR* 
<->  ( ( 0 [,] +oo )  i^i  RR* )  =  ( 0 [,] +oo ) )
6721, 66mpbi 213 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 [,] +oo )  i^i  RR* )  =  ( 0 [,] +oo )
6867eqcomi 2480 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] +oo )  =  ( ( 0 [,] +oo )  i^i  RR* )
69 ovex 6336 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,] +oo )  e. 
_V
70 xrsbas 19061 . . . . . . . . . . . . . . . . . 18  |-  RR*  =  ( Base `  RR*s )
7122, 70ressbas 15257 . . . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . . 15  |-  ( 0 [,] +oo )  =  ( Base `  G
)
74 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
7573, 74gsumsn 17665 . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( G  gsumg  ( x  e.  { y } 
|->  ( F `  x
) ) )  =  ( F `  y
) )
77 simp3 1032 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  = +oo )
7857, 76, 773eqtrrd 2510 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  =  ( G  gsumg  ( F  |`  { y } ) ) )
79 reseq2 5106 . . . . . . . . . . . . . . 15  |-  ( x  =  { y }  ->  ( F  |`  x )  =  ( F  |`  { y } ) )
8079oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( x  =  { y }  ->  ( G  gsumg  ( F  |`  x ) )  =  ( G  gsumg  ( F  |`  { y } ) ) )
8180eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( x  =  { y }  ->  ( +oo  =  ( G  gsumg  ( F  |`  x
) )  <-> +oo  =  ( G  gsumg  ( F  |`  { y } ) ) ) )
8281rspcev 3136 . . . . . . . . . . . 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 673 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  ( G 
gsumg  ( F  |`  x ) ) )
84 pnfxr 11435 . . . . . . . . . . . . 13  |- +oo  e.  RR*
8584a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  RR* )
8639elrnmpt 5087 . . . . . . . . . . . 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 240 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) )
89 supxrpnf 11629 . . . . . . . . . 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 673 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
91903exp 1230 . . . . . . . 8  |-  ( ph  ->  ( y  e.  X  ->  ( ( F `  y )  = +oo  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo ) ) )
9291adantr 472 . . . . . . 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 2870 . . . . . 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 2508 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
969adantr 472 . . . . . 6  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  X  e.  V )
9711adantr 472 . . . . . . 7  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,] +oo ) )
98 simpr 468 . . . . . . 7  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  -. +oo  e.  ran  F )
9997, 98fge0iccico 38326 . . . . . 6  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,) +oo ) )
10096, 99sge0reval 38328 . . . . 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 5933 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x )  =  ( y  e.  x  |->  ( F `  y
) ) )
102101adantlr 729 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x )  =  ( y  e.  x  |->  ( F `  y
) ) )
103102oveq2d 6324 . . . . . . . . 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 5882 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
105 eqid 2471 . . . . . . . . . . . . . 14  |-  ( RR*ss  ( 0 [,] +oo ) )  =  (
RR*ss  ( 0 [,] +oo ) )
106 xrsadd 19062 . . . . . . . . . . . . . 14  |-  +e 
=  ( +g  `  RR*s
)
107105, 106ressplusg 15317 . . . . . . . . . . . . 13  |-  ( ( 0 [,] +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) ) )
10869, 107ax-mp 5 . . . . . . . . . . . 12  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
109108eqcomi 2480 . . . . . . . . . . 11  |-  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )  =  +e
110104, 109eqtr2i 2494 . . . . . . . . . 10  |-  +e 
=  ( +g  `  G
)
11122oveq1i 6318 . . . . . . . . . . 11  |-  ( Gs  ( 0 [,) +oo )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )
112 icossicc 11746 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
11369, 112pm3.2i 462 . . . . . . . . . . . 12  |-  ( ( 0 [,] +oo )  e.  _V  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )
114 ressabs 15266 . . . . . . . . . . . 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 2494 . . . . . . . . . 10  |-  ( RR*ss  ( 0 [,) +oo ) )  =  ( Gs  ( 0 [,) +oo ) )
11759elexi 3041 . . . . . . . . . . 11  |-  G  e. 
_V
118117a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  G  e.  _V )
119 simpr 468 . . . . . . . . . 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 9705 . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  F : X --> ( 0 [,] +oo ) )
12527sselda 3418 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( ~P X  i^i  Fin )  /\  y  e.  x
)  ->  y  e.  X )
126125adantll 728 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  X )
127124, 126ffvelrnd 6038 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( 0 [,] +oo ) )
12821, 127sseldi 3416 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  RR* )
129 iccgelb 11716 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `
 y )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( F `  y
) )
130122, 123, 127, 129syl3anc 1292 . . . . . . . . . . . 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 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  y )  = +oo  -> +oo  =  ( F `  y ) )
133132adantl 473 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  -> +oo  =  ( F `  y )
)
134 ffun 5742 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : X --> ( 0 [,] +oo )  ->  Fun  F )
13511, 134syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  Fun  F )
136135ad2antrr 740 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  Fun  F )
13723, 125sylan 479 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  X )
138 fdm 5745 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( F : X --> ( 0 [,] +oo )  ->  dom  F  =  X )
13911, 138syl 17 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  dom  F  =  X )
140139eqcomd 2477 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  X  =  dom  F
)
141140ad2antrr 740 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  X  =  dom  F )
142137, 141eleqtrd 2551 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  y  e.  dom  F )
143 fvelrn 6030 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  ran  F
)
144136, 142, 143syl2anc 673 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ran  F )
145144adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  ->  ( F `  y )  e.  ran  F )
146133, 145eqeltrd 2549 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x
)  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  F )
147146adantlllr 37424 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\ 
-. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  F )
14898ad3antrrr 744 . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  -.  ( F `  y )  = +oo )
150149neqned 2650 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  =/= +oo )
151 ge0xrre 37729 . . . . . . . . . . . . . 14  |-  ( ( ( F `  y
)  e.  ( 0 [,] +oo )  /\  ( F `  y )  =/= +oo )  -> 
( F `  y
)  e.  RR )
152127, 150, 151syl2anc 673 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  RR )
153152ltpnfd 11446 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  < +oo )
154122, 123, 128, 130, 153elicod 11710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( 0 [,) +oo ) )
155 eqid 2471 . . . . . . . . . . 11  |-  ( y  e.  x  |->  ( F `
 y ) )  =  ( y  e.  x  |->  ( F `  y ) )
156154, 155fmptd 6061 . . . . . . . . . 10  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (
y  e.  x  |->  ( F `  y ) ) : x --> ( 0 [,) +oo ) )
157 0e0icopnf 11768 . . . . . . . . . . 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 3414 . . . . . . . . . . . 12  |-  ( y  e.  ( 0 [,] +oo )  ->  y  e. 
RR* )
160 xaddid2 11557 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( 0 +e y )  =  y )
161 xaddid1 11556 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( y +e 0 )  =  y )
162160, 161jca 541 . . . . . . . . . . . 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 473 . . . . . . . . . 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 16597 . . . . . . . . 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 19101 . . . . . . . . . . . . 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 2471 . . . . . . . . . . . 12  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
169119, 167, 156, 168gsumsubm 16698 . . . . . . . . . . 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 2472 . . . . . . . . . . 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 3034 . . . . . . . . . . . . . 14  |-  x  e. 
_V
172171mptex 6152 . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . 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 11766 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,) +oo )  C_  RR
179 ax-resscn 9614 . . . . . . . . . . . . . . . . 17  |-  RR  C_  CC
180178, 179sstri 3427 . . . . . . . . . . . . . . . 16  |-  ( 0 [,) +oo )  C_  CC
181 cnfldbas 19051 . . . . . . . . . . . . . . . . 17  |-  CC  =  ( Base ` fld )
182168, 181ressbas2 15258 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
183180, 182ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
184183eqcomi 2480 . . . . . . . . . . . . . 14  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  ( 0 [,) +oo )
185112, 21sstri 3427 . . . . . . . . . . . . . . 15  |-  ( 0 [,) +oo )  C_  RR*
186 eqid 2471 . . . . . . . . . . . . . . . 16  |-  ( RR*ss  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
187186, 70ressbas2 15258 . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . 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 19115 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
194 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
195 eqid 2471 . . . . . . . . . . . . . . 15  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  (
Base `  (flds  ( 0 [,) +oo ) ) )
196 eqid 2471 . . . . . . . . . . . . . . 15  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
197195, 196srgacl 17835 . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 2559 . . . . . . . . . . . . . . . 16  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  ( 0 [,) +oo ) )
203200, 202sseldd 3419 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  s  e.  RR )
204203adantr 472 . . . . . . . . . . . . . 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 2559 . . . . . . . . . . . . . . . 16  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  ( 0 [,) +oo ) )
208205, 207sseldd 3419 . . . . . . . . . . . . . . 15  |-  ( t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  ->  t  e.  RR )
209208adantl 473 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( Base `  (flds  ( 0 [,) +oo )
) )  /\  t  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )  ->  t  e.  RR )
210 rexadd 11548 . . . . . . . . . . . . . . . 16  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s +e
t )  =  ( s  +  t ) )
211210eqcomd 2477 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s  +  t )  =  ( s +e t ) )
212166elexi 3041 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,) +oo )  e. 
_V
213 cnfldadd 19052 . . . . . . . . . . . . . . . . . . . . 21  |-  +  =  ( +g  ` fld )
214168, 213ressplusg 15317 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
215212, 214ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
216215, 213eqtr3i 2495 . . . . . . . . . . . . . . . . . 18  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  ( +g  ` fld )
217216, 213eqtr4i 2496 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  (flds  ( 0 [,) +oo )
) )  =  +
218217oveqi 6321 . . . . . . . . . . . . . . . 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 15317 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) )
221212, 220ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )
222221eqcomi 2480 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )  =  +e
223222oveqi 6321 . . . . . . . . . . . . . . . 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 2515 . . . . . . . . . . . . . 14  |-  ( ( s  e.  RR  /\  t  e.  RR )  ->  ( s ( +g  `  (flds  ( 0 [,) +oo )
) ) t )  =  ( s ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) t ) )
226204, 209, 225syl2anc 673 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 5625 . . . . . . . . . . . . 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 2559 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
231230ralrimiva 2809 . . . . . . . . . . . . 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 6068 . . . . . . . . . . . . 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 16595 . . . . . . . . . . 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 2509 . . . . . . . . . 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 473 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
237152recnd 9687 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -. +oo  e.  ran  F
)  /\  x  e.  ( ~P X  i^i  Fin ) )  /\  y  e.  x )  ->  ( F `  y )  e.  CC )
238236, 237gsumfsum 19111 . . . . . . . . . 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 2507 . . . . . . . . 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 2510 . . . . . . . 8  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sum_ y  e.  x  ( F `  y )  =  ( G  gsumg  ( F  |`  x
) ) )
241240mpteq2dva 4482 . . . . . . 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 5068 . . . . . 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 7978 . . . . 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 2505 . . . 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 808 . . 3  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  )
)
24622, 9, 11, 1xrge0tsms 21930 . . 3  |-  ( ph  ->  ( G tsums  F )  =  { sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  ( G  gsumg  ( F  |`  x ) ) ) ,  RR* ,  <  ) } )
247245, 246eleq12d 2543 . 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 240 1  |-  ( ph  ->  (Σ^ `  F )  e.  ( G tsums  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   {csn 3959   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694   +ecxad 11430   [,)cico 11662   [,]cicc 11663   sum_csu 13829   Basecbs 15199   ↾s cress 15200   +g cplusg 15268    gsumg cgsu 15417   RR*scxrs 15476   Mndcmnd 16613  SubMndcsubmnd 16659  CMndccmn 17508  SRingcsrg 17817  ℂfldccnfld 19047   tsums ctsu 21218  Σ^csumge0 38318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-ordt 15477  df-xrs 15478  df-mre 15570  df-mrc 15571  df-acs 15573  df-ps 16524  df-tsr 16525  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-srg 17818  df-ring 17860  df-cring 17861  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-ntr 20112  df-nei 20191  df-cn 20320  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-sumge0 38319
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator