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Theorem gsumesum 26363
Description: Relate a group sum on  ( RR*ss  ( 0 [,] +oo ) ) to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Hypotheses
Ref Expression
gsumesum.0  |-  F/ k
ph
gsumesum.1  |-  ( ph  ->  A  e.  Fin )
gsumesum.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
Assertion
Ref Expression
gsumesum  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  = Σ* k  e.  A B )
Distinct variable group:    A, k
Allowed substitution hints:    ph( k)    B( k)

Proof of Theorem gsumesum
Dummy variables  a  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumesum.0 . . 3  |-  F/ k
ph
2 nfcv 2569 . . 3  |-  F/_ k A
3 gsumesum.1 . . 3  |-  ( ph  ->  A  e.  Fin )
4 gsumesum.2 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
5 eqidd 2434 . . 3  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) )
61, 2, 3, 4, 5esumval 26353 . 2  |-  ( ph  -> Σ* k  e.  A B  =  sup ( ran  (
x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) ,  RR* ,  <  )
)
7 xrltso 11105 . . . 4  |-  <  Or  RR*
87a1i 11 . . 3  |-  ( ph  ->  <  Or  RR* )
9 iccssxr 11365 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
10 xrge0base 25968 . . . . 5  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
11 xrge00 25969 . . . . 5  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
12 xrge0cmn 17698 . . . . . 6  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
1312a1i 11 . . . . 5  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
14 nfcv 2569 . . . . . 6  |-  F/_ k
( 0 [,] +oo )
15 eqid 2433 . . . . . 6  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
161, 2, 14, 4, 15fmptdF 25795 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
173, 16fisuppfi 7616 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { 0 } ) )  e.  Fin )
1810, 11, 13, 3, 16, 17gsumclOLD 16379 . . . 4  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e.  ( 0 [,] +oo ) )
199, 18sseldi 3342 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e. 
RR* )
20 pwidg 3861 . . . . . . 7  |-  ( A  e.  Fin  ->  A  e.  ~P A )
213, 20syl 16 . . . . . 6  |-  ( ph  ->  A  e.  ~P A
)
2221, 3elind 3528 . . . . 5  |-  ( ph  ->  A  e.  ( ~P A  i^i  Fin )
)
23 eqidd 2434 . . . . 5  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
24 mpteq1 4360 . . . . . . . 8  |-  ( x  =  A  ->  (
k  e.  x  |->  B )  =  ( k  e.  A  |->  B ) )
2524oveq2d 6096 . . . . . . 7  |-  ( x  =  A  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
2625eqeq2d 2444 . . . . . 6  |-  ( x  =  A  ->  (
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) )  <->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) ) )
2726rspcev 3062 . . . . 5  |-  ( ( A  e.  ( ~P A  i^i  Fin )  /\  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) )
2822, 23, 27syl2anc 654 . . . 4  |-  ( ph  ->  E. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) )
29 eqid 2433 . . . . 5  |-  ( x  e.  ( ~P A  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )  =  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )
30 ovex 6105 . . . . 5  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
_V
3129, 30elrnmpti 5077 . . . 4  |-  ( ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  A  |->  B ) )  e. 
ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )  <->  E. x  e.  ( ~P A  i^i  Fin )
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) )
3228, 31sylibr 212 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e. 
ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )
33 simpr 458 . . . . . 6  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )
34 mpteq1 4360 . . . . . . . . 9  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
3534oveq2d 6096 . . . . . . . 8  |-  ( x  =  a  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
3635cbvmptv 4371 . . . . . . 7  |-  ( x  e.  ( ~P A  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )  =  ( a  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) )
37 ovex 6105 . . . . . . 7  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
_V
3836, 37elrnmpti 5077 . . . . . 6  |-  ( y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )  <->  E. a  e.  ( ~P A  i^i  Fin )
y  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) )
3933, 38sylib 196 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  E. a  e.  ( ~P A  i^i  Fin ) y  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) ) )
4012a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
41 vex 2965 . . . . . . . . . . . 12  |-  a  e. 
_V
4241a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  _V )
43 nfv 1672 . . . . . . . . . . . . 13  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
441, 43nfan 1859 . . . . . . . . . . . 12  |-  F/ k ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )
45 nfcv 2569 . . . . . . . . . . . 12  |-  F/_ k
a
46 simpll 746 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  ph )
47 inss1 3558 . . . . . . . . . . . . . . . . 17  |-  ( ~P A  i^i  Fin )  C_ 
~P A
4847sseli 3340 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
4948elpwid 3858 . . . . . . . . . . . . . . 15  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  C_  A )
5049ad2antlr 719 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  C_  A )
51 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  a )
5250, 51sseldd 3345 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  A )
5346, 52, 4syl2anc 654 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  B  e.  ( 0 [,] +oo ) )
54 eqid 2433 . . . . . . . . . . . 12  |-  ( k  e.  a  |->  B )  =  ( k  e.  a  |->  B )
5544, 45, 14, 53, 54fmptdF 25795 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  a  |->  B ) : a --> ( 0 [,] +oo )
)
56 inss2 3559 . . . . . . . . . . . . 13  |-  ( ~P A  i^i  Fin )  C_ 
Fin
57 simpr 458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  ( ~P A  i^i  Fin ) )
5856, 57sseldi 3342 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
5958, 55fisuppfi 7616 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  a  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
6010, 11, 40, 42, 55, 59gsumclOLD 16379 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e.  ( 0 [,] +oo ) )
619, 60sseldi 3342 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
RR* )
623adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  A  e.  Fin )
63 difexg 4428 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( A  \  a )  e. 
_V )
6462, 63syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( A  \  a )  e. 
_V )
65 nfcv 2569 . . . . . . . . . . . 12  |-  F/_ k
( A  \  a
)
66 simpll 746 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  ph )
67 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  k  e.  ( A  \  a
) )
6867eldifad 3328 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  k  e.  A )
6966, 68, 4syl2anc 654 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  B  e.  ( 0 [,] +oo ) )
70 eqid 2433 . . . . . . . . . . . 12  |-  ( k  e.  ( A  \ 
a )  |->  B )  =  ( k  e.  ( A  \  a
)  |->  B )
7144, 65, 14, 69, 70fmptdF 25795 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  ( A 
\  a )  |->  B ) : ( A 
\  a ) --> ( 0 [,] +oo )
)
72 diffi 7531 . . . . . . . . . . . . 13  |-  ( A  e.  Fin  ->  ( A  \  a )  e. 
Fin )
7362, 72syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( A  \  a )  e. 
Fin )
7473, 71fisuppfi 7616 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  ( A  \  a
)  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
7510, 11, 40, 64, 71, 74gsumclOLD 16379 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e.  ( 0 [,] +oo ) )
769, 75sseldi 3342 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e. 
RR* )
77 elxrge0 11380 . . . . . . . . . . 11  |-  ( ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e.  ( 0 [,] +oo ) 
<->  ( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e. 
RR*  /\  0  <_  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) )
7877simprbi 461 . . . . . . . . . 10  |-  ( ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e.  ( 0 [,] +oo )  ->  0  <_  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) )
7975, 78syl 16 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  0  <_  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) )
80 xraddge02 25874 . . . . . . . . . 10  |-  ( ( ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  e. 
RR*  /\  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e. 
RR* )  ->  (
0  <_  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  -> 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) ) )
8180imp 429 . . . . . . . . 9  |-  ( ( ( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
RR*  /\  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e. 
RR* )  /\  0  <_  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) )
8261, 76, 79, 81syl21anc 1210 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) )
8382adantlr 707 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) )
84 simpll 746 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ph )
8549adantl 463 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  C_  A )
86 xrge0plusg 25970 . . . . . . . . . 10  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
8712a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
883adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  A  e.  Fin )
8916adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
9017adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  ( `' ( k  e.  A  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
91 disjdif 3739 . . . . . . . . . . 11  |-  ( a  i^i  ( A  \ 
a ) )  =  (/)
9291a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
a  i^i  ( A  \  a ) )  =  (/) )
93 undif 3747 . . . . . . . . . . . . 13  |-  ( a 
C_  A  <->  ( a  u.  ( A  \  a
) )  =  A )
9493biimpi 194 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
a  u.  ( A 
\  a ) )  =  A )
9594eqcomd 2438 . . . . . . . . . . 11  |-  ( a 
C_  A  ->  A  =  ( a  u.  ( A  \  a
) ) )
9695adantl 463 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  A  =  ( a  u.  ( A  \  a
) ) )
9710, 11, 86, 87, 88, 89, 90, 92, 96gsumsplitOLD 16400 . . . . . . . . 9  |-  ( (
ph  /\  a  C_  A )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  =  ( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  a ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  ( A  \  a
) ) ) ) )
98 resmpt 5144 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
( k  e.  A  |->  B )  |`  a
)  =  ( k  e.  a  |->  B ) )
9998oveq2d 6096 . . . . . . . . . . 11  |-  ( a 
C_  A  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  a ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
10099adantl 463 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  a ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
101 difss 3471 . . . . . . . . . . . . 13  |-  ( A 
\  a )  C_  A
102 resmpt 5144 . . . . . . . . . . . . 13  |-  ( ( A  \  a ) 
C_  A  ->  (
( k  e.  A  |->  B )  |`  ( A  \  a ) )  =  ( k  e.  ( A  \  a
)  |->  B ) )
103101, 102ax-mp 5 . . . . . . . . . . . 12  |-  ( ( k  e.  A  |->  B )  |`  ( A  \  a ) )  =  ( k  e.  ( A  \  a ) 
|->  B )
104103oveq2i 6091 . . . . . . . . . . 11  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  ( A  \  a
) ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )
105104a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  ( A  \  a
) ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) )
106100, 105oveq12d 6098 . . . . . . . . 9  |-  ( (
ph  /\  a  C_  A )  ->  (
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e.  A  |->  B )  |`  a ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  ( A  \  a
) ) ) )  =  ( ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) )
10797, 106eqtrd 2465 . . . . . . . 8  |-  ( (
ph  /\  a  C_  A )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  =  ( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) )
10884, 85, 107syl2anc 654 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  =  ( ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) +e ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) ) )
10983, 108breqtrrd 4306 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
110109ralrimiva 2789 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  A. a  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
111 r19.29r 2848 . . . . . 6  |-  ( ( E. a  e.  ( ~P A  i^i  Fin ) y  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) )  /\  A. a  e.  ( ~P A  i^i  Fin )
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )  ->  E. a  e.  ( ~P A  i^i  Fin ) ( y  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  /\  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) ) )
112 breq1 4283 . . . . . . . 8  |-  ( y  =  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  -> 
( y  <_  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  <->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) ) )
113112biimpar 482 . . . . . . 7  |-  ( ( y  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  /\  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )  ->  y  <_  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) ) )
114113rexlimivw 2827 . . . . . 6  |-  ( E. a  e.  ( ~P A  i^i  Fin )
( y  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) )  /\  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )  ->  y  <_  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) ) )
115111, 114syl 16 . . . . 5  |-  ( ( E. a  e.  ( ~P A  i^i  Fin ) y  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) )  /\  A. a  e.  ( ~P A  i^i  Fin )
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) )  <_ 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )  ->  y  <_  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) ) )
11639, 110, 115syl2anc 654 . . . 4  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  y  <_  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  A  |->  B ) ) )
11719adantr 462 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  e. 
RR* )
11812a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
119 simpr 458 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
120 nfv 1672 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
1211, 120nfan 1859 . . . . . . . . . . 11  |-  F/ k ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )
122 nfcv 2569 . . . . . . . . . . 11  |-  F/_ k
x
123 simpll 746 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  ph )
12447sseli 3340 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
125124ad2antlr 719 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  e.  ~P A )
126125elpwid 3858 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  C_  A )
127 simpr 458 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  x )
128126, 127sseldd 3345 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  A )
129123, 128, 4syl2anc 654 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  B  e.  ( 0 [,] +oo ) )
130 eqid 2433 . . . . . . . . . . 11  |-  ( k  e.  x  |->  B )  =  ( k  e.  x  |->  B )
131121, 122, 14, 129, 130fmptdF 25795 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  x  |->  B ) : x --> ( 0 [,] +oo ) )
13256, 119sseldi 3342 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
133132, 131fisuppfi 7616 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  x  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
13410, 11, 118, 119, 131, 133gsumclOLD 16379 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,] +oo ) )
1359, 134sseldi 3342 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
136135ralrimiva 2789 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
13729rnmptss 5859 . . . . . . 7  |-  ( A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  x  |->  B ) )  e. 
RR*  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR* )
138136, 137syl 16 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR* )
139138sselda 3344 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  y  e.  RR* )
140 xrltnle 9430 . . . . . 6  |-  ( ( ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e. 
RR*  /\  y  e.  RR* )  ->  ( (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  < 
y  <->  -.  y  <_  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  A  |->  B ) ) ) )
141140con2bid 329 . . . . 5  |-  ( ( ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e. 
RR*  /\  y  e.  RR* )  ->  ( y  <_  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  <->  -.  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  < 
y ) )
142117, 139, 141syl2anc 654 . . . 4  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  ( y  <_  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  <->  -.  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  < 
y ) )
143116, 142mpbid 210 . . 3  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  -.  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  < 
y )
1448, 19, 32, 143supmax 7703 . 2  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) ,  RR* ,  <  )  =  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) ) )
1456, 144eqtr2d 2466 1  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  = Σ* k  e.  A B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362   F/wnf 1592    e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962    \ cdif 3313    u. cun 3314    i^i cin 3315    C_ wss 3316   (/)c0 3625   ~Pcpw 3848   {csn 3865   class class class wbr 4280    e. cmpt 4338    Or wor 4627   `'ccnv 4826   ran crn 4828    |` cres 4829   "cima 4830   -->wf 5402  (class class class)co 6080   Fincfn 7298   supcsup 7678   0cc0 9269   +oocpnf 9402   RR*cxr 9404    < clt 9405    <_ cle 9406   +ecxad 11074   [,]cicc 11290   ↾s cress 14157    gsumg cgsu 14361   RR*scxrs 14420  CMndccmn 16256  Σ*cesum 26336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-xadd 11077  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-seq 11790  df-hash 12087  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-tset 14239  df-ple 14240  df-ds 14242  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-ordt 14421  df-xrs 14422  df-mre 14506  df-mrc 14507  df-acs 14509  df-ps 15352  df-tsr 15353  df-mnd 15397  df-submnd 15447  df-cntz 15814  df-cmn 16258  df-fbas 17657  df-fg 17658  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-ntr 18465  df-nei 18543  df-cn 18672  df-haus 18760  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-tsms 19538  df-esum 26337
This theorem is referenced by:  esumlub  26364
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