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Theorem gsumesum 28231
Description: Relate a group sum on  ( RR*ss  ( 0 [,] +oo ) ) to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)
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 2619 . . 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 2458 . . 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 28220 . 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 11372 . . . 4  |-  <  Or  RR*
87a1i 11 . . 3  |-  ( ph  ->  <  Or  RR* )
9 iccssxr 11632 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
10 xrge0base 27833 . . . . 5  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
11 xrge0cmn 18587 . . . . . 6  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
1211a1i 11 . . . . 5  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
134ex 434 . . . . . 6  |-  ( ph  ->  ( k  e.  A  ->  B  e.  ( 0 [,] +oo ) ) )
141, 13ralrimi 2857 . . . . 5  |-  ( ph  ->  A. k  e.  A  B  e.  ( 0 [,] +oo ) )
1510, 12, 3, 14gsummptcl 17121 . . . 4  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e.  ( 0 [,] +oo ) )
169, 15sseldi 3497 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e. 
RR* )
17 pwidg 4028 . . . . . . 7  |-  ( A  e.  Fin  ->  A  e.  ~P A )
183, 17syl 16 . . . . . 6  |-  ( ph  ->  A  e.  ~P A
)
1918, 3elind 3684 . . . . 5  |-  ( ph  ->  A  e.  ( ~P A  i^i  Fin )
)
20 eqidd 2458 . . . . 5  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
21 mpteq1 4537 . . . . . . . 8  |-  ( x  =  A  ->  (
k  e.  x  |->  B )  =  ( k  e.  A  |->  B ) )
2221oveq2d 6312 . . . . . . 7  |-  ( x  =  A  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
2322eqeq2d 2471 . . . . . 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 ) ) ) )
2423rspcev 3210 . . . . 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 ) ) )
2519, 20, 24syl2anc 661 . . . 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 ) ) )
26 eqid 2457 . . . . 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 ) ) )
27 ovex 6324 . . . . 5  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
_V
2826, 27elrnmpti 5263 . . . 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 ) ) )
2925, 28sylibr 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 ) ) ) )
30 simpr 461 . . . . . 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 ) ) ) )
31 mpteq1 4537 . . . . . . . . 9  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
3231oveq2d 6312 . . . . . . . 8  |-  ( x  =  a  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
3332cbvmptv 4548 . . . . . . 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 ) ) )
34 ovex 6324 . . . . . . 7  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
_V
3533, 34elrnmpti 5263 . . . . . 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 ) ) )
3630, 35sylib 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 ) ) )
3711a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
38 inss2 3715 . . . . . . . . . . . 12  |-  ( ~P A  i^i  Fin )  C_ 
Fin
39 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  ( ~P A  i^i  Fin ) )
4038, 39sseldi 3497 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
41 nfv 1708 . . . . . . . . . . . . 13  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
421, 41nfan 1929 . . . . . . . . . . . 12  |-  F/ k ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )
43 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  ph )
44 inss1 3714 . . . . . . . . . . . . . . . . . 18  |-  ( ~P A  i^i  Fin )  C_ 
~P A
4544sseli 3495 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
4645elpwid 4025 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  C_  A )
4746ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  C_  A )
48 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  a )
4947, 48sseldd 3500 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  A )
5043, 49, 4syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  B  e.  ( 0 [,] +oo ) )
5150ex 434 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  a  ->  B  e.  ( 0 [,] +oo ) ) )
5242, 51ralrimi 2857 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  A. k  e.  a  B  e.  ( 0 [,] +oo ) )
5310, 37, 40, 52gsummptcl 17121 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e.  ( 0 [,] +oo ) )
549, 53sseldi 3497 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
RR* )
55 diffi 7770 . . . . . . . . . . . . 13  |-  ( A  e.  Fin  ->  ( A  \  a )  e. 
Fin )
563, 55syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  \  a
)  e.  Fin )
5756adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( A  \  a )  e. 
Fin )
58 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  ph )
59 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  k  e.  ( A  \  a
) )
6059eldifad 3483 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  k  e.  A )
6158, 60, 4syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  B  e.  ( 0 [,] +oo ) )
6261ex 434 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  ( A 
\  a )  ->  B  e.  ( 0 [,] +oo ) ) )
6342, 62ralrimi 2857 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  A. k  e.  ( A  \  a
) B  e.  ( 0 [,] +oo )
)
6410, 37, 57, 63gsummptcl 17121 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e.  ( 0 [,] +oo ) )
659, 64sseldi 3497 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e. 
RR* )
66 elxrge0 11654 . . . . . . . . . . 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 ) ) ) )
6766simprbi 464 . . . . . . . . . 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 ) ) )
6864, 67syl 16 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  0  <_  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) )
69 xraddge02 27734 . . . . . . . . . 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 ) ) ) ) )
7069imp 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 ) ) ) )
7154, 65, 68, 70syl21anc 1227 . . . . . . . 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 ) ) ) )
7271adantlr 714 . . . . . . 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 ) ) ) )
73 simpll 753 . . . . . . . 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 )
7446adantl 466 . . . . . . . 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 )
75 xrge00 27834 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
76 xrge0plusg 27835 . . . . . . . . . 10  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
7711a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
783adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  A  e.  Fin )
79 eqid 2457 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
801, 4, 79fmptdf 6057 . . . . . . . . . . 11  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
8180adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
8279fnmpt 5713 . . . . . . . . . . . . 13  |-  ( A. k  e.  A  B  e.  ( 0 [,] +oo )  ->  ( k  e.  A  |->  B )  Fn  A )
8314, 82syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( k  e.  A  |->  B )  Fn  A
)
84 c0ex 9607 . . . . . . . . . . . . 13  |-  0  e.  _V
8584a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  0  e.  _V )
8683, 3, 85fndmfifsupp 7860 . . . . . . . . . . 11  |-  ( ph  ->  ( k  e.  A  |->  B ) finSupp  0 )
8786adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
k  e.  A  |->  B ) finSupp  0 )
88 disjdif 3903 . . . . . . . . . . 11  |-  ( a  i^i  ( A  \ 
a ) )  =  (/)
8988a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
a  i^i  ( A  \  a ) )  =  (/) )
90 undif 3911 . . . . . . . . . . . . 13  |-  ( a 
C_  A  <->  ( a  u.  ( A  \  a
) )  =  A )
9190biimpi 194 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
a  u.  ( A 
\  a ) )  =  A )
9291eqcomd 2465 . . . . . . . . . . 11  |-  ( a 
C_  A  ->  A  =  ( a  u.  ( A  \  a
) ) )
9392adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  A  =  ( a  u.  ( A  \  a
) ) )
9410, 75, 76, 77, 78, 81, 87, 89, 93gsumsplit 17073 . . . . . . . . 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
) ) ) ) )
95 resmpt 5333 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
( k  e.  A  |->  B )  |`  a
)  =  ( k  e.  a  |->  B ) )
9695oveq2d 6312 . . . . . . . . . . 11  |-  ( a 
C_  A  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  a ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
9796adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  a ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
98 difss 3627 . . . . . . . . . . . . 13  |-  ( A 
\  a )  C_  A
99 resmpt 5333 . . . . . . . . . . . . 13  |-  ( ( A  \  a ) 
C_  A  ->  (
( k  e.  A  |->  B )  |`  ( A  \  a ) )  =  ( k  e.  ( A  \  a
)  |->  B ) )
10098, 99ax-mp 5 . . . . . . . . . . . 12  |-  ( ( k  e.  A  |->  B )  |`  ( A  \  a ) )  =  ( k  e.  ( A  \  a ) 
|->  B )
101100oveq2i 6307 . . . . . . . . . . 11  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  ( A  \  a
) ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )
102101a1i 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 ) ) )
10397, 102oveq12d 6314 . . . . . . . . 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 ) ) ) )
10494, 103eqtrd 2498 . . . . . . . 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 ) ) ) )
10573, 74, 104syl2anc 661 . . . . . . 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 ) ) ) )
10672, 105breqtrrd 4482 . . . . . 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 ) ) )
107106ralrimiva 2871 . . . . 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 ) ) )
108 r19.29r 2993 . . . . . 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 ) ) ) )
109 breq1 4459 . . . . . . . 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 ) ) ) )
110109biimpar 485 . . . . . . 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 ) ) )
111110rexlimivw 2946 . . . . . 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 ) ) )
112108, 111syl 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 ) ) )
11336, 107, 112syl2anc 661 . . . 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 ) ) )
11416adantr 465 . . . . 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* )
11511a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
116 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
11738, 116sseldi 3497 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
118 nfv 1708 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
1191, 118nfan 1929 . . . . . . . . . . 11  |-  F/ k ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )
120 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  ph )
12144sseli 3495 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
122121ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  e.  ~P A )
123122elpwid 4025 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  C_  A )
124 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  x )
125123, 124sseldd 3500 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  A )
126120, 125, 4syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  B  e.  ( 0 [,] +oo ) )
127126ex 434 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  x  ->  B  e.  ( 0 [,] +oo ) ) )
128119, 127ralrimi 2857 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A. k  e.  x  B  e.  ( 0 [,] +oo ) )
12910, 115, 117, 128gsummptcl 17121 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,] +oo ) )
1309, 129sseldi 3497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
131130ralrimiva 2871 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
13226rnmptss 6061 . . . . . . 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* )
133131, 132syl 16 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR* )
134133sselda 3499 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  y  e.  RR* )
135 xrltnle 9670 . . . . . 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 ) ) ) )
136135con2bid 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 ) )
137114, 134, 136syl2anc 661 . . . 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 ) )
138113, 137mpbid 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 )
1398, 16, 29, 138supmax 7941 . 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 ) ) )
1406, 139eqtr2d 2499 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 1395   F/wnf 1617    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   class class class wbr 4456    |-> cmpt 4515    Or wor 4808   ran crn 5009    |` cres 5010    Fn wfn 5589   -->wf 5590  (class class class)co 6296   Fincfn 7535   finSupp cfsupp 7847   supcsup 7918   0cc0 9509   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   +ecxad 11341   [,]cicc 11557   ↾s cress 14645    gsumg cgsu 14858   RR*scxrs 14917  CMndccmn 16925  Σ*cesum 28201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-xadd 11344  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-tset 14731  df-ple 14732  df-ds 14734  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-ordt 14918  df-xrs 14919  df-mre 15003  df-mrc 15004  df-acs 15006  df-ps 15957  df-tsr 15958  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-cntz 16482  df-cmn 16927  df-fbas 18543  df-fg 18544  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-ntr 19648  df-nei 19726  df-cn 19855  df-haus 19943  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-tsms 20751  df-esum 28202
This theorem is referenced by:  esumlub  28232
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