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Theorem gsumesum 27707
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 2629 . . 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 2468 . . 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 27697 . 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 11343 . . . 4  |-  <  Or  RR*
87a1i 11 . . 3  |-  ( ph  ->  <  Or  RR* )
9 iccssxr 11603 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
10 xrge0base 27335 . . . . 5  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
11 xrge0cmn 18228 . . . . . 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 2864 . . . . 5  |-  ( ph  ->  A. k  e.  A  B  e.  ( 0 [,] +oo ) )
1510, 12, 3, 14gsummptcl 16785 . . . 4  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e.  ( 0 [,] +oo ) )
169, 15sseldi 3502 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  e. 
RR* )
17 pwidg 4023 . . . . . . 7  |-  ( A  e.  Fin  ->  A  e.  ~P A )
183, 17syl 16 . . . . . 6  |-  ( ph  ->  A  e.  ~P A
)
1918, 3elind 3688 . . . . 5  |-  ( ph  ->  A  e.  ( ~P A  i^i  Fin )
)
20 eqidd 2468 . . . . 5  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
21 mpteq1 4527 . . . . . . . 8  |-  ( x  =  A  ->  (
k  e.  x  |->  B )  =  ( k  e.  A  |->  B ) )
2221oveq2d 6298 . . . . . . 7  |-  ( x  =  A  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
2322eqeq2d 2481 . . . . . 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 3214 . . . . 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 2467 . . . . 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 6307 . . . . 5  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
_V
2826, 27elrnmpti 5251 . . . 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 4527 . . . . . . . . 9  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
3231oveq2d 6298 . . . . . . . 8  |-  ( x  =  a  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
3332cbvmptv 4538 . . . . . . 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 6307 . . . . . . 7  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
_V
3533, 34elrnmpti 5251 . . . . . 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 3719 . . . . . . . . . . . 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 3502 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
41 nfv 1683 . . . . . . . . . . . . 13  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
421, 41nfan 1875 . . . . . . . . . . . 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 3718 . . . . . . . . . . . . . . . . . 18  |-  ( ~P A  i^i  Fin )  C_ 
~P A
4544sseli 3500 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
4645elpwid 4020 . . . . . . . . . . . . . . . 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 3505 . . . . . . . . . . . . . 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 2864 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  A. k  e.  a  B  e.  ( 0 [,] +oo ) )
5310, 37, 40, 52gsummptcl 16785 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e.  ( 0 [,] +oo ) )
549, 53sseldi 3502 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
RR* )
55 diffi 7747 . . . . . . . . . . . . 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 3488 . . . . . . . . . . . . . 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 2864 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  A. k  e.  ( A  \  a
) B  e.  ( 0 [,] +oo )
)
6410, 37, 57, 63gsummptcl 16785 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e.  ( 0 [,] +oo ) )
659, 64sseldi 3502 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )  e. 
RR* )
66 elxrge0 11625 . . . . . . . . . . 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 27245 . . . . . . . . . 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 27336 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
76 xrge0plusg 27337 . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
801, 4, 79fmptdf 6044 . . . . . . . . . . 11  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
8180adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
8279fnmpt 5705 . . . . . . . . . . . . 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 9586 . . . . . . . . . . . . 13  |-  0  e.  _V
8584a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  0  e.  _V )
8683, 3, 85fndmfifsupp 7838 . . . . . . . . . . 11  |-  ( ph  ->  ( k  e.  A  |->  B ) finSupp  0 )
8786adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
k  e.  A  |->  B ) finSupp  0 )
88 disjdif 3899 . . . . . . . . . . 11  |-  ( a  i^i  ( A  \ 
a ) )  =  (/)
8988a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
a  i^i  ( A  \  a ) )  =  (/) )
90 undif 3907 . . . . . . . . . . . . 13  |-  ( a 
C_  A  <->  ( a  u.  ( A  \  a
) )  =  A )
9190biimpi 194 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
a  u.  ( A 
\  a ) )  =  A )
9291eqcomd 2475 . . . . . . . . . . 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 16737 . . . . . . . . 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 5321 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
( k  e.  A  |->  B )  |`  a
)  =  ( k  e.  a  |->  B ) )
9695oveq2d 6298 . . . . . . . . . . 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 3631 . . . . . . . . . . . . 13  |-  ( A 
\  a )  C_  A
99 resmpt 5321 . . . . . . . . . . . . 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 6293 . . . . . . . . . . 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 6300 . . . . . . . . 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 2508 . . . . . . . 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 4473 . . . . . 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 2878 . . . . 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 2998 . . . . . 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 4450 . . . . . . . 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 2952 . . . . . 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 3502 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
118 nfv 1683 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
1191, 118nfan 1875 . . . . . . . . . . 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 3500 . . . . . . . . . . . . . . . 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 4020 . . . . . . . . . . . . . 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 3505 . . . . . . . . . . . . 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 2864 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A. k  e.  x  B  e.  ( 0 [,] +oo ) )
12910, 115, 117, 128gsummptcl 16785 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,] +oo ) )
1309, 129sseldi 3502 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
131130ralrimiva 2878 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
13226rnmptss 6048 . . . . . . 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 3504 . . . . 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 9649 . . . . . 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 7921 . 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 2509 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 1379   F/wnf 1599    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505    Or wor 4799   ran crn 5000    |` cres 5001    Fn wfn 5581   -->wf 5582  (class class class)co 6282   Fincfn 7513   finSupp cfsupp 7825   supcsup 7896   0cc0 9488   +oocpnf 9621   RR*cxr 9623    < clt 9624    <_ cle 9625   +ecxad 11312   [,]cicc 11528   ↾s cress 14487    gsumg cgsu 14692   RR*scxrs 14751  CMndccmn 16594  Σ*cesum 27680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-xadd 11315  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-tset 14570  df-ple 14571  df-ds 14573  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-ordt 14752  df-xrs 14753  df-mre 14837  df-mrc 14838  df-acs 14840  df-ps 15683  df-tsr 15684  df-mnd 15728  df-submnd 15778  df-cntz 16150  df-cmn 16596  df-fbas 18187  df-fg 18188  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-ntr 19287  df-nei 19365  df-cn 19494  df-haus 19582  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-tsms 20360  df-esum 27681
This theorem is referenced by:  esumlub  27708
  Copyright terms: Public domain W3C validator