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Theorem gsumesum 24404
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 2540 . . 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 2405 . . 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 24394 . 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 10690 . . . 4  |-  <  Or  RR*
87a1i 11 . . 3  |-  ( ph  ->  <  Or  RR* )
9 iccssxr 10949 . . . 4  |-  ( 0 [,]  +oo )  C_  RR*
10 xrge0base 24160 . . . . 5  |-  ( 0 [,]  +oo )  =  (
Base `  ( RR* ss  ( 0 [,]  +oo ) ) )
11 xrge00 24161 . . . . 5  |-  0  =  ( 0g `  ( RR* ss  ( 0 [,] 
+oo ) ) )
12 xrge0cmn 16695 . . . . . 6  |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
1312a1i 11 . . . . 5  |-  ( ph  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
)
14 nfcv 2540 . . . . . 6  |-  F/_ k
( 0 [,]  +oo )
15 eqid 2404 . . . . . 6  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
161, 2, 14, 4, 15fmptdF 24022 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,]  +oo ) )
173, 16fisuppfi 14728 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { 0 } ) )  e.  Fin )
1810, 11, 13, 3, 16, 17gsumcl 15476 . . . 4  |-  ( ph  ->  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  A  |->  B ) )  e.  ( 0 [,]  +oo ) )
199, 18sseldi 3306 . . 3  |-  ( ph  ->  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  A  |->  B ) )  e. 
RR* )
20 pwidg 3771 . . . . . . 7  |-  ( A  e.  Fin  ->  A  e.  ~P A )
213, 20syl 16 . . . . . 6  |-  ( ph  ->  A  e.  ~P A
)
22 elin 3490 . . . . . 6  |-  ( A  e.  ( ~P A  i^i  Fin )  <->  ( A  e.  ~P A  /\  A  e.  Fin ) )
2321, 3, 22sylanbrc 646 . . . . 5  |-  ( ph  ->  A  e.  ( ~P A  i^i  Fin )
)
24 eqidd 2405 . . . . 5  |-  ( ph  ->  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  A  |->  B ) ) )
25 mpteq1 4249 . . . . . . . 8  |-  ( x  =  A  ->  (
k  e.  x  |->  B )  =  ( k  e.  A  |->  B ) )
2625oveq2d 6056 . . . . . . 7  |-  ( x  =  A  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  A  |->  B ) ) )
2726eqeq2d 2415 . . . . . 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 ) ) ) )
2827rspcev 3012 . . . . 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 ) ) )
2923, 24, 28syl2anc 643 . . . 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 ) ) )
30 eqid 2404 . . . . 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 ) ) )
31 ovex 6065 . . . . 5  |-  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  _V
3230, 31elrnmpti 5080 . . . 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 ) ) )
3329, 32sylibr 204 . . 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 ) ) ) )
34 simpr 448 . . . . . 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 ) ) ) )
35 mpteq1 4249 . . . . . . . . 9  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
3635oveq2d 6056 . . . . . . . 8  |-  ( x  =  a  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) ) )
3736cbvmptv 4260 . . . . . . 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 ) ) )
38 ovex 6065 . . . . . . 7  |-  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) )  e.  _V
3937, 38elrnmpti 5080 . . . . . 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 ) ) )
4034, 39sylib 189 . . . . 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 ) ) )
4112a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd )
42 vex 2919 . . . . . . . . . . . 12  |-  a  e. 
_V
4342a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  _V )
44 nfv 1626 . . . . . . . . . . . . 13  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
451, 44nfan 1842 . . . . . . . . . . . 12  |-  F/ k ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )
46 nfcv 2540 . . . . . . . . . . . 12  |-  F/_ k
a
47 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  ph )
48 inss1 3521 . . . . . . . . . . . . . . . . 17  |-  ( ~P A  i^i  Fin )  C_ 
~P A
4948sseli 3304 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
5049elpwid 3768 . . . . . . . . . . . . . . 15  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  C_  A )
5150ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  C_  A )
52 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  a )
5351, 52sseldd 3309 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  A )
5447, 53, 4syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  B  e.  ( 0 [,]  +oo ) )
55 eqid 2404 . . . . . . . . . . . 12  |-  ( k  e.  a  |->  B )  =  ( k  e.  a  |->  B )
5645, 46, 14, 54, 55fmptdF 24022 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  a  |->  B ) : a --> ( 0 [,]  +oo )
)
57 inss2 3522 . . . . . . . . . . . . 13  |-  ( ~P A  i^i  Fin )  C_ 
Fin
58 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  ( ~P A  i^i  Fin ) )
5957, 58sseldi 3306 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
6059, 56fisuppfi 14728 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  a  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
6110, 11, 41, 43, 56, 60gsumcl 15476 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) )  e.  ( 0 [,]  +oo ) )
629, 61sseldi 3306 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) )  e.  RR* )
633adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  A  e.  Fin )
64 difexg 4311 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( A  \  a )  e. 
_V )
6563, 64syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( A  \  a )  e. 
_V )
66 nfcv 2540 . . . . . . . . . . . 12  |-  F/_ k
( A  \  a
)
67 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  ph )
68 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  k  e.  ( A  \  a
) )
6968eldifad 3292 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  k  e.  A )
7067, 69, 4syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  ( A  \  a
) )  ->  B  e.  ( 0 [,]  +oo ) )
71 eqid 2404 . . . . . . . . . . . 12  |-  ( k  e.  ( A  \ 
a )  |->  B )  =  ( k  e.  ( A  \  a
)  |->  B )
7245, 66, 14, 70, 71fmptdF 24022 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  ( A 
\  a )  |->  B ) : ( A 
\  a ) --> ( 0 [,]  +oo )
)
73 diffi 7298 . . . . . . . . . . . . 13  |-  ( A  e.  Fin  ->  ( A  \  a )  e. 
Fin )
7463, 73syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( A  \  a )  e. 
Fin )
7574, 72fisuppfi 14728 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  ( A  \  a
)  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
7610, 11, 41, 65, 72, 75gsumcl 15476 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  ( A  \ 
a )  |->  B ) )  e.  ( 0 [,]  +oo ) )
779, 76sseldi 3306 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  ( A  \ 
a )  |->  B ) )  e.  RR* )
78 elxrge0 10964 . . . . . . . . . . 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 ) ) ) )
7978simprbi 451 . . . . . . . . . 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 ) ) )
8076, 79syl 16 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  0  <_  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  ( A  \  a ) 
|->  B ) ) )
81 xraddge02 24076 . . . . . . . . . 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 ) ) ) ) )
8281imp 419 . . . . . . . . 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 ) ) ) )
8362, 77, 80, 82syl21anc 1183 . . . . . . . 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 ) ) ) )
8483adantlr 696 . . . . . . 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 ) ) ) )
85 simpll 731 . . . . . . . 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 )
8650adantl 453 . . . . . . . 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 )
87 xrge0plusg 24162 . . . . . . . . . 10  |-  + e  =  ( +g  `  ( RR* ss  ( 0 [,]  +oo ) ) )
8812a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd )
893adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  A  e.  Fin )
9016adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
k  e.  A  |->  B ) : A --> ( 0 [,]  +oo ) )
9117adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  ( `' ( k  e.  A  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
92 disjdif 3660 . . . . . . . . . . 11  |-  ( a  i^i  ( A  \ 
a ) )  =  (/)
9392a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
a  i^i  ( A  \  a ) )  =  (/) )
94 undif 3668 . . . . . . . . . . . . 13  |-  ( a 
C_  A  <->  ( a  u.  ( A  \  a
) )  =  A )
9594biimpi 187 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
a  u.  ( A 
\  a ) )  =  A )
9695eqcomd 2409 . . . . . . . . . . 11  |-  ( a 
C_  A  ->  A  =  ( a  u.  ( A  \  a
) ) )
9796adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  A  =  ( a  u.  ( A  \  a
) ) )
9810, 11, 87, 88, 89, 90, 91, 93, 97gsumsplit 15485 . . . . . . . . 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 ) ) ) ) )
99 resmpt 5150 . . . . . . . . . . . 12  |-  ( a 
C_  A  ->  (
( k  e.  A  |->  B )  |`  a
)  =  ( k  e.  a  |->  B ) )
10099oveq2d 6056 . . . . . . . . . . 11  |-  ( a 
C_  A  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  a )
)  =  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) ) )
101100adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  a  C_  A )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  a )
)  =  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) ) )
102 difss 3434 . . . . . . . . . . . . 13  |-  ( A 
\  a )  C_  A
103 resmpt 5150 . . . . . . . . . . . . 13  |-  ( ( A  \  a ) 
C_  A  ->  (
( k  e.  A  |->  B )  |`  ( A  \  a ) )  =  ( k  e.  ( A  \  a
)  |->  B ) )
104102, 103ax-mp 8 . . . . . . . . . . . 12  |-  ( ( k  e.  A  |->  B )  |`  ( A  \  a ) )  =  ( k  e.  ( A  \  a ) 
|->  B )
105104oveq2i 6051 . . . . . . . . . . 11  |-  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( ( k  e.  A  |->  B )  |`  ( A  \  a ) ) )  =  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  ( A  \  a ) 
|->  B ) )
106105a1i 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 ) ) )
107101, 106oveq12d 6058 . . . . . . . . 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 ) ) ) )
10898, 107eqtrd 2436 . . . . . . . 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 ) ) ) )
10985, 86, 108syl2anc 643 . . . . . . 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 ) ) ) )
11084, 109breqtrrd 4198 . . . . . 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 ) ) )
111110ralrimiva 2749 . . . . 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 ) ) )
112 r19.29r 2807 . . . . . 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 ) ) ) )
113 breq1 4175 . . . . . . . 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 ) ) ) )
114113biimpar 472 . . . . . . 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 ) ) )
115114rexlimivw 2786 . . . . . 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 ) ) )
116112, 115syl 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 ) ) )
11740, 111, 116syl2anc 643 . . . 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 ) ) )
11819adantr 452 . . . . 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* )
11912a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd )
120 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
121 nfv 1626 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
1221, 121nfan 1842 . . . . . . . . . . 11  |-  F/ k ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )
123 nfcv 2540 . . . . . . . . . . 11  |-  F/_ k
x
124 simpll 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  ph )
12548sseli 3304 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
126125ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  e.  ~P A )
127126elpwid 3768 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  C_  A )
128 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  x )
129127, 128sseldd 3309 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  A )
130124, 129, 4syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  B  e.  ( 0 [,]  +oo ) )
131 eqid 2404 . . . . . . . . . . 11  |-  ( k  e.  x  |->  B )  =  ( k  e.  x  |->  B )
132122, 123, 14, 130, 131fmptdF 24022 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  x  |->  B ) : x --> ( 0 [,]  +oo ) )
13357, 120sseldi 3306 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
134133, 132fisuppfi 14728 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  x  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
13510, 11, 119, 120, 132, 134gsumcl 15476 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,]  +oo ) )
1369, 135sseldi 3306 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  RR* )
137136ralrimiva 2749 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
13830rnmptss 5856 . . . . . . 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* )
139137, 138syl 16 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR* )
140139sselda 3308 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) )  ->  y  e.  RR* )
141 xrltnle 9100 . . . . . 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 ) ) ) )
142141con2bid 320 . . . . 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 ) )
143118, 140, 142syl2anc 643 . . . 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 ) )
144117, 143mpbid 202 . . 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 )
1458, 19, 33, 144supmax 7426 . 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 ) ) )
1466, 145eqtr2d 2437 1  |-  ( ph  ->  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  A  |->  B ) )  = Σ* k  e.  A B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1550    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   class class class wbr 4172    e. cmpt 4226    Or wor 4462   `'ccnv 4836   ran crn 4838    |` cres 4839   "cima 4840   -->wf 5409  (class class class)co 6040   Fincfn 7068   supcsup 7403   0cc0 8946    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077   + ecxad 10664   [,]cicc 10875   ↾s cress 13425   RR* scxrs 13677    gsumg cgsu 13679  CMndccmn 15367  Σ*cesum 24377
This theorem is referenced by:  esumlub  24405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-xadd 10667  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-tset 13503  df-ple 13504  df-ds 13506  df-rest 13605  df-topn 13606  df-topgen 13622  df-ordt 13680  df-xrs 13681  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-ps 14584  df-tsr 14585  df-mnd 14645  df-submnd 14694  df-cntz 15071  df-cmn 15369  df-fbas 16654  df-fg 16655  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-ntr 17039  df-nei 17117  df-cn 17245  df-haus 17333  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-tsms 18109  df-esum 24378
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