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Theorem esumpfinvalf 28971
Description: Same as esumpfinval 28970, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Proof shortened by AV, 25-Jul-2019.)
Hypotheses
Ref Expression
esumpfinvalf.1  |-  F/_ k A
esumpfinvalf.2  |-  F/ k
ph
esumpfinvalf.a  |-  ( ph  ->  A  e.  Fin )
esumpfinvalf.b  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )
Assertion
Ref Expression
esumpfinvalf  |-  ( ph  -> Σ* k  e.  A B  = 
sum_ k  e.  A  B )

Proof of Theorem esumpfinvalf
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 df-esum 28923 . . . 4  |- Σ* k  e.  A B  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )
2 xrge0base 28522 . . . . . 6  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
3 xrge00 28523 . . . . . 6  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
4 xrge0cmn 19087 . . . . . . 7  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
54a1i 11 . . . . . 6  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
6 xrge0tps 28822 . . . . . . 7  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  TopSp
76a1i 11 . . . . . 6  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e.  TopSp )
8 esumpfinvalf.a . . . . . 6  |-  ( ph  ->  A  e.  Fin )
9 esumpfinvalf.2 . . . . . . 7  |-  F/ k
ph
10 esumpfinvalf.1 . . . . . . 7  |-  F/_ k A
11 nfcv 2612 . . . . . . 7  |-  F/_ k
( 0 [,] +oo )
12 icossicc 11746 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
13 esumpfinvalf.b . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )
1412, 13sseldi 3416 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
15 eqid 2471 . . . . . . 7  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
169, 10, 11, 14, 15fmptdF 28331 . . . . . 6  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
17 c0ex 9655 . . . . . . . 8  |-  0  e.  _V
1817a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  _V )
1916, 8, 18fdmfifsupp 7911 . . . . . 6  |-  ( ph  ->  ( k  e.  A  |->  B ) finSupp  0 )
20 xrge0topn 28823 . . . . . . 7  |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
2120eqcomi 2480 . . . . . 6  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  (
TopOpen `  ( RR*ss  (
0 [,] +oo )
) )
22 xrhaus 28430 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Haus
23 ovex 6336 . . . . . . . 8  |-  ( 0 [,] +oo )  e. 
_V
24 resthaus 20461 . . . . . . . 8  |-  ( ( (ordTop `  <_  )  e. 
Haus  /\  ( 0 [,] +oo )  e.  _V )  ->  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Haus )
2522, 23, 24mp2an 686 . . . . . . 7  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Haus
2625a1i 11 . . . . . 6  |-  ( ph  ->  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)  e.  Haus )
272, 3, 5, 7, 8, 16, 19, 21, 26haustsmsid 21233 . . . . 5  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e.  A  |->  B ) )  =  { ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) ) } )
2827unieqd 4200 . . . 4  |-  ( ph  ->  U. ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )  =  U. {
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) } )
291, 28syl5eq 2517 . . 3  |-  ( ph  -> Σ* k  e.  A B  = 
U. { ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) ) } )
30 ovex 6336 . . . 4  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )  e. 
_V
3130unisn 4205 . . 3  |-  U. {
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) }  =  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  A  |->  B ) )
3229, 31syl6eq 2521 . 2  |-  ( ph  -> Σ* k  e.  A B  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
33 nfcv 2612 . . . 4  |-  F/_ k
( 0 [,) +oo )
349, 10, 33, 13, 15fmptdF 28331 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,) +oo ) )
35 esumpfinvallem 28969 . . 3  |-  ( ( A  e.  Fin  /\  ( k  e.  A  |->  B ) : A --> ( 0 [,) +oo ) )  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
368, 34, 35syl2anc 673 . 2  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  A  |->  B ) ) )
37 rge0ssre 11766 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
38 ax-resscn 9614 . . . . . . . 8  |-  RR  C_  CC
3937, 38sstri 3427 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
4039, 13sseldi 3416 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
4140sbt 2268 . . . . 5  |-  [ l  /  k ] ( ( ph  /\  k  e.  A )  ->  B  e.  CC )
42 sbim 2244 . . . . . 6  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  A )  ->  B  e.  CC )  <-> 
( [ l  / 
k ] ( ph  /\  k  e.  A )  ->  [ l  / 
k ] B  e.  CC ) )
43 sban 2248 . . . . . . . 8  |-  ( [ l  /  k ] ( ph  /\  k  e.  A )  <->  ( [
l  /  k ]
ph  /\  [ l  /  k ] k  e.  A ) )
449sbf 2229 . . . . . . . . 9  |-  ( [ l  /  k ]
ph 
<-> 
ph )
4510clelsb3f 28195 . . . . . . . . 9  |-  ( [ l  /  k ] k  e.  A  <->  l  e.  A )
4644, 45anbi12i 711 . . . . . . . 8  |-  ( ( [ l  /  k ] ph  /\  [ l  /  k ] k  e.  A )  <->  ( ph  /\  l  e.  A ) )
4743, 46bitri 257 . . . . . . 7  |-  ( [ l  /  k ] ( ph  /\  k  e.  A )  <->  ( ph  /\  l  e.  A ) )
48 sbsbc 3259 . . . . . . . 8  |-  ( [ l  /  k ] B  e.  CC  <->  [. l  / 
k ]. B  e.  CC )
49 vex 3034 . . . . . . . . 9  |-  l  e. 
_V
50 sbcel1g 3780 . . . . . . . . 9  |-  ( l  e.  _V  ->  ( [. l  /  k ]. B  e.  CC  <->  [_ l  /  k ]_ B  e.  CC )
)
5149, 50ax-mp 5 . . . . . . . 8  |-  ( [. l  /  k ]. B  e.  CC  <->  [_ l  /  k ]_ B  e.  CC )
5248, 51bitri 257 . . . . . . 7  |-  ( [ l  /  k ] B  e.  CC  <->  [_ l  / 
k ]_ B  e.  CC )
5347, 52imbi12i 333 . . . . . 6  |-  ( ( [ l  /  k ] ( ph  /\  k  e.  A )  ->  [ l  /  k ] B  e.  CC ) 
<->  ( ( ph  /\  l  e.  A )  ->  [_ l  /  k ]_ B  e.  CC ) )
5442, 53bitri 257 . . . . 5  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  A )  ->  B  e.  CC )  <-> 
( ( ph  /\  l  e.  A )  ->  [_ l  /  k ]_ B  e.  CC ) )
5541, 54mpbi 213 . . . 4  |-  ( (
ph  /\  l  e.  A )  ->  [_ l  /  k ]_ B  e.  CC )
568, 55gsumfsum 19111 . . 3  |-  ( ph  ->  (fld 
gsumg  ( l  e.  A  |-> 
[_ l  /  k ]_ B ) )  = 
sum_ l  e.  A  [_ l  /  k ]_ B )
57 nfcv 2612 . . . . 5  |-  F/_ l A
58 nfcv 2612 . . . . 5  |-  F/_ l B
59 nfcsb1v 3365 . . . . 5  |-  F/_ k [_ l  /  k ]_ B
60 csbeq1a 3358 . . . . 5  |-  ( k  =  l  ->  B  =  [_ l  /  k ]_ B )
6110, 57, 58, 59, 60cbvmptf 4486 . . . 4  |-  ( k  e.  A  |->  B )  =  ( l  e.  A  |->  [_ l  /  k ]_ B )
6261oveq2i 6319 . . 3  |-  (fld  gsumg  ( k  e.  A  |->  B ) )  =  (fld 
gsumg  ( l  e.  A  |-> 
[_ l  /  k ]_ B ) )
6360, 57, 10, 58, 59cbvsum 13838 . . 3  |-  sum_ k  e.  A  B  =  sum_ l  e.  A  [_ l  /  k ]_ B
6456, 62, 633eqtr4g 2530 . 2  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
6532, 36, 643eqtr2d 2511 1  |-  ( ph  -> Σ* k  e.  A B  = 
sum_ k  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   F/wnf 1675   [wsb 1805    e. wcel 1904   F/_wnfc 2599   _Vcvv 3031   [.wsbc 3255   [_csb 3349   {csn 3959   U.cuni 4190    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   +oocpnf 9690    <_ cle 9694   [,)cico 11662   [,]cicc 11663   sum_csu 13829   ↾s cress 15200   ↾t crest 15397   TopOpenctopn 15398    gsumg cgsu 15417  ordTopcordt 15475   RR*scxrs 15476  CMndccmn 17508  ℂfldccnfld 19047   TopSpctps 19996   Hauscha 20401   tsums ctsu 21218  Σ*cesum 28922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-xadd 11433  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-ordt 15477  df-xrs 15478  df-ps 16524  df-tsr 16525  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-esum 28923
This theorem is referenced by:  volfiniune  29126
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