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Theorem esumsn 27698
Description: The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Hypotheses
Ref Expression
esumsn.1  |-  ( (
ph  /\  k  =  M )  ->  A  =  B )
esumsn.2  |-  ( ph  ->  M  e.  V )
esumsn.3  |-  ( ph  ->  B  e.  ( 0 [,] +oo ) )
Assertion
Ref Expression
esumsn  |-  ( ph  -> Σ* k  e.  { M } A  =  B )
Distinct variable groups:    B, k    k, M    k, V    ph, k
Allowed substitution hint:    A( k)

Proof of Theorem esumsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-esum 27667 . . 3  |- Σ* k  e.  { M } A  =  U. ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e. 
{ M }  |->  A ) )
21a1i 11 . 2  |-  ( ph  -> Σ* k  e.  { M } A  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  { M }  |->  A ) ) )
3 eqid 2460 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  =  (
RR*ss  ( 0 [,] +oo ) )
4 snfi 7586 . . . . 5  |-  { M }  e.  Fin
54a1i 11 . . . 4  |-  ( ph  ->  { M }  e.  Fin )
6 elsni 4045 . . . . . . . . 9  |-  ( k  e.  { M }  ->  k  =  M )
7 esumsn.1 . . . . . . . . 9  |-  ( (
ph  /\  k  =  M )  ->  A  =  B )
86, 7sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { M } )  ->  A  =  B )
98mpteq2dva 4526 . . . . . . 7  |-  ( ph  ->  ( k  e.  { M }  |->  A )  =  ( k  e. 
{ M }  |->  B ) )
10 esumsn.2 . . . . . . . 8  |-  ( ph  ->  M  e.  V )
11 esumsn.3 . . . . . . . 8  |-  ( ph  ->  B  e.  ( 0 [,] +oo ) )
12 fmptsn 6072 . . . . . . . 8  |-  ( ( M  e.  V  /\  B  e.  ( 0 [,] +oo ) )  ->  { <. M ,  B >. }  =  ( k  e.  { M }  |->  B ) )
1310, 11, 12syl2anc 661 . . . . . . 7  |-  ( ph  ->  { <. M ,  B >. }  =  ( k  e.  { M }  |->  B ) )
149, 13eqtr4d 2504 . . . . . 6  |-  ( ph  ->  ( k  e.  { M }  |->  A )  =  { <. M ,  B >. } )
15 fsng 6051 . . . . . . 7  |-  ( ( M  e.  V  /\  B  e.  ( 0 [,] +oo ) )  ->  ( ( k  e.  { M }  |->  A ) : { M } --> { B }  <->  ( k  e.  { M }  |->  A )  =  { <. M ,  B >. } ) )
1610, 11, 15syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( k  e. 
{ M }  |->  A ) : { M }
--> { B }  <->  ( k  e.  { M }  |->  A )  =  { <. M ,  B >. } ) )
1714, 16mpbird 232 . . . . 5  |-  ( ph  ->  ( k  e.  { M }  |->  A ) : { M } --> { B } )
1811snssd 4165 . . . . 5  |-  ( ph  ->  { B }  C_  ( 0 [,] +oo ) )
19 fss 5730 . . . . 5  |-  ( ( ( k  e.  { M }  |->  A ) : { M } --> { B }  /\  { B }  C_  ( 0 [,] +oo ) )  ->  ( k  e. 
{ M }  |->  A ) : { M }
--> ( 0 [,] +oo ) )
2017, 18, 19syl2anc 661 . . . 4  |-  ( ph  ->  ( k  e.  { M }  |->  A ) : { M } --> ( 0 [,] +oo ) )
21 xrltso 11336 . . . . . . 7  |-  <  Or  RR*
2221a1i 11 . . . . . 6  |-  ( ph  ->  <  Or  RR* )
23 0xr 9629 . . . . . . 7  |-  0  e.  RR*
2423a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
25 elxrge0 11618 . . . . . . . 8  |-  ( B  e.  ( 0 [,] +oo )  <->  ( B  e. 
RR*  /\  0  <_  B ) )
2611, 25sylib 196 . . . . . . 7  |-  ( ph  ->  ( B  e.  RR*  /\  0  <_  B )
)
2726simpld 459 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
28 suppr 7918 . . . . . 6  |-  ( (  <  Or  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  sup ( { 0 ,  B } ,  RR* ,  <  )  =  if ( B  <  0 ,  0 ,  B ) )
2922, 24, 27, 28syl3anc 1223 . . . . 5  |-  ( ph  ->  sup ( { 0 ,  B } ,  RR* ,  <  )  =  if ( B  <  0 ,  0 ,  B ) )
30 0fin 7737 . . . . . . . . . . 11  |-  (/)  e.  Fin
3130a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  Fin )
32 reseq2 5259 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( ( k  e.  { M }  |->  A )  |`  x )  =  ( ( k  e.  { M }  |->  A )  |`  (/) ) )
33 res0 5269 . . . . . . . . . . . . . 14  |-  ( ( k  e.  { M }  |->  A )  |`  (/) )  =  (/)
3432, 33syl6eq 2517 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( ( k  e.  { M }  |->  A )  |`  x )  =  (/) )
3534oveq2d 6291 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  (/) ) )
36 xrge00 27322 . . . . . . . . . . . . 13  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
3736gsum0 15816 . . . . . . . . . . . 12  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  (/) )  =  0
3835, 37syl6eq 2517 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  0 )
3938adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  (/) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  0 )
40 reseq2 5259 . . . . . . . . . . . . 13  |-  ( x  =  { M }  ->  ( ( k  e. 
{ M }  |->  A )  |`  x )  =  ( ( k  e.  { M }  |->  A )  |`  { M } ) )
41 ssid 3516 . . . . . . . . . . . . . 14  |-  { M }  C_  { M }
42 resmpt 5314 . . . . . . . . . . . . . 14  |-  ( { M }  C_  { M }  ->  ( ( k  e.  { M }  |->  A )  |`  { M } )  =  ( k  e.  { M }  |->  A ) )
4341, 42ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( k  e.  { M }  |->  A )  |`  { M } )  =  ( k  e.  { M }  |->  A )
4440, 43syl6eq 2517 . . . . . . . . . . . 12  |-  ( x  =  { M }  ->  ( ( k  e. 
{ M }  |->  A )  |`  x )  =  ( k  e. 
{ M }  |->  A ) )
4544oveq2d 6291 . . . . . . . . . . 11  |-  ( x  =  { M }  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  { M }  |->  A ) ) )
46 xrge0base 27321 . . . . . . . . . . . 12  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
47 xrge0cmn 18221 . . . . . . . . . . . . 13  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
48 cmnmnd 16602 . . . . . . . . . . . . 13  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e. CMnd  ->  (
RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
4947, 48ax-mp 5 . . . . . . . . . . . 12  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd
5046, 49, 7, 10, 11gsumsn2 27418 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  { M }  |->  A ) )  =  B )
5145, 50sylan9eqr 2523 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  { M } )  -> 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  B )
5231, 5, 24, 11, 39, 51fmptpr 6077 . . . . . . . . 9  |-  ( ph  ->  { <. (/) ,  0 >. ,  <. { M } ,  B >. }  =  ( x  e.  { (/) ,  { M } }  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
53 pwsn 4232 . . . . . . . . . . . . 13  |-  ~P { M }  =  { (/)
,  { M } }
54 prssi 4176 . . . . . . . . . . . . . 14  |-  ( (
(/)  e.  Fin  /\  { M }  e.  Fin )  ->  { (/) ,  { M } }  C_  Fin )
5530, 4, 54mp2an 672 . . . . . . . . . . . . 13  |-  { (/) ,  { M } }  C_ 
Fin
5653, 55eqsstri 3527 . . . . . . . . . . . 12  |-  ~P { M }  C_  Fin
57 df-ss 3483 . . . . . . . . . . . 12  |-  ( ~P { M }  C_  Fin 
<->  ( ~P { M }  i^i  Fin )  =  ~P { M }
)
5856, 57mpbi 208 . . . . . . . . . . 11  |-  ( ~P { M }  i^i  Fin )  =  ~P { M }
5958, 53eqtri 2489 . . . . . . . . . 10  |-  ( ~P { M }  i^i  Fin )  =  { (/) ,  { M } }
60 eqid 2460 . . . . . . . . . 10  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)
6159, 60mpteq12i 4524 . . . . . . . . 9  |-  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) )  =  ( x  e.  { (/) ,  { M } }  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) )
6252, 61syl6eqr 2519 . . . . . . . 8  |-  ( ph  ->  { <. (/) ,  0 >. ,  <. { M } ,  B >. }  =  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
6362rneqd 5221 . . . . . . 7  |-  ( ph  ->  ran  { <. (/) ,  0
>. ,  <. { M } ,  B >. }  =  ran  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
64 rnpropg 5479 . . . . . . . 8  |-  ( (
(/)  e.  Fin  /\  { M }  e.  Fin )  ->  ran  { <. (/) ,  0
>. ,  <. { M } ,  B >. }  =  { 0 ,  B } )
6531, 5, 64syl2anc 661 . . . . . . 7  |-  ( ph  ->  ran  { <. (/) ,  0
>. ,  <. { M } ,  B >. }  =  { 0 ,  B } )
6663, 65eqtr3d 2503 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) )  =  {
0 ,  B }
)
6766supeq1d 7895 . . . . 5  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) ,  RR* ,  <  )  =  sup ( { 0 ,  B } ,  RR* ,  <  ) )
6826simprd 463 . . . . . . . . 9  |-  ( ph  ->  0  <_  B )
69 xrlenlt 9641 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  B  e.  RR* )  ->  (
0  <_  B  <->  -.  B  <  0 ) )
7024, 27, 69syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
7168, 70mpbid 210 . . . . . . . 8  |-  ( ph  ->  -.  B  <  0
)
72 eqidd 2461 . . . . . . . 8  |-  ( ph  ->  B  =  B )
7371, 72jca 532 . . . . . . 7  |-  ( ph  ->  ( -.  B  <  0  /\  B  =  B ) )
7473olcd 393 . . . . . 6  |-  ( ph  ->  ( ( B  <  0  /\  B  =  0 )  \/  ( -.  B  <  0  /\  B  =  B
) ) )
75 eqif 3970 . . . . . 6  |-  ( B  =  if ( B  <  0 ,  0 ,  B )  <->  ( ( B  <  0  /\  B  =  0 )  \/  ( -.  B  <  0  /\  B  =  B ) ) )
7674, 75sylibr 212 . . . . 5  |-  ( ph  ->  B  =  if ( B  <  0 ,  0 ,  B ) )
7729, 67, 763eqtr4rd 2512 . . . 4  |-  ( ph  ->  B  =  sup ( ran  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) ,  RR* ,  <  ) )
783, 5, 20, 77xrge0tsmsd 27424 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e. 
{ M }  |->  A ) )  =  { B } )
7978unieqd 4248 . 2  |-  ( ph  ->  U. ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  { M }  |->  A ) )  =  U. { B } )
80 unisng 4254 . . 3  |-  ( B  e.  ( 0 [,] +oo )  ->  U. { B }  =  B
)
8111, 80syl 16 . 2  |-  ( ph  ->  U. { B }  =  B )
822, 79, 813eqtrd 2505 1  |-  ( ph  -> Σ* k  e.  { M } A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469   (/)c0 3778   ifcif 3932   ~Pcpw 4003   {csn 4020   {cpr 4022   <.cop 4026   U.cuni 4238   class class class wbr 4440    |-> cmpt 4498    Or wor 4792   ran crn 4993    |` cres 4994   -->wf 5575  (class class class)co 6275   Fincfn 7506   supcsup 7889   0cc0 9481   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618   [,]cicc 11521   ↾s cress 14480    gsumg cgsu 14685   RR*scxrs 14744   Mndcmnd 15715  CMndccmn 16587   tsums ctsu 20352  Σ*cesum 27666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-xadd 11308  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-tset 14563  df-ple 14564  df-ds 14566  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-ordt 14745  df-xrs 14746  df-mre 14830  df-mrc 14831  df-acs 14833  df-ps 15676  df-tsr 15677  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-fbas 18180  df-fg 18181  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-ntr 19280  df-nei 19358  df-cn 19487  df-haus 19575  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-tsms 20353  df-esum 27667
This theorem is referenced by:  esumpr  27699  esumpr2  27700  esumfzf  27701  ddemeas  27834  oms0  27892
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