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Theorem esumsn 26515
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 26484 . . 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 2443 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  =  (
RR*ss  ( 0 [,] +oo ) )
4 snfi 7390 . . . . 5  |-  { M }  e.  Fin
54a1i 11 . . . 4  |-  ( ph  ->  { M }  e.  Fin )
6 elsni 3902 . . . . . . . . 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 4378 . . . . . . 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 5899 . . . . . . . 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 2478 . . . . . 6  |-  ( ph  ->  ( k  e.  { M }  |->  A )  =  { <. M ,  B >. } )
15 fsng 5882 . . . . . . 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 4018 . . . . 5  |-  ( ph  ->  { B }  C_  ( 0 [,] +oo ) )
19 fss 5567 . . . . 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 11118 . . . . . . 7  |-  <  Or  RR*
2221a1i 11 . . . . . 6  |-  ( ph  ->  <  Or  RR* )
23 0xr 9430 . . . . . . 7  |-  0  e.  RR*
2423a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
25 elxrge0 11394 . . . . . . . 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 7718 . . . . . 6  |-  ( (  <  Or  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  sup ( { 0 ,  B } ,  RR* ,  <  )  =  if ( B  <  0 ,  0 ,  B ) )
2922, 24, 27, 28syl3anc 1218 . . . . 5  |-  ( ph  ->  sup ( { 0 ,  B } ,  RR* ,  <  )  =  if ( B  <  0 ,  0 ,  B ) )
30 0fin 7540 . . . . . . . . . . 11  |-  (/)  e.  Fin
3130a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  Fin )
32 reseq2 5105 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( ( k  e.  { M }  |->  A )  |`  x )  =  ( ( k  e.  { M }  |->  A )  |`  (/) ) )
33 res0 5115 . . . . . . . . . . . . . 14  |-  ( ( k  e.  { M }  |->  A )  |`  (/) )  =  (/)
3432, 33syl6eq 2491 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( ( k  e.  { M }  |->  A )  |`  x )  =  (/) )
3534oveq2d 6107 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  (/) ) )
36 xrge00 26147 . . . . . . . . . . . . 13  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
3736gsum0 15510 . . . . . . . . . . . 12  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  (/) )  =  0
3835, 37syl6eq 2491 . . . . . . . . . . 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 5105 . . . . . . . . . . . . 13  |-  ( x  =  { M }  ->  ( ( k  e. 
{ M }  |->  A )  |`  x )  =  ( ( k  e.  { M }  |->  A )  |`  { M } ) )
41 ssid 3375 . . . . . . . . . . . . . 14  |-  { M }  C_  { M }
42 resmpt 5156 . . . . . . . . . . . . . 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 2491 . . . . . . . . . . . 12  |-  ( x  =  { M }  ->  ( ( k  e. 
{ M }  |->  A )  |`  x )  =  ( k  e. 
{ M }  |->  A ) )
4544oveq2d 6107 . . . . . . . . . . 11  |-  ( x  =  { M }  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  { M }  |->  A ) ) )
46 xrge0base 26146 . . . . . . . . . . . 12  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
47 xrge0cmn 17855 . . . . . . . . . . . . 13  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
48 cmnmnd 16292 . . . . . . . . . . . . 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 26243 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  { M }  |->  A ) )  =  B )
5145, 50sylan9eqr 2497 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  { M } )  -> 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  B )
5231, 5, 24, 11, 39, 51fmptpr 5903 . . . . . . . . 9  |-  ( ph  ->  { <. (/) ,  0 >. ,  <. { M } ,  B >. }  =  ( x  e.  { (/) ,  { M } }  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
53 pwsn 4085 . . . . . . . . . . . . 13  |-  ~P { M }  =  { (/)
,  { M } }
54 prssi 4029 . . . . . . . . . . . . . 14  |-  ( (
(/)  e.  Fin  /\  { M }  e.  Fin )  ->  { (/) ,  { M } }  C_  Fin )
5530, 4, 54mp2an 672 . . . . . . . . . . . . 13  |-  { (/) ,  { M } }  C_ 
Fin
5653, 55eqsstri 3386 . . . . . . . . . . . 12  |-  ~P { M }  C_  Fin
57 df-ss 3342 . . . . . . . . . . . 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 2463 . . . . . . . . . 10  |-  ( ~P { M }  i^i  Fin )  =  { (/) ,  { M } }
60 eqid 2443 . . . . . . . . . 10  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)
6159, 60mpteq12i 4376 . . . . . . . . 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 2493 . . . . . . . 8  |-  ( ph  ->  { <. (/) ,  0 >. ,  <. { M } ,  B >. }  =  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
6362rneqd 5067 . . . . . . 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 5319 . . . . . . . 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 2477 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) )  =  {
0 ,  B }
)
6766supeq1d 7696 . . . . 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 9442 . . . . . . . . . 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 2444 . . . . . . . 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 3827 . . . . . 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 2486 . . . 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 26253 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e. 
{ M }  |->  A ) )  =  { B } )
7978unieqd 4101 . 2  |-  ( ph  ->  U. ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  { M }  |->  A ) )  =  U. { B } )
80 unisng 4107 . . 3  |-  ( B  e.  ( 0 [,] +oo )  ->  U. { B }  =  B
)
8111, 80syl 16 . 2  |-  ( ph  ->  U. { B }  =  B )
822, 79, 813eqtrd 2479 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 1369    e. wcel 1756    i^i cin 3327    C_ wss 3328   (/)c0 3637   ifcif 3791   ~Pcpw 3860   {csn 3877   {cpr 3879   <.cop 3883   U.cuni 4091   class class class wbr 4292    e. cmpt 4350    Or wor 4640   ran crn 4841    |` cres 4842   -->wf 5414  (class class class)co 6091   Fincfn 7310   supcsup 7690   0cc0 9282   +oocpnf 9415   RR*cxr 9417    < clt 9418    <_ cle 9419   [,]cicc 11303   ↾s cress 14175    gsumg cgsu 14379   RR*scxrs 14438   Mndcmnd 15409  CMndccmn 16277   tsums ctsu 19696  Σ*cesum 26483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-xadd 11090  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-tset 14257  df-ple 14258  df-ds 14260  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-ordt 14439  df-xrs 14440  df-mre 14524  df-mrc 14525  df-acs 14527  df-ps 15370  df-tsr 15371  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-fbas 17814  df-fg 17815  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-ntr 18624  df-nei 18702  df-cn 18831  df-haus 18919  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-tsms 19697  df-esum 26484
This theorem is referenced by:  esumpr  26516  esumpr2  26517  esumfzf  26518  ddemeas  26652  oms0  26710
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