Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumsnf Structured version   Visualization version   Unicode version

Theorem esumsnf 28959
Description: The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 2-May-2020.)
Hypotheses
Ref Expression
esumsnf.0  |-  F/_ k B
esumsnf.1  |-  ( (
ph  /\  k  =  M )  ->  A  =  B )
esumsnf.2  |-  ( ph  ->  M  e.  V )
esumsnf.3  |-  ( ph  ->  B  e.  ( 0 [,] +oo ) )
Assertion
Ref Expression
esumsnf  |-  ( ph  -> Σ* k  e.  { M } A  =  B )
Distinct variable groups:    k, M    ph, k
Allowed substitution hints:    A( k)    B( k)    V( k)

Proof of Theorem esumsnf
Dummy variables  x  l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-esum 28923 . . 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 2471 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  =  (
RR*ss  ( 0 [,] +oo ) )
4 snfi 7668 . . . . 5  |-  { M }  e.  Fin
54a1i 11 . . . 4  |-  ( ph  ->  { M }  e.  Fin )
6 elsni 3985 . . . . . . . . 9  |-  ( k  e.  { M }  ->  k  =  M )
7 esumsnf.1 . . . . . . . . 9  |-  ( (
ph  /\  k  =  M )  ->  A  =  B )
86, 7sylan2 482 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { M } )  ->  A  =  B )
98mpteq2dva 4482 . . . . . . 7  |-  ( ph  ->  ( k  e.  { M }  |->  A )  =  ( k  e. 
{ M }  |->  B ) )
10 esumsnf.2 . . . . . . . 8  |-  ( ph  ->  M  e.  V )
11 esumsnf.3 . . . . . . . 8  |-  ( ph  ->  B  e.  ( 0 [,] +oo ) )
12 fmptsn 6100 . . . . . . . . 9  |-  ( ( M  e.  V  /\  B  e.  ( 0 [,] +oo ) )  ->  { <. M ,  B >. }  =  ( l  e.  { M }  |->  B ) )
13 nfcv 2612 . . . . . . . . . 10  |-  F/_ l B
14 esumsnf.0 . . . . . . . . . 10  |-  F/_ k B
15 eqidd 2472 . . . . . . . . . 10  |-  ( k  =  l  ->  B  =  B )
1613, 14, 15cbvmpt 4487 . . . . . . . . 9  |-  ( k  e.  { M }  |->  B )  =  ( l  e.  { M }  |->  B )
1712, 16syl6eqr 2523 . . . . . . . 8  |-  ( ( M  e.  V  /\  B  e.  ( 0 [,] +oo ) )  ->  { <. M ,  B >. }  =  ( k  e.  { M }  |->  B ) )
1810, 11, 17syl2anc 673 . . . . . . 7  |-  ( ph  ->  { <. M ,  B >. }  =  ( k  e.  { M }  |->  B ) )
199, 18eqtr4d 2508 . . . . . 6  |-  ( ph  ->  ( k  e.  { M }  |->  A )  =  { <. M ,  B >. } )
20 fsng 6079 . . . . . . 7  |-  ( ( M  e.  V  /\  B  e.  ( 0 [,] +oo ) )  ->  ( ( k  e.  { M }  |->  A ) : { M } --> { B }  <->  ( k  e.  { M }  |->  A )  =  { <. M ,  B >. } ) )
2110, 11, 20syl2anc 673 . . . . . 6  |-  ( ph  ->  ( ( k  e. 
{ M }  |->  A ) : { M }
--> { B }  <->  ( k  e.  { M }  |->  A )  =  { <. M ,  B >. } ) )
2219, 21mpbird 240 . . . . 5  |-  ( ph  ->  ( k  e.  { M }  |->  A ) : { M } --> { B } )
2311snssd 4108 . . . . 5  |-  ( ph  ->  { B }  C_  ( 0 [,] +oo ) )
2422, 23fssd 5750 . . . 4  |-  ( ph  ->  ( k  e.  { M }  |->  A ) : { M } --> ( 0 [,] +oo ) )
25 xrltso 11463 . . . . . . 7  |-  <  Or  RR*
2625a1i 11 . . . . . 6  |-  ( ph  ->  <  Or  RR* )
27 0xr 9705 . . . . . . 7  |-  0  e.  RR*
2827a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
29 elxrge0 11767 . . . . . . . 8  |-  ( B  e.  ( 0 [,] +oo )  <->  ( B  e. 
RR*  /\  0  <_  B ) )
3011, 29sylib 201 . . . . . . 7  |-  ( ph  ->  ( B  e.  RR*  /\  0  <_  B )
)
3130simpld 466 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
32 suppr 8005 . . . . . 6  |-  ( (  <  Or  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  sup ( { 0 ,  B } ,  RR* ,  <  )  =  if ( B  <  0 ,  0 ,  B ) )
3326, 28, 31, 32syl3anc 1292 . . . . 5  |-  ( ph  ->  sup ( { 0 ,  B } ,  RR* ,  <  )  =  if ( B  <  0 ,  0 ,  B ) )
34 0fin 7817 . . . . . . . . . . 11  |-  (/)  e.  Fin
3534a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  Fin )
36 reseq2 5106 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( ( k  e.  { M }  |->  A )  |`  x )  =  ( ( k  e.  { M }  |->  A )  |`  (/) ) )
37 res0 5115 . . . . . . . . . . . . . 14  |-  ( ( k  e.  { M }  |->  A )  |`  (/) )  =  (/)
3836, 37syl6eq 2521 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( ( k  e.  { M }  |->  A )  |`  x )  =  (/) )
3938oveq2d 6324 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  (/) ) )
40 xrge00 28523 . . . . . . . . . . . . 13  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
4140gsum0 16599 . . . . . . . . . . . 12  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  (/) )  =  0
4239, 41syl6eq 2521 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  0 )
4342adantl 473 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  (/) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  0 )
44 reseq2 5106 . . . . . . . . . . . . 13  |-  ( x  =  { M }  ->  ( ( k  e. 
{ M }  |->  A )  |`  x )  =  ( ( k  e.  { M }  |->  A )  |`  { M } ) )
45 ssid 3437 . . . . . . . . . . . . . 14  |-  { M }  C_  { M }
46 resmpt 5160 . . . . . . . . . . . . . 14  |-  ( { M }  C_  { M }  ->  ( ( k  e.  { M }  |->  A )  |`  { M } )  =  ( k  e.  { M }  |->  A ) )
4745, 46ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( k  e.  { M }  |->  A )  |`  { M } )  =  ( k  e.  { M }  |->  A )
4844, 47syl6eq 2521 . . . . . . . . . . . 12  |-  ( x  =  { M }  ->  ( ( k  e. 
{ M }  |->  A )  |`  x )  =  ( k  e. 
{ M }  |->  A ) )
4948oveq2d 6324 . . . . . . . . . . 11  |-  ( x  =  { M }  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  { M }  |->  A ) ) )
50 xrge0base 28522 . . . . . . . . . . . 12  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
51 xrge0cmn 19087 . . . . . . . . . . . . . 14  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
52 cmnmnd 17523 . . . . . . . . . . . . . 14  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e. CMnd  ->  (
RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
5351, 52ax-mp 5 . . . . . . . . . . . . 13  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd
5453a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
55 nfv 1769 . . . . . . . . . . . 12  |-  F/ k
ph
5650, 54, 10, 11, 7, 55, 14gsumsnfd 17662 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  { M }  |->  A ) )  =  B )
5749, 56sylan9eqr 2527 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  { M } )  -> 
( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  B )
5835, 5, 28, 11, 43, 57fmptpr 6105 . . . . . . . . 9  |-  ( ph  ->  { <. (/) ,  0 >. ,  <. { M } ,  B >. }  =  ( x  e.  { (/) ,  { M } }  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
59 pwsn 4184 . . . . . . . . . . . . 13  |-  ~P { M }  =  { (/)
,  { M } }
60 prssi 4119 . . . . . . . . . . . . . 14  |-  ( (
(/)  e.  Fin  /\  { M }  e.  Fin )  ->  { (/) ,  { M } }  C_  Fin )
6134, 4, 60mp2an 686 . . . . . . . . . . . . 13  |-  { (/) ,  { M } }  C_ 
Fin
6259, 61eqsstri 3448 . . . . . . . . . . . 12  |-  ~P { M }  C_  Fin
63 df-ss 3404 . . . . . . . . . . . 12  |-  ( ~P { M }  C_  Fin 
<->  ( ~P { M }  i^i  Fin )  =  ~P { M }
)
6462, 63mpbi 213 . . . . . . . . . . 11  |-  ( ~P { M }  i^i  Fin )  =  ~P { M }
6564, 59eqtri 2493 . . . . . . . . . 10  |-  ( ~P { M }  i^i  Fin )  =  { (/) ,  { M } }
66 eqid 2471 . . . . . . . . . 10  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
)
6765, 66mpteq12i 4480 . . . . . . . . 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 )
) )
6858, 67syl6eqr 2523 . . . . . . . 8  |-  ( ph  ->  { <. (/) ,  0 >. ,  <. { M } ,  B >. }  =  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
6968rneqd 5068 . . . . . . 7  |-  ( ph  ->  ran  { <. (/) ,  0
>. ,  <. { M } ,  B >. }  =  ran  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) )
70 rnpropg 5323 . . . . . . . 8  |-  ( (
(/)  e.  Fin  /\  { M }  e.  Fin )  ->  ran  { <. (/) ,  0
>. ,  <. { M } ,  B >. }  =  { 0 ,  B } )
7135, 5, 70syl2anc 673 . . . . . . 7  |-  ( ph  ->  ran  { <. (/) ,  0
>. ,  <. { M } ,  B >. }  =  { 0 ,  B } )
7269, 71eqtr3d 2507 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) )  =  {
0 ,  B }
)
7372supeq1d 7978 . . . . 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* ,  <  ) )
7430simprd 470 . . . . . . . . 9  |-  ( ph  ->  0  <_  B )
75 xrlenlt 9717 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  B  e.  RR* )  ->  (
0  <_  B  <->  -.  B  <  0 ) )
7628, 31, 75syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
7774, 76mpbid 215 . . . . . . . 8  |-  ( ph  ->  -.  B  <  0
)
78 eqidd 2472 . . . . . . . 8  |-  ( ph  ->  B  =  B )
7977, 78jca 541 . . . . . . 7  |-  ( ph  ->  ( -.  B  <  0  /\  B  =  B ) )
8079olcd 400 . . . . . 6  |-  ( ph  ->  ( ( B  <  0  /\  B  =  0 )  \/  ( -.  B  <  0  /\  B  =  B
) ) )
81 eqif 3910 . . . . . 6  |-  ( B  =  if ( B  <  0 ,  0 ,  B )  <->  ( ( B  <  0  /\  B  =  0 )  \/  ( -.  B  <  0  /\  B  =  B ) ) )
8280, 81sylibr 217 . . . . 5  |-  ( ph  ->  B  =  if ( B  <  0 ,  0 ,  B ) )
8333, 73, 823eqtr4rd 2516 . . . 4  |-  ( ph  ->  B  =  sup ( ran  ( x  e.  ( ~P { M }  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( ( k  e. 
{ M }  |->  A )  |`  x )
) ) ,  RR* ,  <  ) )
843, 5, 24, 83xrge0tsmsd 28622 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e. 
{ M }  |->  A ) )  =  { B } )
8584unieqd 4200 . 2  |-  ( ph  ->  U. ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  { M }  |->  A ) )  =  U. { B } )
86 unisng 4206 . . 3  |-  ( B  e.  ( 0 [,] +oo )  ->  U. { B }  =  B
)
8711, 86syl 17 . 2  |-  ( ph  ->  U. { B }  =  B )
882, 85, 873eqtrd 2509 1  |-  ( ph  -> Σ* k  e.  { M } A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   F/_wnfc 2599    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   {csn 3959   {cpr 3961   <.cop 3965   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454    Or wor 4759   ran crn 4840    |` cres 4841   -->wf 5585  (class class class)co 6308   Fincfn 7587   supcsup 7972   0cc0 9557   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694   [,]cicc 11663   ↾s cress 15200    gsumg cgsu 15417   RR*scxrs 15476   Mndcmnd 16613  CMndccmn 17508   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
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  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-of 6550  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-inf 7975  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-q 11288  df-xadd 11433  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  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-tset 15287  df-ple 15288  df-ds 15290  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-ordt 15477  df-xrs 15478  df-mre 15570  df-mrc 15571  df-acs 15573  df-ps 16524  df-tsr 16525  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-ntr 20112  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:  esumsn  28960  esum2dlem  28987
  Copyright terms: Public domain W3C validator