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Theorem sge0supre 38019
Description: If the arbitrary sum of nonnegative extended reals is real, then it is the supremum (in the real numbers) of finite subsums. Similar to sge0sup 38021, but here we can use  sup with respect to  RR instead of  RR* (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0supre.x  |-  ( ph  ->  X  e.  V )
sge0supre.f  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
sge0supre.re  |-  ( ph  ->  (Σ^ `  F )  e.  RR )
Assertion
Ref Expression
sge0supre  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR ,  <  ) )
Distinct variable groups:    x, F, y    x, X, y    ph, x, y
Allowed substitution hints:    V( x, y)

Proof of Theorem sge0supre
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0supre.x . . 3  |-  ( ph  ->  X  e.  V )
2 sge0supre.f . . . 4  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
31adantr 466 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  X  e.  V )
42adantr 466 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F : X
--> ( 0 [,] +oo ) )
5 simpr 462 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
63, 4, 5sge0pnfval 38003 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  = +oo )
7 sge0supre.re . . . . . . 7  |-  ( ph  ->  (Σ^ `  F )  e.  RR )
81, 2sge0repnf 38016 . . . . . . 7  |-  ( ph  ->  ( (Σ^ `  F )  e.  RR  <->  -.  (Σ^ `  F )  = +oo ) )
97, 8mpbid 213 . . . . . 6  |-  ( ph  ->  -.  (Σ^ `  F )  = +oo )
109adantr 466 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  -.  (Σ^ `  F
)  = +oo )
116, 10pm2.65da 578 . . . 4  |-  ( ph  ->  -. +oo  e.  ran  F )
122, 11fge0iccico 38000 . . 3  |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )
131, 12sge0reval 38002 . 2  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
1412sge0rnre 37994 . . 3  |-  ( ph  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  C_  RR )
15 sge0rnn0 37998 . . . 4  |-  ran  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =/=  (/)
1615a1i 11 . . 3  |-  ( ph  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =/=  (/) )
17 simpr 462 . . . . . . 7  |-  ( (
ph  /\  w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )  ->  w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )
18 eqid 2422 . . . . . . . . 9  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )
1918elrnmpt 5097 . . . . . . . 8  |-  ( w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  -> 
( w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  <->  E. x  e.  ( ~P X  i^i  Fin ) w  =  sum_ y  e.  x  ( F `  y )
) )
2019adantl 467 . . . . . . 7  |-  ( (
ph  /\  w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )  ->  (
w  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  <->  E. x  e.  ( ~P X  i^i  Fin ) w  =  sum_ y  e.  x  ( F `  y )
) )
2117, 20mpbid 213 . . . . . 6  |-  ( (
ph  /\  w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )  ->  E. x  e.  ( ~P X  i^i  Fin ) w  =  sum_ y  e.  x  ( F `  y )
)
22 simp3 1007 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin )  /\  w  =  sum_ y  e.  x  ( F `  y )
)  ->  w  =  sum_ y  e.  x  ( F `  y ) )
23 ressxr 9685 . . . . . . . . . . . . . . . 16  |-  RR  C_  RR*
2423a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  C_  RR* )
2514, 24sstrd 3474 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  C_  RR* )
2625adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  C_  RR* )
27 id 23 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  ( ~P X  i^i  Fin ) )
28 sumex 13742 . . . . . . . . . . . . . . . 16  |-  sum_ y  e.  x  ( F `  y )  e.  _V
2928a1i 11 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  sum_ y  e.  x  ( F `  y )  e.  _V )
3018elrnmpt1 5099 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( ~P X  i^i  Fin )  /\  sum_ y  e.  x  ( F `  y )  e.  _V )  ->  sum_ y  e.  x  ( F `  y )  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )
3127, 29, 30syl2anc 665 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  sum_ y  e.  x  ( F `  y )  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) )
3231adantl 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sum_ y  e.  x  ( F `  y )  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) )
33 supxrub 11611 . . . . . . . . . . . . 13  |-  ( ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  C_  RR*  /\  sum_ y  e.  x  ( F `  y )  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )  ->  sum_ y  e.  x  ( F `  y )  <_  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
3426, 32, 33syl2anc 665 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sum_ y  e.  x  ( F `  y )  <_  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
3513eqcomd 2430 . . . . . . . . . . . . 13  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  )  =  (Σ^ `  F ) )
3635adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  )  =  (Σ^ `  F ) )
3734, 36breqtrd 4445 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sum_ y  e.  x  ( F `  y )  <_  (Σ^ `  F
) )
38373adant3 1025 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin )  /\  w  =  sum_ y  e.  x  ( F `  y )
)  ->  sum_ y  e.  x  ( F `  y )  <_  (Σ^ `  F
) )
3922, 38eqbrtrd 4441 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin )  /\  w  =  sum_ y  e.  x  ( F `  y )
)  ->  w  <_  (Σ^ `  F
) )
40393exp 1204 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( ~P X  i^i  Fin )  ->  ( w  = 
sum_ y  e.  x  ( F `  y )  ->  w  <_  (Σ^ `  F
) ) ) )
4140rexlimdv 2915 . . . . . . 7  |-  ( ph  ->  ( E. x  e.  ( ~P X  i^i  Fin ) w  =  sum_ y  e.  x  ( F `  y )  ->  w  <_  (Σ^ `  F ) ) )
4241adantr 466 . . . . . 6  |-  ( (
ph  /\  w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )  ->  ( E. x  e.  ( ~P X  i^i  Fin )
w  =  sum_ y  e.  x  ( F `  y )  ->  w  <_  (Σ^ `  F ) ) )
4321, 42mpd 15 . . . . 5  |-  ( (
ph  /\  w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )  ->  w  <_  (Σ^ `  F ) )
4443ralrimiva 2839 . . . 4  |-  ( ph  ->  A. w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) w  <_  (Σ^ `  F
) )
45 breq2 4424 . . . . . 6  |-  ( z  =  (Σ^ `  F )  ->  (
w  <_  z  <->  w  <_  (Σ^ `  F
) ) )
4645ralbidv 2864 . . . . 5  |-  ( z  =  (Σ^ `  F )  ->  ( A. w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) w  <_  z  <->  A. w  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) w  <_  (Σ^ `  F
) ) )
4746rspcev 3182 . . . 4  |-  ( ( (Σ^ `  F )  e.  RR  /\ 
A. w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) w  <_  (Σ^ `  F
) )  ->  E. z  e.  RR  A. w  e. 
ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) w  <_  z
)
487, 44, 47syl2anc 665 . . 3  |-  ( ph  ->  E. z  e.  RR  A. w  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) w  <_  z
)
49 supxrre 11614 . . 3  |-  ( ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  C_  RR  /\  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =/=  (/)  /\  E. z  e.  RR  A. w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) w  <_  z
)  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR ,  <  ) )
5014, 16, 48, 49syl3anc 1264 . 2  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR ,  <  ) )
5113, 50eqtrd 2463 1  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081    i^i cin 3435    C_ wss 3436   (/)c0 3761   ~Pcpw 3979   class class class wbr 4420    |-> cmpt 4479   ran crn 4851   -->wf 5594   ` cfv 5598  (class class class)co 6302   Fincfn 7574   supcsup 7957   RRcr 9539   0cc0 9540   +oocpnf 9673   RR*cxr 9675    < clt 9676    <_ cle 9677   [,]cicc 11639   sum_csu 13740  Σ^csumge0 37992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-sumge0 37993
This theorem is referenced by:  sge0ltfirp  38030  sge0resplit  38036
  Copyright terms: Public domain W3C validator