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Theorem sge0sup 38243
Description: The arbitrary sum of nonnegative extended reals is the supremum of finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0sup.x  |-  ( ph  ->  X  e.  V )
sge0sup.f  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
Assertion
Ref Expression
sge0sup  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
Distinct variable groups:    x, F    x, X    ph, x
Allowed substitution hint:    V( x)

Proof of Theorem sge0sup
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqidd 2454 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  = +oo )
2 sge0sup.x . . . . 5  |-  ( ph  ->  X  e.  V )
32adantr 467 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  X  e.  V )
4 sge0sup.f . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
54adantr 467 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F : X
--> ( 0 [,] +oo ) )
6 simpr 463 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
73, 5, 6sge0pnfval 38225 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  = +oo )
8 vex 3050 . . . . . . . . 9  |-  x  e. 
_V
98a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  _V )
104adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
11 elinel1 3621 . . . . . . . . . . 11  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  ~P X )
12 elpwi 3962 . . . . . . . . . . 11  |-  ( x  e.  ~P X  ->  x  C_  X )
1311, 12syl 17 . . . . . . . . . 10  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  C_  X )
1413adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  C_  X )
1510, 14fssresd 5755 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
169, 15sge0xrcl 38237 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x ) )  e.  RR* )
1716adantlr 722 . . . . . 6  |-  ( ( ( ph  /\ +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x
) )  e.  RR* )
1817ralrimiva 2804 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  A. x  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  x
) )  e.  RR* )
19 eqid 2453 . . . . . 6  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
2019rnmptss 6057 . . . . 5  |-  ( A. x  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  x ) )  e.  RR*  ->  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  C_  RR* )
2118, 20syl 17 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  C_  RR* )
22 ffn 5733 . . . . . . . . 9  |-  ( F : X --> ( 0 [,] +oo )  ->  F  Fn  X )
234, 22syl 17 . . . . . . . 8  |-  ( ph  ->  F  Fn  X )
24 fvelrnb 5917 . . . . . . . 8  |-  ( F  Fn  X  ->  ( +oo  e.  ran  F  <->  E. y  e.  X  ( F `  y )  = +oo ) )
2523, 24syl 17 . . . . . . 7  |-  ( ph  ->  ( +oo  e.  ran  F  <->  E. y  e.  X  ( F `  y )  = +oo ) )
2625adantr 467 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( +oo  e.  ran  F  <->  E. y  e.  X  ( F `  y )  = +oo ) )
276, 26mpbid 214 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  E. y  e.  X  ( F `  y )  = +oo )
28 snelpwi 4648 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  { y }  e.  ~P X
)
29 snfi 7655 . . . . . . . . . . . . 13  |-  { y }  e.  Fin
3029a1i 11 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  { y }  e.  Fin )
3128, 30elind 3620 . . . . . . . . . . 11  |-  ( y  e.  X  ->  { y }  e.  ( ~P X  i^i  Fin )
)
32313ad2ant2 1031 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  { y }  e.  ( ~P X  i^i  Fin ) )
33 simp2 1010 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  y  e.  X
)
3443ad2ant1 1030 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  F : X --> ( 0 [,] +oo ) )
3533snssd 4120 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  { y } 
C_  X )
3634, 35fssresd 5755 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F  |`  { y } ) : { y } --> ( 0 [,] +oo ) )
3733, 36sge0sn 38231 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  (Σ^ `  ( F  |`  { y } ) )  =  ( ( F  |`  { y } ) `
 y ) )
38 ssnid 3999 . . . . . . . . . . . . 13  |-  y  e. 
{ y }
39 fvres 5884 . . . . . . . . . . . . 13  |-  ( y  e.  { y }  ->  ( ( F  |`  { y } ) `
 y )  =  ( F `  y
) )
4038, 39ax-mp 5 . . . . . . . . . . . 12  |-  ( ( F  |`  { y } ) `  y
)  =  ( F `
 y )
4140a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( ( F  |`  { y } ) `
 y )  =  ( F `  y
) )
42 simp3 1011 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  = +oo )
4337, 41, 423eqtrrd 2492 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  =  (Σ^ `  ( F  |`  { y } ) ) )
44 reseq2 5103 . . . . . . . . . . . . 13  |-  ( x  =  { y }  ->  ( F  |`  x )  =  ( F  |`  { y } ) )
4544fveq2d 5874 . . . . . . . . . . . 12  |-  ( x  =  { y }  ->  (Σ^ `  ( F  |`  x
) )  =  (Σ^ `  ( F  |`  { y } ) ) )
4645eqeq2d 2463 . . . . . . . . . . 11  |-  ( x  =  { y }  ->  ( +oo  =  (Σ^ `  ( F  |`  x ) )  <-> +oo  =  (Σ^ `  ( F  |`  { y } ) ) ) )
4746rspcev 3152 . . . . . . . . . 10  |-  ( ( { y }  e.  ( ~P X  i^i  Fin )  /\ +oo  =  (Σ^ `  ( F  |`  { y } ) ) )  ->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  (Σ^ `  ( F  |`  x
) ) )
4832, 43, 47syl2anc 667 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  (Σ^ `  ( F  |`  x
) ) )
49 pnfex 11420 . . . . . . . . . 10  |- +oo  e.  _V
5049a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  _V )
5119, 48, 50elrnmptd 37463 . . . . . . . 8  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) )
52513exp 1208 . . . . . . 7  |-  ( ph  ->  ( y  e.  X  ->  ( ( F `  y )  = +oo  -> +oo  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) ) ) )
5352rexlimdv 2879 . . . . . 6  |-  ( ph  ->  ( E. y  e.  X  ( F `  y )  = +oo  -> +oo  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) ) )
5453adantr 467 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( E. y  e.  X  ( F `  y )  = +oo  -> +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) ) )
5527, 54mpd 15 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) )
56 supxrpnf 11611 . . . 4  |-  ( ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  C_  RR* 
/\ +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) )  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
5721, 55, 56syl2anc 667 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
581, 7, 573eqtr4d 2497 . 2  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
592adantr 467 . . . 4  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  X  e.  V )
604adantr 467 . . . . 5  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,] +oo ) )
61 simpr 463 . . . . 5  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  -. +oo  e.  ran  F )
6260, 61fge0iccico 38222 . . . 4  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,) +oo ) )
6359, 62sge0reval 38224 . . 3  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  (Σ^ `  F
)  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
64 elinel2 3622 . . . . . . . . 9  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  Fin )
6564adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
6615adantlr 722 . . . . . . . . 9  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
67 nelrnres 37472 . . . . . . . . . 10  |-  ( -. +oo  e.  ran  F  ->  -. +oo  e.  ran  ( F  |`  x ) )
6867ad2antlr 734 . . . . . . . . 9  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  -. +oo  e.  ran  ( F  |`  x ) )
6966, 68fge0iccico 38222 . . . . . . . 8  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,) +oo ) )
7065, 69sge0fsum 38239 . . . . . . 7  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x ) )  =  sum_ y  e.  x  ( ( F  |`  x ) `  y
) )
71 simpr 463 . . . . . . . . . 10  |-  ( ( x  e.  ( ~P X  i^i  Fin )  /\  y  e.  x
)  ->  y  e.  x )
72 fvres 5884 . . . . . . . . . 10  |-  ( y  e.  x  ->  (
( F  |`  x
) `  y )  =  ( F `  y ) )
7371, 72syl 17 . . . . . . . . 9  |-  ( ( x  e.  ( ~P X  i^i  Fin )  /\  y  e.  x
)  ->  ( ( F  |`  x ) `  y )  =  ( F `  y ) )
7473sumeq2dv 13781 . . . . . . . 8  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  sum_ y  e.  x  ( ( F  |`  x ) `  y )  =  sum_ y  e.  x  ( F `  y )
)
7574adantl 468 . . . . . . 7  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  sum_ y  e.  x  ( ( F  |`  x ) `  y )  =  sum_ y  e.  x  ( F `  y )
)
7670, 75eqtrd 2487 . . . . . 6  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x ) )  =  sum_ y  e.  x  ( F `  y ) )
7776mpteq2dva 4492 . . . . 5  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )
7877rneqd 5065 . . . 4  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  =  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) )
7978supeq1d 7965 . . 3  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  =  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
8063, 79eqtr4d 2490 . 2  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  (Σ^ `  F
)  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
8158, 80pm2.61dan 801 1  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740   _Vcvv 3047    i^i cin 3405    C_ wss 3406   ~Pcpw 3953   {csn 3970    |-> cmpt 4464   ran crn 4838    |` cres 4839    Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295   Fincfn 7574   supcsup 7959   0cc0 9544   +oocpnf 9677   RR*cxr 9679    < clt 9680   [,]cicc 11645   sum_csu 13764  Σ^csumge0 38214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  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 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-sumge0 38215
This theorem is referenced by:  sge0gerp  38247  sge0pnffigt  38248  sge0lefi  38250
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