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Theorem sge0sup 38347
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 2472 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  = +oo )
2 sge0sup.x . . . . 5  |-  ( ph  ->  X  e.  V )
32adantr 472 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  X  e.  V )
4 sge0sup.f . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
54adantr 472 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  F : X
--> ( 0 [,] +oo ) )
6 simpr 468 . . . 4  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
73, 5, 6sge0pnfval 38329 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  = +oo )
8 vex 3034 . . . . . . . . 9  |-  x  e. 
_V
98a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  _V )
104adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
11 elinel1 3610 . . . . . . . . . . 11  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  ~P X )
12 elpwi 3951 . . . . . . . . . . 11  |-  ( x  e.  ~P X  ->  x  C_  X )
1311, 12syl 17 . . . . . . . . . 10  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  C_  X )
1413adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  C_  X )
1510, 14fssresd 5762 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
169, 15sge0xrcl 38341 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x ) )  e.  RR* )
1716adantlr 729 . . . . . 6  |-  ( ( ( ph  /\ +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x
) )  e.  RR* )
1817ralrimiva 2809 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  A. x  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  x
) )  e.  RR* )
19 eqid 2471 . . . . . 6  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
2019rnmptss 6068 . . . . 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 5739 . . . . . . . . 9  |-  ( F : X --> ( 0 [,] +oo )  ->  F  Fn  X )
234, 22syl 17 . . . . . . . 8  |-  ( ph  ->  F  Fn  X )
24 fvelrnb 5926 . . . . . . . 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 472 . . . . . 6  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( +oo  e.  ran  F  <->  E. y  e.  X  ( F `  y )  = +oo ) )
276, 26mpbid 215 . . . . 5  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  E. y  e.  X  ( F `  y )  = +oo )
28 snelpwi 4645 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  { y }  e.  ~P X
)
29 snfi 7668 . . . . . . . . . . . . 13  |-  { y }  e.  Fin
3029a1i 11 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  { y }  e.  Fin )
3128, 30elind 3609 . . . . . . . . . . 11  |-  ( y  e.  X  ->  { y }  e.  ( ~P X  i^i  Fin )
)
32313ad2ant2 1052 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  { y }  e.  ( ~P X  i^i  Fin ) )
33 simp2 1031 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  y  e.  X
)
3443ad2ant1 1051 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  F : X --> ( 0 [,] +oo ) )
3533snssd 4108 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  { y } 
C_  X )
3634, 35fssresd 5762 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F  |`  { y } ) : { y } --> ( 0 [,] +oo ) )
3733, 36sge0sn 38335 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  (Σ^ `  ( F  |`  { y } ) )  =  ( ( F  |`  { y } ) `
 y ) )
38 ssnid 3989 . . . . . . . . . . . . 13  |-  y  e. 
{ y }
39 fvres 5893 . . . . . . . . . . . . 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 1032 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  ( F `  y )  = +oo )
4337, 41, 423eqtrrd 2510 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  =  (Σ^ `  ( F  |`  { y } ) ) )
44 reseq2 5106 . . . . . . . . . . . . 13  |-  ( x  =  { y }  ->  ( F  |`  x )  =  ( F  |`  { y } ) )
4544fveq2d 5883 . . . . . . . . . . . 12  |-  ( x  =  { y }  ->  (Σ^ `  ( F  |`  x
) )  =  (Σ^ `  ( F  |`  { y } ) ) )
4645eqeq2d 2481 . . . . . . . . . . 11  |-  ( x  =  { y }  ->  ( +oo  =  (Σ^ `  ( F  |`  x ) )  <-> +oo  =  (Σ^ `  ( F  |`  { y } ) ) ) )
4746rspcev 3136 . . . . . . . . . 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 673 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  ->  E. x  e.  ( ~P X  i^i  Fin ) +oo  =  (Σ^ `  ( F  |`  x
) ) )
49 pnfex 11436 . . . . . . . . . 10  |- +oo  e.  _V
5049a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  _V )
5119, 48, 50elrnmptd 37524 . . . . . . . 8  |-  ( (
ph  /\  y  e.  X  /\  ( F `  y )  = +oo )  -> +oo  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) )
52513exp 1230 . . . . . . 7  |-  ( ph  ->  ( y  e.  X  ->  ( ( F `  y )  = +oo  -> +oo  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) ) ) )
5352rexlimdv 2870 . . . . . 6  |-  ( ph  ->  ( E. y  e.  X  ( F `  y )  = +oo  -> +oo  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) ) )
5453adantr 472 . . . . 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 11629 . . . 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 673 . . 3  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
581, 7, 573eqtr4d 2515 . 2  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
592adantr 472 . . . 4  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  X  e.  V )
604adantr 472 . . . . 5  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,] +oo ) )
61 simpr 468 . . . . 5  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  -. +oo  e.  ran  F )
6260, 61fge0iccico 38326 . . . 4  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  F : X --> ( 0 [,) +oo ) )
6359, 62sge0reval 38328 . . 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 3611 . . . . . . . . 9  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  e.  Fin )
6564adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  Fin )
6615adantlr 729 . . . . . . . . 9  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
67 nelrnres 37533 . . . . . . . . . 10  |-  ( -. +oo  e.  ran  F  ->  -. +oo  e.  ran  ( F  |`  x ) )
6867ad2antlr 741 . . . . . . . . 9  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  -. +oo  e.  ran  ( F  |`  x ) )
6966, 68fge0iccico 38326 . . . . . . . 8  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,) +oo ) )
7065, 69sge0fsum 38343 . . . . . . 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 468 . . . . . . . . . 10  |-  ( ( x  e.  ( ~P X  i^i  Fin )  /\  y  e.  x
)  ->  y  e.  x )
72 fvres 5893 . . . . . . . . . 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 13846 . . . . . . . 8  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  sum_ y  e.  x  ( ( F  |`  x ) `  y )  =  sum_ y  e.  x  ( F `  y )
)
7574adantl 473 . . . . . . 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 2505 . . . . . 6  |-  ( ( ( ph  /\  -. +oo  e.  ran  F )  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x ) )  =  sum_ y  e.  x  ( F `  y ) )
7776mpteq2dva 4482 . . . . 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 5068 . . . 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 7978 . . 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 2508 . 2  |-  ( (
ph  /\  -. +oo  e.  ran  F )  ->  (Σ^ `  F
)  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
8158, 80pm2.61dan 808 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   {csn 3959    |-> cmpt 4454   ran crn 4840    |` cres 4841    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   0cc0 9557   +oocpnf 9690   RR*cxr 9692    < clt 9693   [,]cicc 11663   sum_csu 13829  Σ^csumge0 38318
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-fal 1458  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-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-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  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-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-sumge0 38319
This theorem is referenced by:  sge0gerp  38351  sge0pnffigt  38352  sge0lefi  38354
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