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Theorem sge0gerp 38025
Description: The arbitrary sum of nonnegative extended reals is larger or equal to a given extended real number if this number can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0gerp.x  |-  ( ph  ->  X  e.  V )
sge0gerp.f  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
sge0gerp.a  |-  ( ph  ->  A  e.  RR* )
sge0gerp.z  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. z  e.  ( ~P X  i^i  Fin ) A  <_  (
(Σ^ `  ( F  |`  z
) ) +e
x ) )
Assertion
Ref Expression
sge0gerp  |-  ( ph  ->  A  <_  (Σ^ `  F ) )
Distinct variable groups:    x, A, z    x, F, z    x, X, z    ph, x, z
Allowed substitution hints:    V( x, z)

Proof of Theorem sge0gerp
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfv 1751 . . 3  |-  F/ x ph
2 simpr 462 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  z  e.  ( ~P X  i^i  Fin ) )
3 sge0gerp.f . . . . . . . 8  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
43adantr 466 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
5 elinel1 3651 . . . . . . . . 9  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  z  e.  ~P X )
6 elpwi 3988 . . . . . . . . 9  |-  ( z  e.  ~P X  -> 
z  C_  X )
75, 6syl 17 . . . . . . . 8  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  z  C_  X )
87adantl 467 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  z  C_  X )
94, 8fssresd 5764 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  z ) : z --> ( 0 [,] +oo ) )
102, 9sge0xrcl 38015 . . . . 5  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  z ) )  e.  RR* )
1110ralrimiva 2839 . . . 4  |-  ( ph  ->  A. z  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  z
) )  e.  RR* )
12 eqid 2422 . . . . 5  |-  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )  =  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )
1312rnmptss 6064 . . . 4  |-  ( A. z  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  z ) )  e.  RR*  ->  ran  (
z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )  C_  RR* )
1411, 13syl 17 . . 3  |-  ( ph  ->  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )  C_  RR* )
15 sge0gerp.a . . 3  |-  ( ph  ->  A  e.  RR* )
16 sge0gerp.z . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. z  e.  ( ~P X  i^i  Fin ) A  <_  (
(Σ^ `  ( F  |`  z
) ) +e
x ) )
17 nfv 1751 . . . . 5  |-  F/ z ( ph  /\  x  e.  RR+ )
18 nfmpt1 4510 . . . . . . 7  |-  F/_ z
( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )
1918nfrn 5093 . . . . . 6  |-  F/_ z ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )
20 nfv 1751 . . . . . 6  |-  F/ z  A  <_  ( y +e x )
2119, 20nfrex 2888 . . . . 5  |-  F/ z E. y  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) A  <_  ( y +e x )
22 id 23 . . . . . . . . 9  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  z  e.  ( ~P X  i^i  Fin ) )
23 fvex 5888 . . . . . . . . . 10  |-  (Σ^ `  ( F  |`  z
) )  e.  _V
2423a1i 11 . . . . . . . . 9  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  (Σ^ `  ( F  |`  z ) )  e.  _V )
2512elrnmpt1 5099 . . . . . . . . 9  |-  ( ( z  e.  ( ~P X  i^i  Fin )  /\  (Σ^ `  ( F  |`  z
) )  e.  _V )  ->  (Σ^ `  ( F  |`  z
) )  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) )
2622, 24, 25syl2anc 665 . . . . . . . 8  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  (Σ^ `  ( F  |`  z ) )  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) )
27263ad2ant2 1027 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  z  e.  ( ~P X  i^i  Fin )  /\  A  <_ 
( (Σ^ `  ( F  |`  z
) ) +e
x ) )  -> 
(Σ^ `  ( F  |`  z
) )  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) )
28 simp3 1007 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  z  e.  ( ~P X  i^i  Fin )  /\  A  <_ 
( (Σ^ `  ( F  |`  z
) ) +e
x ) )  ->  A  <_  ( (Σ^ `  ( F  |`  z
) ) +e
x ) )
29 nfv 1751 . . . . . . . 8  |-  F/ y  A  <_  ( (Σ^ `  ( F  |`  z ) ) +e x )
30 oveq1 6309 . . . . . . . . 9  |-  ( y  =  (Σ^ `  ( F  |`  z
) )  ->  (
y +e x )  =  ( (Σ^ `  ( F  |`  z ) ) +e x ) )
3130breq2d 4432 . . . . . . . 8  |-  ( y  =  (Σ^ `  ( F  |`  z
) )  ->  ( A  <_  ( y +e x )  <->  A  <_  ( (Σ^ `  ( F  |`  z
) ) +e
x ) ) )
3229, 31rspce 3177 . . . . . . 7  |-  ( ( (Σ^ `  ( F  |`  z
) )  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )  /\  A  <_  ( (Σ^ `  ( F  |`  z
) ) +e
x ) )  ->  E. y  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) A  <_  ( y +e x ) )
3327, 28, 32syl2anc 665 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  z  e.  ( ~P X  i^i  Fin )  /\  A  <_ 
( (Σ^ `  ( F  |`  z
) ) +e
x ) )  ->  E. y  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) A  <_  ( y +e x ) )
34333exp 1204 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( z  e.  ( ~P X  i^i  Fin )  ->  ( A  <_  ( (Σ^ `  ( F  |`  z
) ) +e
x )  ->  E. y  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) A  <_  ( y +e x ) ) ) )
3517, 21, 34rexlimd 2909 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( E. z  e.  ( ~P X  i^i  Fin ) A  <_  ( (Σ^ `  ( F  |`  z
) ) +e
x )  ->  E. y  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) A  <_  ( y +e x ) ) )
3616, 35mpd 15 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) A  <_  ( y +e x ) )
371, 14, 15, 36supxrge 37403 . 2  |-  ( ph  ->  A  <_  sup ( ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) , 
RR* ,  <  ) )
38 sge0gerp.x . . . 4  |-  ( ph  ->  X  e.  V )
3938, 3sge0sup 38021 . . 3  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) , 
RR* ,  <  ) )
4039eqcomd 2430 . 2  |-  ( ph  ->  sup ( ran  (
z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) , 
RR* ,  <  )  =  (Σ^ `  F ) )
4137, 40breqtrd 4445 1  |-  ( ph  ->  A  <_  (Σ^ `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   _Vcvv 3081    i^i cin 3435    C_ wss 3436   ~Pcpw 3979   class class class wbr 4420    |-> cmpt 4479   ran crn 4851    |` cres 4852   -->wf 5594   ` cfv 5598  (class class class)co 6302   Fincfn 7574   supcsup 7957   0cc0 9540   +oocpnf 9673   RR*cxr 9675    < clt 9676    <_ cle 9677   RR+crp 11303   +ecxad 11408   [,]cicc 11639  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-xadd 11411  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:  sge0gerpmpt  38032
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