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Theorem sge0gerp 38351
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 1769 . . 3  |-  F/ x ph
2 simpr 468 . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
5 elinel1 3610 . . . . . . . . 9  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  z  e.  ~P X )
6 elpwi 3951 . . . . . . . . 9  |-  ( z  e.  ~P X  -> 
z  C_  X )
75, 6syl 17 . . . . . . . 8  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  z  C_  X )
87adantl 473 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  z  C_  X )
94, 8fssresd 5762 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  z ) : z --> ( 0 [,] +oo ) )
102, 9sge0xrcl 38341 . . . . 5  |-  ( (
ph  /\  z  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  z ) )  e.  RR* )
1110ralrimiva 2809 . . . 4  |-  ( ph  ->  A. z  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  z
) )  e.  RR* )
12 eqid 2471 . . . . 5  |-  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )  =  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )
1312rnmptss 6068 . . . 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 1769 . . . . 5  |-  F/ z ( ph  /\  x  e.  RR+ )
18 nfmpt1 4485 . . . . . . 7  |-  F/_ z
( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )
1918nfrn 5083 . . . . . 6  |-  F/_ z ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) )
20 nfv 1769 . . . . . 6  |-  F/ z  A  <_  ( y +e x )
2119, 20nfrex 2848 . . . . 5  |-  F/ z E. y  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) A  <_  ( y +e x )
22 id 22 . . . . . . . . 9  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  z  e.  ( ~P X  i^i  Fin ) )
23 fvex 5889 . . . . . . . . . 10  |-  (Σ^ `  ( F  |`  z
) )  e.  _V
2423a1i 11 . . . . . . . . 9  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  (Σ^ `  ( F  |`  z ) )  e.  _V )
2512elrnmpt1 5089 . . . . . . . . 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 673 . . . . . . . 8  |-  ( z  e.  ( ~P X  i^i  Fin )  ->  (Σ^ `  ( F  |`  z ) )  e.  ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) )
27263ad2ant2 1052 . . . . . . 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 1032 . . . . . . 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 1769 . . . . . . . 8  |-  F/ y  A  <_  ( (Σ^ `  ( F  |`  z ) ) +e x )
30 oveq1 6315 . . . . . . . . 9  |-  ( y  =  (Σ^ `  ( F  |`  z
) )  ->  (
y +e x )  =  ( (Σ^ `  ( F  |`  z ) ) +e x ) )
3130breq2d 4407 . . . . . . . 8  |-  ( y  =  (Σ^ `  ( F  |`  z
) )  ->  ( A  <_  ( y +e x )  <->  A  <_  ( (Σ^ `  ( F  |`  z
) ) +e
x ) ) )
3229, 31rspce 3131 . . . . . . 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 673 . . . . . 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 1230 . . . . 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 2866 . . . 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 37648 . 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 38347 . . 3  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) , 
RR* ,  <  ) )
4039eqcomd 2477 . 2  |-  ( ph  ->  sup ( ran  (
z  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  z
) ) ) , 
RR* ,  <  )  =  (Σ^ `  F ) )
4137, 40breqtrd 4420 1  |-  ( ph  ->  A  <_  (Σ^ `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ 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   class class class wbr 4395    |-> cmpt 4454   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   0cc0 9557   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694   RR+crp 11325   +ecxad 11430   [,]cicc 11663  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-xadd 11433  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:  sge0gerpmpt  38358
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