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Theorem sge0pnffigt 38275
Description: If the sum of nonnegative extended reals is +oo, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0pnffigt.x  |-  ( ph  ->  X  e.  V )
sge0pnffigt.f  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
sge0pnffigt.pnf  |-  ( ph  ->  (Σ^ `  F )  = +oo )
sge0pnffigt.y  |-  ( ph  ->  Y  e.  RR )
Assertion
Ref Expression
sge0pnffigt  |-  ( ph  ->  E. x  e.  ( ~P X  i^i  Fin ) Y  <  (Σ^ `  ( F  |`  x
) ) )
Distinct variable groups:    x, F    x, X    x, Y    ph, x
Allowed substitution hint:    V( x)

Proof of Theorem sge0pnffigt
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0pnffigt.y . . 3  |-  ( ph  ->  Y  e.  RR )
2 sge0pnffigt.x . . . . . 6  |-  ( ph  ->  X  e.  V )
3 sge0pnffigt.f . . . . . 6  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
42, 3sge0sup 38270 . . . . 5  |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  ) )
5 sge0pnffigt.pnf . . . . 5  |-  ( ph  ->  (Σ^ `  F )  = +oo )
64, 5eqtr3d 2497 . . . 4  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
7 vex 3059 . . . . . . . . 9  |-  x  e. 
_V
87a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  _V )
93adantr 471 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
10 elpwinss 37424 . . . . . . . . . 10  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  C_  X )
1110adantl 472 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  C_  X )
129, 11fssresd 5772 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
138, 12sge0xrcl 38264 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x ) )  e.  RR* )
1413ralrimiva 2813 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  x
) )  e.  RR* )
15 eqid 2461 . . . . . . 7  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
1615rnmptss 6075 . . . . . 6  |-  ( 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* )
1714, 16syl 17 . . . . 5  |-  ( ph  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  C_  RR* )
18 supxrunb2 11634 . . . . 5  |-  ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  C_  RR* 
->  ( A. y  e.  RR  E. z  e. 
ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) y  <  z  <->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo ) )
1917, 18syl 17 . . . 4  |-  ( ph  ->  ( A. y  e.  RR  E. z  e. 
ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) y  <  z  <->  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo ) )
206, 19mpbird 240 . . 3  |-  ( ph  ->  A. y  e.  RR  E. z  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) y  <  z )
21 breq1 4418 . . . . 5  |-  ( y  =  Y  ->  (
y  <  z  <->  Y  <  z ) )
2221rexbidv 2912 . . . 4  |-  ( y  =  Y  ->  ( E. z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) y  <  z  <->  E. z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) Y  <  z ) )
2322rspcva 3159 . . 3  |-  ( ( Y  e.  RR  /\  A. y  e.  RR  E. z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) y  <  z )  ->  E. z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) Y  <  z )
241, 20, 23syl2anc 671 . 2  |-  ( ph  ->  E. z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) Y  <  z )
25 vex 3059 . . . . . . . 8  |-  z  e. 
_V
2615elrnmpt 5099 . . . . . . . 8  |-  ( z  e.  _V  ->  (
z  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  <->  E. x  e.  ( ~P X  i^i  Fin ) z  =  (Σ^ `  ( F  |`  x ) ) ) )
2725, 26ax-mp 5 . . . . . . 7  |-  ( z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  <->  E. x  e.  ( ~P X  i^i  Fin ) z  =  (Σ^ `  ( F  |`  x ) ) )
2827biimpi 199 . . . . . 6  |-  ( z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  ->  E. x  e.  ( ~P X  i^i  Fin )
z  =  (Σ^ `  ( F  |`  x
) ) )
29283ad2ant2 1036 . . . . 5  |-  ( (
ph  /\  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  /\  Y  <  z )  ->  E. x  e.  ( ~P X  i^i  Fin )
z  =  (Σ^ `  ( F  |`  x
) ) )
30 nfv 1771 . . . . . . 7  |-  F/ x ph
31 nfcv 2602 . . . . . . . 8  |-  F/_ x
z
32 nfmpt1 4505 . . . . . . . . 9  |-  F/_ x
( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
3332nfrn 5095 . . . . . . . 8  |-  F/_ x ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
3431, 33nfel 2614 . . . . . . 7  |-  F/ x  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
35 nfv 1771 . . . . . . 7  |-  F/ x  Y  <  z
3630, 34, 35nf3an 2023 . . . . . 6  |-  F/ x
( ph  /\  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  /\  Y  <  z )
37 simpl 463 . . . . . . . . . . 11  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  ->  Y  <  z )
38 simpr 467 . . . . . . . . . . . 12  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  -> 
z  =  (Σ^ `  ( F  |`  x
) ) )
3938breq2d 4427 . . . . . . . . . . 11  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  -> 
( Y  <  z  <->  Y  <  (Σ^ `  ( F  |`  x
) ) ) )
4037, 39mpbid 215 . . . . . . . . . 10  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) )
4140ex 440 . . . . . . . . 9  |-  ( Y  <  z  ->  (
z  =  (Σ^ `  ( F  |`  x
) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) ) )
4241adantl 472 . . . . . . . 8  |-  ( (
ph  /\  Y  <  z )  ->  ( z  =  (Σ^ `  ( F  |`  x
) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) ) )
4342a1d 26 . . . . . . 7  |-  ( (
ph  /\  Y  <  z )  ->  ( x  e.  ( ~P X  i^i  Fin )  ->  ( z  =  (Σ^ `  ( F  |`  x
) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) ) ) )
44433adant2 1033 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  /\  Y  <  z )  -> 
( x  e.  ( ~P X  i^i  Fin )  ->  ( z  =  (Σ^ `  ( F  |`  x
) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) ) ) )
4536, 44reximdai 2867 . . . . 5  |-  ( (
ph  /\  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  /\  Y  <  z )  -> 
( E. x  e.  ( ~P X  i^i  Fin ) z  =  (Σ^ `  ( F  |`  x ) )  ->  E. x  e.  ( ~P X  i^i  Fin ) Y  <  (Σ^ `  ( F  |`  x
) ) ) )
4629, 45mpd 15 . . . 4  |-  ( (
ph  /\  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  /\  Y  <  z )  ->  E. x  e.  ( ~P X  i^i  Fin ) Y  <  (Σ^ `  ( F  |`  x
) ) )
47463exp 1214 . . 3  |-  ( ph  ->  ( z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  -> 
( Y  <  z  ->  E. x  e.  ( ~P X  i^i  Fin ) Y  <  (Σ^ `  ( F  |`  x
) ) ) ) )
4847rexlimdv 2888 . 2  |-  ( ph  ->  ( E. z  e. 
ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) Y  <  z  ->  E. x  e.  ( ~P X  i^i  Fin ) Y  <  (Σ^ `  ( F  |`  x ) ) ) )
4924, 48mpd 15 1  |-  ( ph  ->  E. x  e.  ( ~P X  i^i  Fin ) Y  <  (Σ^ `  ( F  |`  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   A.wral 2748   E.wrex 2749   _Vcvv 3056    i^i cin 3414    C_ wss 3415   ~Pcpw 3962   class class class wbr 4415    |-> cmpt 4474   ran crn 4853    |` cres 4854   -->wf 5596   ` cfv 5600  (class class class)co 6314   Fincfn 7594   supcsup 7979   RRcr 9563   0cc0 9564   +oocpnf 9697   RR*cxr 9699    < clt 9700   [,]cicc 11666  Σ^csumge0 38241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-rp 11331  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-sum 13801  df-sumge0 38242
This theorem is referenced by:  sge0pnffigtmpt  38319  omeiunltfirp  38377
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