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Theorem sge0pnffigt 38070
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 38065 . . . . 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 2466 . . . 4  |-  ( ph  ->  sup ( ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) , 
RR* ,  <  )  = +oo )
7 vex 3085 . . . . . . . . 9  |-  x  e. 
_V
87a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  e.  _V )
93adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,] +oo ) )
10 elpwinss 37293 . . . . . . . . . 10  |-  ( x  e.  ( ~P X  i^i  Fin )  ->  x  C_  X )
1110adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  x  C_  X )
129, 11fssresd 5765 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  x ) : x --> ( 0 [,] +oo ) )
138, 12sge0xrcl 38059 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  x ) )  e.  RR* )
1413ralrimiva 2840 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  x
) )  e.  RR* )
15 eqid 2423 . . . . . . 7  |-  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  =  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
1615rnmptss 6065 . . . . . 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 11608 . . . . 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 236 . . 3  |-  ( ph  ->  A. y  e.  RR  E. z  e.  ran  (
x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) y  <  z )
21 breq1 4424 . . . . 5  |-  ( y  =  Y  ->  (
y  <  z  <->  Y  <  z ) )
2221rexbidv 2940 . . . 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 3181 . . 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 666 . 2  |-  ( ph  ->  E. z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) ) Y  <  z )
25 vex 3085 . . . . . . . 8  |-  z  e. 
_V
2615elrnmpt 5098 . . . . . . . 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 198 . . . . . 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 1028 . . . . 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 1752 . . . . . . 7  |-  F/ x ph
31 nfcv 2585 . . . . . . . 8  |-  F/_ x
z
32 nfmpt1 4511 . . . . . . . . 9  |-  F/_ x
( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
3332nfrn 5094 . . . . . . . 8  |-  F/_ x ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
3431, 33nfel 2598 . . . . . . 7  |-  F/ x  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )
35 nfv 1752 . . . . . . 7  |-  F/ x  Y  <  z
3630, 34, 35nf3an 1987 . . . . . 6  |-  F/ x
( ph  /\  z  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x
) ) )  /\  Y  <  z )
37 simpl 459 . . . . . . . . . . 11  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  ->  Y  <  z )
38 simpr 463 . . . . . . . . . . . 12  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  -> 
z  =  (Σ^ `  ( F  |`  x
) ) )
3938breq2d 4433 . . . . . . . . . . 11  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  -> 
( Y  <  z  <->  Y  <  (Σ^ `  ( F  |`  x
) ) ) )
4037, 39mpbid 214 . . . . . . . . . 10  |-  ( ( Y  <  z  /\  z  =  (Σ^ `  ( F  |`  x
) ) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) )
4140ex 436 . . . . . . . . 9  |-  ( Y  <  z  ->  (
z  =  (Σ^ `  ( F  |`  x
) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) ) )
4241adantl 468 . . . . . . . 8  |-  ( (
ph  /\  Y  <  z )  ->  ( z  =  (Σ^ `  ( F  |`  x
) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) ) )
4342a1d 27 . . . . . . 7  |-  ( (
ph  /\  Y  <  z )  ->  ( x  e.  ( ~P X  i^i  Fin )  ->  ( z  =  (Σ^ `  ( F  |`  x
) )  ->  Y  <  (Σ^ `  ( F  |`  x
) ) ) ) )
44433adant2 1025 . . . . . 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 2895 . . . . 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 1205 . . 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 2916 . 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   E.wrex 2777   _Vcvv 3082    i^i cin 3436    C_ wss 3437   ~Pcpw 3980   class class class wbr 4421    |-> cmpt 4480   ran crn 4852    |` cres 4853   -->wf 5595   ` cfv 5599  (class class class)co 6303   Fincfn 7575   supcsup 7958   RRcr 9540   0cc0 9541   +oocpnf 9674   RR*cxr 9676    < clt 9677   [,]cicc 11640  Σ^csumge0 38036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-sup 7960  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-rp 11305  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-seq 12215  df-exp 12274  df-hash 12517  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-sum 13746  df-sumge0 38037
This theorem is referenced by:  omeiunltfirp  38163
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