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Theorem esumlub 28232
Description: The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)
Hypotheses
Ref Expression
esumlub.f  |-  F/ k
ph
esumlub.0  |-  ( ph  ->  A  e.  V )
esumlub.1  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
esumlub.2  |-  ( ph  ->  X  e.  RR* )
esumlub.3  |-  ( ph  ->  X  < Σ* k  e.  A B )
Assertion
Ref Expression
esumlub  |-  ( ph  ->  E. a  e.  ( ~P A  i^i  Fin ) X  < Σ* k  e.  a B )
Distinct variable groups:    k, a, A    B, a    X, a    ph, a
Allowed substitution hints:    ph( k)    B( k)    V( k, a)    X( k)

Proof of Theorem esumlub
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 esumlub.3 . . . 4  |-  ( ph  ->  X  < Σ* k  e.  A B )
2 esumlub.f . . . . . . 7  |-  F/ k
ph
3 nfcv 2619 . . . . . . 7  |-  F/_ k A
4 esumlub.0 . . . . . . 7  |-  ( ph  ->  A  e.  V )
5 esumlub.1 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
6 eqidd 2458 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) )
72, 3, 4, 5, 6esumval 28220 . . . . . 6  |-  ( ph  -> Σ* k  e.  A B  =  sup ( ran  (
x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) ,  RR* ,  <  )
)
87breq2d 4468 . . . . 5  |-  ( ph  ->  ( X  < Σ* k  e.  A B 
<->  X  <  sup ( ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) ,  RR* ,  <  )
) )
9 iccssxr 11632 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
10 xrge0base 27833 . . . . . . . . . 10  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
11 xrge0cmn 18587 . . . . . . . . . . 11  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
1211a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
13 inss2 3715 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
Fin
14 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
1513, 14sseldi 3497 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
16 nfv 1708 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
172, 16nfan 1929 . . . . . . . . . . 11  |-  F/ k ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )
18 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  ph )
19 inss1 3714 . . . . . . . . . . . . . . . . 17  |-  ( ~P A  i^i  Fin )  C_ 
~P A
2019sseli 3495 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2120ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  e.  ~P A )
2221elpwid 4025 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  C_  A )
23 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  x )
2422, 23sseldd 3500 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  A )
2518, 24, 5syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  B  e.  ( 0 [,] +oo ) )
2625ex 434 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  x  ->  B  e.  ( 0 [,] +oo ) ) )
2717, 26ralrimi 2857 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A. k  e.  x  B  e.  ( 0 [,] +oo ) )
2810, 12, 15, 27gsummptcl 17121 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,] +oo ) )
299, 28sseldi 3497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
3029ralrimiva 2871 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
31 eqid 2457 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )  =  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )
3231rnmptss 6061 . . . . . . 7  |-  ( A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  x  |->  B ) )  e. 
RR*  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR* )
3330, 32syl 16 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR* )
34 esumlub.2 . . . . . 6  |-  ( ph  ->  X  e.  RR* )
35 supxrlub 11542 . . . . . 6  |-  ( ( ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR*  /\  X  e. 
RR* )  ->  ( X  <  sup ( ran  (
x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) ,  RR* ,  <  )  <->  E. y  e.  ran  (
x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) X  <  y ) )
3633, 34, 35syl2anc 661 . . . . 5  |-  ( ph  ->  ( X  <  sup ( ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) ,  RR* ,  <  )  <->  E. y  e.  ran  (
x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) X  <  y ) )
378, 36bitrd 253 . . . 4  |-  ( ph  ->  ( X  < Σ* k  e.  A B 
<->  E. y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) X  <  y ) )
381, 37mpbid 210 . . 3  |-  ( ph  ->  E. y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) ) X  <  y )
39 ovex 6324 . . . . 5  |-  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
_V
4039a1i 11 . . . 4  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  e. 
_V )
41 mpteq1 4537 . . . . . . . 8  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
4241oveq2d 6312 . . . . . . 7  |-  ( x  =  a  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
4342cbvmptv 4548 . . . . . 6  |-  ( x  e.  ( ~P A  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )  =  ( a  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) )
4443, 39elrnmpti 5263 . . . . 5  |-  ( y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) )  <->  E. a  e.  ( ~P A  i^i  Fin )
y  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) )
4544a1i 11 . . . 4  |-  ( ph  ->  ( y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  x  |->  B ) ) )  <->  E. a  e.  ( ~P A  i^i  Fin )
y  =  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) ) )
46 simpr 461 . . . . 5  |-  ( (
ph  /\  y  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )  ->  y  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) ) )
4746breq2d 4468 . . . 4  |-  ( (
ph  /\  y  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )  ->  ( X  < 
y  <->  X  <  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) ) )
4840, 45, 47rexxfr2d 4673 . . 3  |-  ( ph  ->  ( E. y  e. 
ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) X  <  y  <->  E. a  e.  ( ~P A  i^i  Fin ) X  <  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) ) )
4938, 48mpbid 210 . 2  |-  ( ph  ->  E. a  e.  ( ~P A  i^i  Fin ) X  <  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) )
50 nfv 1708 . . . . . . 7  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
512, 50nfan 1929 . . . . . 6  |-  F/ k ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )
52 simpr 461 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  ( ~P A  i^i  Fin ) )
5313, 52sseldi 3497 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
54 simpll 753 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  ph )
5519sseli 3495 . . . . . . . . . 10  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
5655ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  e.  ~P A )
5756elpwid 4025 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  C_  A )
58 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  a )
5957, 58sseldd 3500 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  A )
6054, 59, 5syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  B  e.  ( 0 [,] +oo ) )
6151, 53, 60gsumesum 28231 . . . . 5  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  = Σ* k  e.  a B )
6261breq2d 4468 . . . 4  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( X  <  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  <->  X  < Σ* k  e.  a B ) )
6362biimpd 207 . . 3  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  ( X  <  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  ->  X  < Σ* k  e.  a B ) )
6463reximdva 2932 . 2  |-  ( ph  ->  ( E. a  e.  ( ~P A  i^i  Fin ) X  <  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  ->  E. a  e.  ( ~P A  i^i  Fin ) X  < Σ* k  e.  a B ) )
6549, 64mpd 15 1  |-  ( ph  ->  E. a  e.  ( ~P A  i^i  Fin ) X  < Σ* k  e.  a B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   F/wnf 1617    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   class class class wbr 4456    |-> cmpt 4515   ran crn 5009  (class class class)co 6296   Fincfn 7535   supcsup 7918   0cc0 9509   +oocpnf 9642   RR*cxr 9644    < clt 9645   [,]cicc 11557   ↾s cress 14645    gsumg cgsu 14858   RR*scxrs 14917  CMndccmn 16925  Σ*cesum 28201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-xadd 11344  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-tset 14731  df-ple 14732  df-ds 14734  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-ordt 14918  df-xrs 14919  df-mre 15003  df-mrc 15004  df-acs 15006  df-ps 15957  df-tsr 15958  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-cntz 16482  df-cmn 16927  df-fbas 18543  df-fg 18544  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-ntr 19648  df-nei 19726  df-cn 19855  df-haus 19943  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-tsms 20751  df-esum 28202
This theorem is referenced by:  esumfsup  28242  esum2d  28265
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