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Theorem esumlub 24444
Description: The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.)
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 2571 . . . . . . 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 2436 . . . . . . 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 24433 . . . . . 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 4216 . . . . 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 10985 . . . . . . . . 9  |-  ( 0 [,]  +oo )  C_  RR*
10 xrge0base 24199 . . . . . . . . . 10  |-  ( 0 [,]  +oo )  =  (
Base `  ( RR* ss  ( 0 [,]  +oo ) ) )
11 xrge00 24200 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR* ss  ( 0 [,] 
+oo ) ) )
12 xrge0cmn 16732 . . . . . . . . . . 11  |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
1312a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd )
14 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
15 nfv 1629 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
162, 15nfan 1846 . . . . . . . . . . 11  |-  F/ k ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )
17 nfcv 2571 . . . . . . . . . . 11  |-  F/_ k
x
18 nfcv 2571 . . . . . . . . . . 11  |-  F/_ k
( 0 [,]  +oo )
19 simpll 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  ph )
20 inss1 3553 . . . . . . . . . . . . . . . 16  |-  ( ~P A  i^i  Fin )  C_ 
~P A
2120sseli 3336 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2221ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  e.  ~P A )
2322elpwid 3800 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  C_  A )
24 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  x )
2523, 24sseldd 3341 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  A )
2619, 25, 5syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  B  e.  ( 0 [,]  +oo ) )
27 eqid 2435 . . . . . . . . . . 11  |-  ( k  e.  x  |->  B )  =  ( k  e.  x  |->  B )
2816, 17, 18, 26, 27fmptdF 24061 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  x  |->  B ) : x --> ( 0 [,]  +oo ) )
29 inss2 3554 . . . . . . . . . . . 12  |-  ( ~P A  i^i  Fin )  C_ 
Fin
3029, 14sseldi 3338 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
3130, 28fisuppfi 14765 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  x  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
3210, 11, 13, 14, 28, 31gsumcl 15513 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,]  +oo ) )
339, 32sseldi 3338 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  RR* )
3433ralrimiva 2781 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
35 eqid 2435 . . . . . . . 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 ) ) )
3635rnmptss 5889 . . . . . . 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* )
3734, 36syl 16 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  x  |->  B ) ) ) 
C_  RR* )
38 esumlub.2 . . . . . 6  |-  ( ph  ->  X  e.  RR* )
39 supxrlub 10896 . . . . . 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 ) )
4037, 38, 39syl2anc 643 . . . . 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 ) )
418, 40bitrd 245 . . . 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 ) )
421, 41mpbid 202 . . 3  |-  ( ph  ->  E. y  e.  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  x  |->  B ) ) ) X  <  y )
43 ovex 6098 . . . . 5  |-  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) )  e.  _V
4443a1i 11 . . . 4  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) )  e.  _V )
45 mpteq1 4281 . . . . . . . 8  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
4645oveq2d 6089 . . . . . . 7  |-  ( x  =  a  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) ) )
4746cbvmptv 4292 . . . . . 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 ) ) )
4847, 43elrnmpti 5113 . . . . 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 ) ) )
4948a1i 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 ) ) ) )
50 simpr 448 . . . . 5  |-  ( (
ph  /\  y  =  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) ) )  ->  y  =  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  a  |->  B ) ) )
5150breq2d 4216 . . . 4  |-  ( (
ph  /\  y  =  ( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) ) )  ->  ( X  < 
y  <->  X  <  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) ) ) )
5244, 49, 51rexxfr2d 4732 . . 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 ) ) ) )
5342, 52mpbid 202 . 2  |-  ( ph  ->  E. a  e.  ( ~P A  i^i  Fin ) X  <  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) ) )
54 nfv 1629 . . . . . . 7  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
552, 54nfan 1846 . . . . . 6  |-  F/ k ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )
56 simpr 448 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  ( ~P A  i^i  Fin ) )
5729, 56sseldi 3338 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
58 simpll 731 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  ph )
5920sseli 3336 . . . . . . . . . 10  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
6059ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  e.  ~P A )
6160elpwid 3800 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  C_  A )
62 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  a )
6361, 62sseldd 3341 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  A )
6458, 63, 5syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  B  e.  ( 0 [,]  +oo ) )
6555, 57, 64gsumesum 24443 . . . . 5  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )  gsumg  ( k  e.  a  |->  B ) )  = Σ* k  e.  a B )
6665breq2d 4216 . . . 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 ) )
6766biimpd 199 . . 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 ) )
6867reximdva 2810 . 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 ) )
6953, 68mpd 15 1  |-  ( ph  ->  E. a  e.  ( ~P A  i^i  Fin ) X  < Σ* k  e.  a B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   class class class wbr 4204    e. cmpt 4258   ran crn 4871  (class class class)co 6073   Fincfn 7101   supcsup 7437   0cc0 8982    +oocpnf 9109   RR*cxr 9111    < clt 9112   [,]cicc 10911   ↾s cress 13462   RR* scxrs 13714    gsumg cgsu 13716  CMndccmn 15404  Σ*cesum 24416
This theorem is referenced by:  esumfsup  24452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-xadd 10703  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-tset 13540  df-ple 13541  df-ds 13543  df-rest 13642  df-topn 13643  df-topgen 13659  df-ordt 13717  df-xrs 13718  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-ps 14621  df-tsr 14622  df-mnd 14682  df-submnd 14731  df-cntz 15108  df-cmn 15406  df-fbas 16691  df-fg 16692  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-ntr 17076  df-nei 17154  df-cn 17283  df-haus 17371  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-tsms 18148  df-esum 24417
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