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Theorem esumlub 26364
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 2569 . . . . . . 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 2434 . . . . . . 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 26353 . . . . . 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 4292 . . . . 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 11365 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
10 xrge0base 25968 . . . . . . . . . 10  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
11 xrge00 25969 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
12 xrge0cmn 17698 . . . . . . . . . . 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 458 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
15 nfv 1672 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
162, 15nfan 1859 . . . . . . . . . . 11  |-  F/ k ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )
17 nfcv 2569 . . . . . . . . . . 11  |-  F/_ k
x
18 nfcv 2569 . . . . . . . . . . 11  |-  F/_ k
( 0 [,] +oo )
19 simpll 746 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  ph )
20 inss1 3558 . . . . . . . . . . . . . . . 16  |-  ( ~P A  i^i  Fin )  C_ 
~P A
2120sseli 3340 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2221ad2antlr 719 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  e.  ~P A )
2322elpwid 3858 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  C_  A )
24 simpr 458 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  x )
2523, 24sseldd 3345 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  A )
2619, 25, 5syl2anc 654 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  B  e.  ( 0 [,] +oo ) )
27 eqid 2433 . . . . . . . . . . 11  |-  ( k  e.  x  |->  B )  =  ( k  e.  x  |->  B )
2816, 17, 18, 26, 27fmptdF 25795 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  x  |->  B ) : x --> ( 0 [,] +oo ) )
29 inss2 3559 . . . . . . . . . . . 12  |-  ( ~P A  i^i  Fin )  C_ 
Fin
3029, 14sseldi 3342 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
3130, 28fisuppfi 7616 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  x  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
3210, 11, 13, 14, 28, 31gsumclOLD 16379 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,] +oo ) )
339, 32sseldi 3342 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
3433ralrimiva 2789 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
35 eqid 2433 . . . . . . . 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 5859 . . . . . . 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 11275 . . . . . 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 654 . . . . 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 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 ) )
421, 41mpbid 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 )
43 ovex 6105 . . . . 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 4360 . . . . . . . 8  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
4645oveq2d 6096 . . . . . . 7  |-  ( x  =  a  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
4746cbvmptv 4371 . . . . . 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 5077 . . . . 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 458 . . . . 5  |-  ( (
ph  /\  y  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )  ->  y  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) ) )
5150breq2d 4292 . . . 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 4497 . . 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 210 . 2  |-  ( ph  ->  E. a  e.  ( ~P A  i^i  Fin ) X  <  ( (
RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) ) )
54 nfv 1672 . . . . . . 7  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
552, 54nfan 1859 . . . . . 6  |-  F/ k ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )
56 simpr 458 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  ( ~P A  i^i  Fin ) )
5729, 56sseldi 3342 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
58 simpll 746 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  ph )
5920sseli 3340 . . . . . . . . . 10  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
6059ad2antlr 719 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  e.  ~P A )
6160elpwid 3858 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  C_  A )
62 simpr 458 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  a )
6361, 62sseldd 3345 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  A )
6458, 63, 5syl2anc 654 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  B  e.  ( 0 [,] +oo ) )
6555, 57, 64gsumesum 26363 . . . . 5  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  = Σ* k  e.  a B )
6665breq2d 4292 . . . 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 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 ) )
6867reximdva 2818 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362   F/wnf 1592    e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962    \ cdif 3313    i^i cin 3315    C_ wss 3316   ~Pcpw 3848   {csn 3865   class class class wbr 4280    e. cmpt 4338   ran crn 4828  (class class class)co 6080   Fincfn 7298   supcsup 7678   0cc0 9269   +oocpnf 9402   RR*cxr 9404    < clt 9405   [,]cicc 11290   ↾s cress 14157    gsumg cgsu 14361   RR*scxrs 14420  CMndccmn 16256  Σ*cesum 26336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-xadd 11077  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-seq 11790  df-hash 12087  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-tset 14239  df-ple 14240  df-ds 14242  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-ordt 14421  df-xrs 14422  df-mre 14506  df-mrc 14507  df-acs 14509  df-ps 15352  df-tsr 15353  df-mnd 15397  df-submnd 15447  df-cntz 15814  df-cmn 16258  df-fbas 17657  df-fg 17658  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-ntr 18465  df-nei 18543  df-cn 18672  df-haus 18760  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-tsms 19538  df-esum 26337
This theorem is referenced by:  esumfsup  26372
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