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Theorem esumlub 26516
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 2584 . . . . . . 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 2444 . . . . . . 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 26505 . . . . . 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 4309 . . . . 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 11383 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
10 xrge0base 26151 . . . . . . . . . 10  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
11 xrge00 26152 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
12 xrge0cmn 17860 . . . . . . . . . . 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 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
15 nfv 1673 . . . . . . . . . . . 12  |-  F/ k  x  e.  ( ~P A  i^i  Fin )
162, 15nfan 1861 . . . . . . . . . . 11  |-  F/ k ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )
17 nfcv 2584 . . . . . . . . . . 11  |-  F/_ k
x
18 nfcv 2584 . . . . . . . . . . 11  |-  F/_ k
( 0 [,] +oo )
19 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  ph )
20 inss1 3575 . . . . . . . . . . . . . . . 16  |-  ( ~P A  i^i  Fin )  C_ 
~P A
2120sseli 3357 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2221ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  e.  ~P A )
2322elpwid 3875 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  x  C_  A )
24 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  x )
2523, 24sseldd 3362 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  k  e.  A )
2619, 25, 5syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  x )  ->  B  e.  ( 0 [,] +oo ) )
27 eqid 2443 . . . . . . . . . . 11  |-  ( k  e.  x  |->  B )  =  ( k  e.  x  |->  B )
2816, 17, 18, 26, 27fmptdF 25977 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
k  e.  x  |->  B ) : x --> ( 0 [,] +oo ) )
29 inss2 3576 . . . . . . . . . . . 12  |-  ( ~P A  i^i  Fin )  C_ 
Fin
3029, 14sseldi 3359 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
3130, 28fisuppfi 7633 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( k  e.  x  |->  B ) "
( _V  \  {
0 } ) )  e.  Fin )
3210, 11, 13, 14, 28, 31gsumclOLD 16405 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e.  ( 0 [,] +oo ) )
339, 32sseldi 3359 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
3433ralrimiva 2804 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( ~P A  i^i  Fin ) ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  e. 
RR* )
35 eqid 2443 . . . . . . . 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 5877 . . . . . . 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 11293 . . . . . 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 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 ) )
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 6121 . . . . 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 4377 . . . . . . . 8  |-  ( x  =  a  ->  (
k  e.  x  |->  B )  =  ( k  e.  a  |->  B ) )
4645oveq2d 6112 . . . . . . 7  |-  ( x  =  a  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )
4746cbvmptv 4388 . . . . . 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 5095 . . . . 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 461 . . . . 5  |-  ( (
ph  /\  y  =  ( ( RR*ss  (
0 [,] +oo )
)  gsumg  ( k  e.  a 
|->  B ) ) )  ->  y  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  ( k  e.  a 
|->  B ) ) )
5150breq2d 4309 . . . 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 4514 . . 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 1673 . . . . . . 7  |-  F/ k  a  e.  ( ~P A  i^i  Fin )
552, 54nfan 1861 . . . . . 6  |-  F/ k ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )
56 simpr 461 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  ( ~P A  i^i  Fin ) )
5729, 56sseldi 3359 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  a  e.  Fin )
58 simpll 753 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  ph )
5920sseli 3357 . . . . . . . . . 10  |-  ( a  e.  ( ~P A  i^i  Fin )  ->  a  e.  ~P A )
6059ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  e.  ~P A )
6160elpwid 3875 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  a  C_  A )
62 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  a )
6361, 62sseldd 3362 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  k  e.  A )
6458, 63, 5syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ~P A  i^i  Fin ) )  /\  k  e.  a )  ->  B  e.  ( 0 [,] +oo ) )
6555, 57, 64gsumesum 26515 . . . . 5  |-  ( (
ph  /\  a  e.  ( ~P A  i^i  Fin ) )  ->  (
( RR*ss  ( 0 [,] +oo ) )  gsumg  ( k  e.  a 
|->  B ) )  = Σ* k  e.  a B )
6665breq2d 4309 . . . 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 2833 . 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 1369   F/wnf 1589    e. wcel 1756   A.wral 2720   E.wrex 2721   _Vcvv 2977    \ cdif 3330    i^i cin 3332    C_ wss 3333   ~Pcpw 3865   {csn 3882   class class class wbr 4297    e. cmpt 4355   ran crn 4846  (class class class)co 6096   Fincfn 7315   supcsup 7695   0cc0 9287   +oocpnf 9420   RR*cxr 9422    < clt 9423   [,]cicc 11308   ↾s cress 14180    gsumg cgsu 14384   RR*scxrs 14443  CMndccmn 16282  Σ*cesum 26488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-xadd 11095  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-tset 14262  df-ple 14263  df-ds 14265  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-ordt 14444  df-xrs 14445  df-mre 14529  df-mrc 14530  df-acs 14532  df-ps 15375  df-tsr 15376  df-mnd 15420  df-submnd 15470  df-cntz 15840  df-cmn 16284  df-fbas 17819  df-fg 17820  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-ntr 18629  df-nei 18707  df-cn 18836  df-haus 18924  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-tsms 19702  df-esum 26489
This theorem is referenced by:  esumfsup  26524
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