MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xrge0gsumle Structured version   Unicode version

Theorem xrge0gsumle 21632
Description: A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)
Hypotheses
Ref Expression
xrge0gsumle.g  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
xrge0gsumle.a  |-  ( ph  ->  A  e.  V )
xrge0gsumle.f  |-  ( ph  ->  F : A --> ( 0 [,] +oo ) )
xrge0gsumle.b  |-  ( ph  ->  B  e.  ( ~P A  i^i  Fin )
)
xrge0gsumle.c  |-  ( ph  ->  C  C_  B )
Assertion
Ref Expression
xrge0gsumle  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  <_  ( G  gsumg  ( F  |`  B )
) )

Proof of Theorem xrge0gsumle
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 iccssxr 11663 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
2 xrge0gsumle.g . . . . . . . . . 10  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
3 xrsbas 18756 . . . . . . . . . 10  |-  RR*  =  ( Base `  RR*s )
42, 3ressbas2 14901 . . . . . . . . 9  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( 0 [,] +oo )  =  ( Base `  G ) )
51, 4ax-mp 5 . . . . . . . 8  |-  ( 0 [,] +oo )  =  ( Base `  G
)
6 eqid 2404 . . . . . . . . . 10  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
76xrge0subm 18781 . . . . . . . . 9  |-  ( 0 [,] +oo )  e.  (SubMnd `  ( RR*ss  ( RR*  \  { -oo } ) ) )
8 xrex 11264 . . . . . . . . . . . . 13  |-  RR*  e.  _V
9 difexg 4544 . . . . . . . . . . . . 13  |-  ( RR*  e.  _V  ->  ( RR*  \  { -oo } )  e.  _V )
108, 9ax-mp 5 . . . . . . . . . . . 12  |-  ( RR*  \  { -oo } )  e.  _V
11 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  0  <_  x )  ->  x  e.  RR* )
12 ge0nemnf 11429 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  0  <_  x )  ->  x  =/= -oo )
1311, 12jca 532 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  0  <_  x )  ->  (
x  e.  RR*  /\  x  =/= -oo ) )
14 elxrge0 11685 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,] +oo )  <->  ( x  e. 
RR*  /\  0  <_  x ) )
15 eldifsn 4099 . . . . . . . . . . . . . 14  |-  ( x  e.  ( RR*  \  { -oo } )  <->  ( x  e.  RR*  /\  x  =/= -oo ) )
1613, 14, 153imtr4i 268 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0 [,] +oo )  ->  x  e.  ( RR*  \  { -oo } ) )
1716ssriv 3448 . . . . . . . . . . . 12  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
18 ressabs 14909 . . . . . . . . . . . 12  |-  ( ( ( RR*  \  { -oo } )  e.  _V  /\  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } ) )  -> 
( ( RR*ss  ( RR*  \  { -oo }
) )s  ( 0 [,] +oo ) )  =  (
RR*ss  ( 0 [,] +oo ) ) )
1910, 17, 18mp2an 672 . . . . . . . . . . 11  |-  ( (
RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )  =  ( RR*ss  (
0 [,] +oo )
)
202, 19eqtr4i 2436 . . . . . . . . . 10  |-  G  =  ( ( RR*ss  ( RR*  \  { -oo }
) )s  ( 0 [,] +oo ) )
216xrs10 18779 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
2220, 21subm0 16313 . . . . . . . . 9  |-  ( ( 0 [,] +oo )  e.  (SubMnd `  ( RR*ss  ( RR*  \  { -oo } ) ) )  -> 
0  =  ( 0g
`  G ) )
237, 22ax-mp 5 . . . . . . . 8  |-  0  =  ( 0g `  G )
24 xrge0cmn 18782 . . . . . . . . . 10  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
252, 24eqeltri 2488 . . . . . . . . 9  |-  G  e. CMnd
2625a1i 11 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
27 elfpw 7858 . . . . . . . . . 10  |-  ( s  e.  ( ~P A  i^i  Fin )  <->  ( s  C_  A  /\  s  e. 
Fin ) )
2827simprbi 464 . . . . . . . . 9  |-  ( s  e.  ( ~P A  i^i  Fin )  ->  s  e.  Fin )
2928adantl 466 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  s  e.  Fin )
30 xrge0gsumle.f . . . . . . . . 9  |-  ( ph  ->  F : A --> ( 0 [,] +oo ) )
3127simplbi 460 . . . . . . . . 9  |-  ( s  e.  ( ~P A  i^i  Fin )  ->  s  C_  A )
32 fssres 5736 . . . . . . . . 9  |-  ( ( F : A --> ( 0 [,] +oo )  /\  s  C_  A )  -> 
( F  |`  s
) : s --> ( 0 [,] +oo )
)
3330, 31, 32syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  s ) : s --> ( 0 [,] +oo ) )
34 c0ex 9622 . . . . . . . . . 10  |-  0  e.  _V
3534a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  0  e.  _V )
3633, 29, 35fdmfifsupp 7875 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  s ) finSupp  0
)
375, 23, 26, 29, 33, 36gsumcl 17249 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  s
) )  e.  ( 0 [,] +oo )
)
381, 37sseldi 3442 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  s
) )  e.  RR* )
39 eqid 2404 . . . . . 6  |-  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  s ) ) )  =  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s
) ) )
4038, 39fmptd 6035 . . . . 5  |-  ( ph  ->  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) : ( ~P A  i^i  Fin ) --> RR* )
41 frn 5722 . . . . 5  |-  ( ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s
) ) ) : ( ~P A  i^i  Fin ) --> RR*  ->  ran  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  s ) ) )  C_  RR* )
4240, 41syl 17 . . . 4  |-  ( ph  ->  ran  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) 
C_  RR* )
43 0ss 3770 . . . . . . 7  |-  (/)  C_  A
44 0fin 7784 . . . . . . 7  |-  (/)  e.  Fin
45 elfpw 7858 . . . . . . 7  |-  ( (/)  e.  ( ~P A  i^i  Fin )  <->  ( (/)  C_  A  /\  (/)  e.  Fin )
)
4643, 44, 45mpbir2an 923 . . . . . 6  |-  (/)  e.  ( ~P A  i^i  Fin )
47 0cn 9620 . . . . . 6  |-  0  e.  CC
48 reseq2 5091 . . . . . . . . . 10  |-  ( s  =  (/)  ->  ( F  |`  s )  =  ( F  |`  (/) ) )
49 res0 5100 . . . . . . . . . 10  |-  ( F  |`  (/) )  =  (/)
5048, 49syl6eq 2461 . . . . . . . . 9  |-  ( s  =  (/)  ->  ( F  |`  s )  =  (/) )
5150oveq2d 6296 . . . . . . . 8  |-  ( s  =  (/)  ->  ( G 
gsumg  ( F  |`  s ) )  =  ( G 
gsumg  (/) ) )
5223gsum0 16231 . . . . . . . 8  |-  ( G 
gsumg  (/) )  =  0
5351, 52syl6eq 2461 . . . . . . 7  |-  ( s  =  (/)  ->  ( G 
gsumg  ( F  |`  s ) )  =  0 )
5439, 53elrnmpt1s 5073 . . . . . 6  |-  ( (
(/)  e.  ( ~P A  i^i  Fin )  /\  0  e.  CC )  ->  0  e.  ran  (
s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s
) ) ) )
5546, 47, 54mp2an 672 . . . . 5  |-  0  e.  ran  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) )
5655a1i 11 . . . 4  |-  ( ph  ->  0  e.  ran  (
s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s
) ) ) )
5742, 56sseldd 3445 . . 3  |-  ( ph  ->  0  e.  RR* )
5825a1i 11 . . . . 5  |-  ( ph  ->  G  e. CMnd )
59 xrge0gsumle.b . . . . . . 7  |-  ( ph  ->  B  e.  ( ~P A  i^i  Fin )
)
60 elfpw 7858 . . . . . . . 8  |-  ( B  e.  ( ~P A  i^i  Fin )  <->  ( B  C_  A  /\  B  e. 
Fin ) )
6160simprbi 464 . . . . . . 7  |-  ( B  e.  ( ~P A  i^i  Fin )  ->  B  e.  Fin )
6259, 61syl 17 . . . . . 6  |-  ( ph  ->  B  e.  Fin )
63 diffi 7788 . . . . . 6  |-  ( B  e.  Fin  ->  ( B  \  C )  e. 
Fin )
6462, 63syl 17 . . . . 5  |-  ( ph  ->  ( B  \  C
)  e.  Fin )
6560simplbi 460 . . . . . . . 8  |-  ( B  e.  ( ~P A  i^i  Fin )  ->  B  C_  A )
6659, 65syl 17 . . . . . . 7  |-  ( ph  ->  B  C_  A )
6766ssdifssd 3583 . . . . . 6  |-  ( ph  ->  ( B  \  C
)  C_  A )
6830, 67fssresd 5737 . . . . 5  |-  ( ph  ->  ( F  |`  ( B  \  C ) ) : ( B  \  C ) --> ( 0 [,] +oo ) )
6934a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  _V )
7068, 64, 69fdmfifsupp 7875 . . . . 5  |-  ( ph  ->  ( F  |`  ( B  \  C ) ) finSupp 
0 )
715, 23, 58, 64, 68, 70gsumcl 17249 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  ( 0 [,] +oo ) )
721, 71sseldi 3442 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  RR* )
73 xrge0gsumle.c . . . . . 6  |-  ( ph  ->  C  C_  B )
74 ssfi 7777 . . . . . 6  |-  ( ( B  e.  Fin  /\  C  C_  B )  ->  C  e.  Fin )
7562, 73, 74syl2anc 661 . . . . 5  |-  ( ph  ->  C  e.  Fin )
7673, 66sstrd 3454 . . . . . 6  |-  ( ph  ->  C  C_  A )
7730, 76fssresd 5737 . . . . 5  |-  ( ph  ->  ( F  |`  C ) : C --> ( 0 [,] +oo ) )
7877, 75, 69fdmfifsupp 7875 . . . . 5  |-  ( ph  ->  ( F  |`  C ) finSupp 
0 )
795, 23, 58, 75, 77, 78gsumcl 17249 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  e.  ( 0 [,] +oo ) )
801, 79sseldi 3442 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  e.  RR* )
81 elxrge0 11685 . . . . 5  |-  ( ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  ( 0 [,] +oo )  <->  ( ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  RR*  /\  0  <_  ( G  gsumg  ( F  |`  ( B  \  C ) ) ) ) )
8281simprbi 464 . . . 4  |-  ( ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  ( 0 [,] +oo )  -> 
0  <_  ( G  gsumg  ( F  |`  ( B  \  C ) ) ) )
8371, 82syl 17 . . 3  |-  ( ph  ->  0  <_  ( G  gsumg  ( F  |`  ( B  \  C ) ) ) )
84 xleadd2a 11501 . . 3  |-  ( ( ( 0  e.  RR*  /\  ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  RR*  /\  ( G  gsumg  ( F  |`  C ) )  e.  RR* )  /\  0  <_  ( G 
gsumg  ( F  |`  ( B 
\  C ) ) ) )  ->  (
( G  gsumg  ( F  |`  C ) ) +e 0 )  <_  ( ( G  gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C ) ) ) ) )
8557, 72, 80, 83, 84syl31anc 1235 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e 0 )  <_ 
( ( G  gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C
) ) ) ) )
86 xaddid1 11493 . . 3  |-  ( ( G  gsumg  ( F  |`  C ) )  e.  RR*  ->  ( ( G  gsumg  ( F  |`  C ) ) +e 0 )  =  ( G 
gsumg  ( F  |`  C ) ) )
8780, 86syl 17 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e 0 )  =  ( G  gsumg  ( F  |`  C ) ) )
88 ovex 6308 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
89 xrsadd 18757 . . . . . 6  |-  +e 
=  ( +g  `  RR*s
)
902, 89ressplusg 14957 . . . . 5  |-  ( ( 0 [,] +oo )  e.  _V  ->  +e 
=  ( +g  `  G
) )
9188, 90ax-mp 5 . . . 4  |-  +e 
=  ( +g  `  G
)
9230, 66fssresd 5737 . . . 4  |-  ( ph  ->  ( F  |`  B ) : B --> ( 0 [,] +oo ) )
9392, 62, 69fdmfifsupp 7875 . . . 4  |-  ( ph  ->  ( F  |`  B ) finSupp 
0 )
94 disjdif 3846 . . . . 5  |-  ( C  i^i  ( B  \  C ) )  =  (/)
9594a1i 11 . . . 4  |-  ( ph  ->  ( C  i^i  ( B  \  C ) )  =  (/) )
96 undif2 3850 . . . . 5  |-  ( C  u.  ( B  \  C ) )  =  ( C  u.  B
)
97 ssequn1 3615 . . . . . 6  |-  ( C 
C_  B  <->  ( C  u.  B )  =  B )
9873, 97sylib 198 . . . . 5  |-  ( ph  ->  ( C  u.  B
)  =  B )
9996, 98syl5req 2458 . . . 4  |-  ( ph  ->  B  =  ( C  u.  ( B  \  C ) ) )
1005, 23, 91, 58, 59, 92, 93, 95, 99gsumsplit 17272 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  B ) )  =  ( ( G  gsumg  ( ( F  |`  B )  |`  C ) ) +e ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C ) ) ) ) )
10173resabs1d 5125 . . . . 5  |-  ( ph  ->  ( ( F  |`  B )  |`  C )  =  ( F  |`  C ) )
102101oveq2d 6296 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( F  |`  B )  |`  C ) )  =  ( G 
gsumg  ( F  |`  C ) ) )
103 difss 3572 . . . . . 6  |-  ( B 
\  C )  C_  B
104 resabs1 5124 . . . . . 6  |-  ( ( B  \  C ) 
C_  B  ->  (
( F  |`  B )  |`  ( B  \  C
) )  =  ( F  |`  ( B  \  C ) ) )
105103, 104mp1i 13 . . . . 5  |-  ( ph  ->  ( ( F  |`  B )  |`  ( B  \  C ) )  =  ( F  |`  ( B  \  C ) ) )
106105oveq2d 6296 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C ) ) )  =  ( G 
gsumg  ( F  |`  ( B 
\  C ) ) ) )
107102, 106oveq12d 6298 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( ( F  |`  B )  |`  C ) ) +e ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C
) ) ) )  =  ( ( G 
gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C ) ) ) ) )
108100, 107eqtr2d 2446 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C
) ) ) )  =  ( G  gsumg  ( F  |`  B ) ) )
10985, 87, 1083brtr3d 4426 1  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  <_  ( G  gsumg  ( F  |`  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   _Vcvv 3061    \ cdif 3413    u. cun 3414    i^i cin 3415    C_ wss 3416   (/)c0 3740   ~Pcpw 3957   {csn 3974   class class class wbr 4397    |-> cmpt 4455   ran crn 4826    |` cres 4827   -->wf 5567   ` cfv 5571  (class class class)co 6280   Fincfn 7556   CCcc 9522   0cc0 9524   +oocpnf 9657   -oocmnf 9658   RR*cxr 9659    <_ cle 9661   +ecxad 11371   [,]cicc 11587   Basecbs 14843   ↾s cress 14844   +g cplusg 14911   0gc0g 15056    gsumg cgsu 15057   RR*scxrs 15116  SubMndcsubmnd 16291  CMndccmn 17124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-xadd 11374  df-icc 11591  df-fz 11729  df-fzo 11857  df-seq 12154  df-hash 12455  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-tset 14930  df-ple 14931  df-ds 14933  df-0g 15058  df-gsum 15059  df-xrs 15118  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-cntz 16681  df-cmn 17126
This theorem is referenced by:  xrge0tsms  21633  xrge0tsmsd  28241
  Copyright terms: Public domain W3C validator