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

Theorem xrge0gsumle 21206
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 11619 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
2 xrge0gsumle.g . . . . . . . . . 10  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
3 xrsbas 18304 . . . . . . . . . 10  |-  RR*  =  ( Base `  RR*s )
42, 3ressbas2 14563 . . . . . . . . 9  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( 0 [,] +oo )  =  ( Base `  G ) )
51, 4ax-mp 5 . . . . . . . 8  |-  ( 0 [,] +oo )  =  ( Base `  G
)
6 eqid 2467 . . . . . . . . . 10  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
76xrge0subm 18329 . . . . . . . . 9  |-  ( 0 [,] +oo )  e.  (SubMnd `  ( RR*ss  ( RR*  \  { -oo } ) ) )
8 xrex 11229 . . . . . . . . . . . . 13  |-  RR*  e.  _V
9 difexg 4601 . . . . . . . . . . . . 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 11386 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  0  <_  x )  ->  x  =/= -oo )
1311, 12jca 532 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  0  <_  x )  ->  (
x  e.  RR*  /\  x  =/= -oo ) )
14 elxrge0 11641 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,] +oo )  <->  ( x  e. 
RR*  /\  0  <_  x ) )
15 eldifsn 4158 . . . . . . . . . . . . . 14  |-  ( x  e.  ( RR*  \  { -oo } )  <->  ( x  e.  RR*  /\  x  =/= -oo ) )
1613, 14, 153imtr4i 266 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0 [,] +oo )  ->  x  e.  ( RR*  \  { -oo } ) )
1716ssriv 3513 . . . . . . . . . . . 12  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
18 ressabs 14570 . . . . . . . . . . . 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 2499 . . . . . . . . . 10  |-  G  =  ( ( RR*ss  ( RR*  \  { -oo }
) )s  ( 0 [,] +oo ) )
216xrs10 18327 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
2220, 21subm0 15859 . . . . . . . . 9  |-  ( ( 0 [,] +oo )  e.  (SubMnd `  ( RR*ss  ( RR*  \  { -oo } ) ) )  -> 
0  =  ( 0g
`  G ) )
237, 22ax-mp 5 . . . . . . . 8  |-  0  =  ( 0g `  G )
24 xrge0cmn 18330 . . . . . . . . . 10  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
252, 24eqeltri 2551 . . . . . . . . 9  |-  G  e. CMnd
2625a1i 11 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
27 elfpw 7834 . . . . . . . . . 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 5757 . . . . . . . . 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 9602 . . . . . . . . . 10  |-  0  e.  _V
3534a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  0  e.  _V )
3633, 29, 35fdmfifsupp 7851 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  s ) finSupp  0
)
375, 23, 26, 29, 33, 36gsumcl 16796 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  s
) )  e.  ( 0 [,] +oo )
)
381, 37sseldi 3507 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  s
) )  e.  RR* )
39 eqid 2467 . . . . . 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 6056 . . . . 5  |-  ( ph  ->  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) : ( ~P A  i^i  Fin ) --> RR* )
41 frn 5743 . . . . 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 16 . . . 4  |-  ( ph  ->  ran  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) 
C_  RR* )
43 0ss 3819 . . . . . . 7  |-  (/)  C_  A
44 0fin 7759 . . . . . . 7  |-  (/)  e.  Fin
45 elfpw 7834 . . . . . . 7  |-  ( (/)  e.  ( ~P A  i^i  Fin )  <->  ( (/)  C_  A  /\  (/)  e.  Fin )
)
4643, 44, 45mpbir2an 918 . . . . . 6  |-  (/)  e.  ( ~P A  i^i  Fin )
47 0cn 9600 . . . . . 6  |-  0  e.  CC
48 reseq2 5274 . . . . . . . . . 10  |-  ( s  =  (/)  ->  ( F  |`  s )  =  ( F  |`  (/) ) )
49 res0 5284 . . . . . . . . . 10  |-  ( F  |`  (/) )  =  (/)
5048, 49syl6eq 2524 . . . . . . . . 9  |-  ( s  =  (/)  ->  ( F  |`  s )  =  (/) )
5150oveq2d 6311 . . . . . . . 8  |-  ( s  =  (/)  ->  ( G 
gsumg  ( F  |`  s ) )  =  ( G 
gsumg  (/) ) )
5223gsum0 15779 . . . . . . . 8  |-  ( G 
gsumg  (/) )  =  0
5351, 52syl6eq 2524 . . . . . . 7  |-  ( s  =  (/)  ->  ( G 
gsumg  ( F  |`  s ) )  =  0 )
5439, 53elrnmpt1s 5256 . . . . . 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 3510 . . 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 7834 . . . . . . . 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 16 . . . . . 6  |-  ( ph  ->  B  e.  Fin )
63 diffi 7763 . . . . . 6  |-  ( B  e.  Fin  ->  ( B  \  C )  e. 
Fin )
6462, 63syl 16 . . . . 5  |-  ( ph  ->  ( B  \  C
)  e.  Fin )
6560simplbi 460 . . . . . . . 8  |-  ( B  e.  ( ~P A  i^i  Fin )  ->  B  C_  A )
6659, 65syl 16 . . . . . . 7  |-  ( ph  ->  B  C_  A )
6766ssdifssd 3647 . . . . . 6  |-  ( ph  ->  ( B  \  C
)  C_  A )
68 fssres 5757 . . . . . 6  |-  ( ( F : A --> ( 0 [,] +oo )  /\  ( B  \  C ) 
C_  A )  -> 
( F  |`  ( B  \  C ) ) : ( B  \  C ) --> ( 0 [,] +oo ) )
6930, 67, 68syl2anc 661 . . . . 5  |-  ( ph  ->  ( F  |`  ( B  \  C ) ) : ( B  \  C ) --> ( 0 [,] +oo ) )
7034a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  _V )
7169, 64, 70fdmfifsupp 7851 . . . . 5  |-  ( ph  ->  ( F  |`  ( B  \  C ) ) finSupp 
0 )
725, 23, 58, 64, 69, 71gsumcl 16796 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  ( 0 [,] +oo ) )
731, 72sseldi 3507 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  RR* )
74 xrge0gsumle.c . . . . . 6  |-  ( ph  ->  C  C_  B )
75 ssfi 7752 . . . . . 6  |-  ( ( B  e.  Fin  /\  C  C_  B )  ->  C  e.  Fin )
7662, 74, 75syl2anc 661 . . . . 5  |-  ( ph  ->  C  e.  Fin )
7774, 66sstrd 3519 . . . . . 6  |-  ( ph  ->  C  C_  A )
78 fssres 5757 . . . . . 6  |-  ( ( F : A --> ( 0 [,] +oo )  /\  C  C_  A )  -> 
( F  |`  C ) : C --> ( 0 [,] +oo ) )
7930, 77, 78syl2anc 661 . . . . 5  |-  ( ph  ->  ( F  |`  C ) : C --> ( 0 [,] +oo ) )
8079, 76, 70fdmfifsupp 7851 . . . . 5  |-  ( ph  ->  ( F  |`  C ) finSupp 
0 )
815, 23, 58, 76, 79, 80gsumcl 16796 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  e.  ( 0 [,] +oo ) )
821, 81sseldi 3507 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  e.  RR* )
83 elxrge0 11641 . . . . 5  |-  ( ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  ( 0 [,] +oo )  <->  ( ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  RR*  /\  0  <_  ( G  gsumg  ( F  |`  ( B  \  C ) ) ) ) )
8483simprbi 464 . . . 4  |-  ( ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  ( 0 [,] +oo )  -> 
0  <_  ( G  gsumg  ( F  |`  ( B  \  C ) ) ) )
8572, 84syl 16 . . 3  |-  ( ph  ->  0  <_  ( G  gsumg  ( F  |`  ( B  \  C ) ) ) )
86 xleadd2a 11458 . . 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 ) ) ) ) )
8757, 73, 82, 85, 86syl31anc 1231 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e 0 )  <_ 
( ( G  gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C
) ) ) ) )
88 xaddid1 11450 . . 3  |-  ( ( G  gsumg  ( F  |`  C ) )  e.  RR*  ->  ( ( G  gsumg  ( F  |`  C ) ) +e 0 )  =  ( G 
gsumg  ( F  |`  C ) ) )
8982, 88syl 16 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e 0 )  =  ( G  gsumg  ( F  |`  C ) ) )
90 ovex 6320 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
91 xrsadd 18305 . . . . . 6  |-  +e 
=  ( +g  `  RR*s
)
922, 91ressplusg 14614 . . . . 5  |-  ( ( 0 [,] +oo )  e.  _V  ->  +e 
=  ( +g  `  G
) )
9390, 92ax-mp 5 . . . 4  |-  +e 
=  ( +g  `  G
)
94 fssres 5757 . . . . 5  |-  ( ( F : A --> ( 0 [,] +oo )  /\  B  C_  A )  -> 
( F  |`  B ) : B --> ( 0 [,] +oo ) )
9530, 66, 94syl2anc 661 . . . 4  |-  ( ph  ->  ( F  |`  B ) : B --> ( 0 [,] +oo ) )
9695, 62, 70fdmfifsupp 7851 . . . 4  |-  ( ph  ->  ( F  |`  B ) finSupp 
0 )
97 disjdif 3905 . . . . 5  |-  ( C  i^i  ( B  \  C ) )  =  (/)
9897a1i 11 . . . 4  |-  ( ph  ->  ( C  i^i  ( B  \  C ) )  =  (/) )
99 undif2 3909 . . . . 5  |-  ( C  u.  ( B  \  C ) )  =  ( C  u.  B
)
100 ssequn1 3679 . . . . . 6  |-  ( C 
C_  B  <->  ( C  u.  B )  =  B )
10174, 100sylib 196 . . . . 5  |-  ( ph  ->  ( C  u.  B
)  =  B )
10299, 101syl5req 2521 . . . 4  |-  ( ph  ->  B  =  ( C  u.  ( B  \  C ) ) )
1035, 23, 93, 58, 59, 95, 96, 98, 102gsumsplit 16819 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  B ) )  =  ( ( G  gsumg  ( ( F  |`  B )  |`  C ) ) +e ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C ) ) ) ) )
104 resabs1 5308 . . . . . 6  |-  ( C 
C_  B  ->  (
( F  |`  B )  |`  C )  =  ( F  |`  C )
)
10574, 104syl 16 . . . . 5  |-  ( ph  ->  ( ( F  |`  B )  |`  C )  =  ( F  |`  C ) )
106105oveq2d 6311 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( F  |`  B )  |`  C ) )  =  ( G 
gsumg  ( F  |`  C ) ) )
107 difss 3636 . . . . . 6  |-  ( B 
\  C )  C_  B
108 resabs1 5308 . . . . . 6  |-  ( ( B  \  C ) 
C_  B  ->  (
( F  |`  B )  |`  ( B  \  C
) )  =  ( F  |`  ( B  \  C ) ) )
109107, 108mp1i 12 . . . . 5  |-  ( ph  ->  ( ( F  |`  B )  |`  ( B  \  C ) )  =  ( F  |`  ( B  \  C ) ) )
110109oveq2d 6311 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C ) ) )  =  ( G 
gsumg  ( F  |`  ( B 
\  C ) ) ) )
111106, 110oveq12d 6313 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( ( F  |`  B )  |`  C ) ) +e ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C
) ) ) )  =  ( ( G 
gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C ) ) ) ) )
112103, 111eqtr2d 2509 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C
) ) ) )  =  ( G  gsumg  ( F  |`  B ) ) )
11387, 89, 1123brtr3d 4482 1  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  <_  ( G  gsumg  ( F  |`  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    \ cdif 3478    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   class class class wbr 4453    |-> cmpt 4511   ran crn 5006    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   0cc0 9504   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    <_ cle 9641   +ecxad 11328   [,]cicc 11544   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   0gc0g 14712    gsumg cgsu 14713   RR*scxrs 14772  SubMndcsubmnd 15838  CMndccmn 16671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-xadd 11331  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-tset 14591  df-ple 14592  df-ds 14594  df-0g 14714  df-gsum 14715  df-xrs 14774  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-cntz 16227  df-cmn 16673
This theorem is referenced by:  xrge0tsms  21207  xrge0tsmsd  27600
  Copyright terms: Public domain W3C validator