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Theorem xrge0gsumle 20429
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 11397 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
2 xrge0gsumle.g . . . . . . . . . 10  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
3 xrsbas 17851 . . . . . . . . . 10  |-  RR*  =  ( Base `  RR*s )
42, 3ressbas2 14248 . . . . . . . . 9  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( 0 [,] +oo )  =  ( Base `  G ) )
51, 4ax-mp 5 . . . . . . . 8  |-  ( 0 [,] +oo )  =  ( Base `  G
)
6 eqid 2443 . . . . . . . . . 10  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
76xrge0subm 17873 . . . . . . . . 9  |-  ( 0 [,] +oo )  e.  (SubMnd `  ( RR*ss  ( RR*  \  { -oo } ) ) )
8 xrex 11007 . . . . . . . . . . . . 13  |-  RR*  e.  _V
9 difexg 4459 . . . . . . . . . . . . 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 11164 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  0  <_  x )  ->  x  =/= -oo )
1311, 12jca 532 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  0  <_  x )  ->  (
x  e.  RR*  /\  x  =/= -oo ) )
14 elxrge0 11413 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,] +oo )  <->  ( x  e. 
RR*  /\  0  <_  x ) )
15 eldifsn 4019 . . . . . . . . . . . . . 14  |-  ( x  e.  ( RR*  \  { -oo } )  <->  ( x  e.  RR*  /\  x  =/= -oo ) )
1613, 14, 153imtr4i 266 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0 [,] +oo )  ->  x  e.  ( RR*  \  { -oo } ) )
1716ssriv 3379 . . . . . . . . . . . 12  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
18 ressabs 14255 . . . . . . . . . . . 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 2466 . . . . . . . . . 10  |-  G  =  ( ( RR*ss  ( RR*  \  { -oo }
) )s  ( 0 [,] +oo ) )
216xrs10 17871 . . . . . . . . . 10  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
2220, 21subm0 15503 . . . . . . . . 9  |-  ( ( 0 [,] +oo )  e.  (SubMnd `  ( RR*ss  ( RR*  \  { -oo } ) ) )  -> 
0  =  ( 0g
`  G ) )
237, 22ax-mp 5 . . . . . . . 8  |-  0  =  ( 0g `  G )
24 xrge0cmn 17874 . . . . . . . . . 10  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
252, 24eqeltri 2513 . . . . . . . . 9  |-  G  e. CMnd
2625a1i 11 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
27 elfpw 7632 . . . . . . . . . 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 5597 . . . . . . . . 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 9399 . . . . . . . . . 10  |-  0  e.  _V
3534a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  0  e.  _V )
3633, 29, 35fdmfifsupp 7649 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  s ) finSupp  0
)
375, 23, 26, 29, 33, 36gsumcl 16416 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  s
) )  e.  ( 0 [,] +oo )
)
381, 37sseldi 3373 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  s
) )  e.  RR* )
39 eqid 2443 . . . . . 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 5886 . . . . 5  |-  ( ph  ->  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) : ( ~P A  i^i  Fin ) --> RR* )
41 frn 5584 . . . . 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 3685 . . . . . . 7  |-  (/)  C_  A
44 0fin 7559 . . . . . . 7  |-  (/)  e.  Fin
45 elfpw 7632 . . . . . . 7  |-  ( (/)  e.  ( ~P A  i^i  Fin )  <->  ( (/)  C_  A  /\  (/)  e.  Fin )
)
4643, 44, 45mpbir2an 911 . . . . . 6  |-  (/)  e.  ( ~P A  i^i  Fin )
47 0cn 9397 . . . . . 6  |-  0  e.  CC
48 reseq2 5124 . . . . . . . . . 10  |-  ( s  =  (/)  ->  ( F  |`  s )  =  ( F  |`  (/) ) )
49 res0 5134 . . . . . . . . . 10  |-  ( F  |`  (/) )  =  (/)
5048, 49syl6eq 2491 . . . . . . . . 9  |-  ( s  =  (/)  ->  ( F  |`  s )  =  (/) )
5150oveq2d 6126 . . . . . . . 8  |-  ( s  =  (/)  ->  ( G 
gsumg  ( F  |`  s ) )  =  ( G 
gsumg  (/) ) )
5223gsum0 15529 . . . . . . . 8  |-  ( G 
gsumg  (/) )  =  0
5351, 52syl6eq 2491 . . . . . . 7  |-  ( s  =  (/)  ->  ( G 
gsumg  ( F  |`  s ) )  =  0 )
5439, 53elrnmpt1s 5106 . . . . . 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 3376 . . 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 7632 . . . . . . . 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 7562 . . . . . 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 3513 . . . . . 6  |-  ( ph  ->  ( B  \  C
)  C_  A )
68 fssres 5597 . . . . . 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 7649 . . . . 5  |-  ( ph  ->  ( F  |`  ( B  \  C ) ) finSupp 
0 )
725, 23, 58, 64, 69, 71gsumcl 16416 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  ( 0 [,] +oo ) )
731, 72sseldi 3373 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( B  \  C ) ) )  e.  RR* )
74 xrge0gsumle.c . . . . . 6  |-  ( ph  ->  C  C_  B )
75 ssfi 7552 . . . . . 6  |-  ( ( B  e.  Fin  /\  C  C_  B )  ->  C  e.  Fin )
7662, 74, 75syl2anc 661 . . . . 5  |-  ( ph  ->  C  e.  Fin )
7774, 66sstrd 3385 . . . . . 6  |-  ( ph  ->  C  C_  A )
78 fssres 5597 . . . . . 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 7649 . . . . 5  |-  ( ph  ->  ( F  |`  C ) finSupp 
0 )
815, 23, 58, 76, 79, 80gsumcl 16416 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  e.  ( 0 [,] +oo ) )
821, 81sseldi 3373 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  e.  RR* )
83 elxrge0 11413 . . . . 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 11236 . . 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 1221 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e 0 )  <_ 
( ( G  gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C
) ) ) ) )
88 xaddid1 11228 . . 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 6135 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
91 xrsadd 17852 . . . . . 6  |-  +e 
=  ( +g  `  RR*s
)
922, 91ressplusg 14299 . . . . 5  |-  ( ( 0 [,] +oo )  e.  _V  ->  +e 
=  ( +g  `  G
) )
9390, 92ax-mp 5 . . . 4  |-  +e 
=  ( +g  `  G
)
94 fssres 5597 . . . . 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 7649 . . . 4  |-  ( ph  ->  ( F  |`  B ) finSupp 
0 )
97 disjdif 3770 . . . . 5  |-  ( C  i^i  ( B  \  C ) )  =  (/)
9897a1i 11 . . . 4  |-  ( ph  ->  ( C  i^i  ( B  \  C ) )  =  (/) )
99 undif2 3774 . . . . 5  |-  ( C  u.  ( B  \  C ) )  =  ( C  u.  B
)
100 ssequn1 3545 . . . . . 6  |-  ( C 
C_  B  <->  ( C  u.  B )  =  B )
10174, 100sylib 196 . . . . 5  |-  ( ph  ->  ( C  u.  B
)  =  B )
10299, 101syl5req 2488 . . . 4  |-  ( ph  ->  B  =  ( C  u.  ( B  \  C ) ) )
1035, 23, 93, 58, 59, 95, 96, 98, 102gsumsplit 16439 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  B ) )  =  ( ( G  gsumg  ( ( F  |`  B )  |`  C ) ) +e ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C ) ) ) ) )
104 resabs1 5158 . . . . . 6  |-  ( C 
C_  B  ->  (
( F  |`  B )  |`  C )  =  ( F  |`  C )
)
10574, 104syl 16 . . . . 5  |-  ( ph  ->  ( ( F  |`  B )  |`  C )  =  ( F  |`  C ) )
106105oveq2d 6126 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( F  |`  B )  |`  C ) )  =  ( G 
gsumg  ( F  |`  C ) ) )
107 difss 3502 . . . . . 6  |-  ( B 
\  C )  C_  B
108 resabs1 5158 . . . . . 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 6126 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( F  |`  B )  |`  ( B  \  C ) ) )  =  ( G 
gsumg  ( F  |`  ( B 
\  C ) ) ) )
111106, 110oveq12d 6128 . . 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 2476 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) ) +e ( G  gsumg  ( F  |`  ( B  \  C
) ) ) )  =  ( G  gsumg  ( F  |`  B ) ) )
11387, 89, 1123brtr3d 4340 1  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  <_  ( G  gsumg  ( F  |`  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2991    \ cdif 3344    u. cun 3345    i^i cin 3346    C_ wss 3347   (/)c0 3656   ~Pcpw 3879   {csn 3896   class class class wbr 4311    e. cmpt 4369   ran crn 4860    |` cres 4861   -->wf 5433   ` cfv 5437  (class class class)co 6110   Fincfn 7329   CCcc 9299   0cc0 9301   +oocpnf 9434   -oocmnf 9435   RR*cxr 9436    <_ cle 9438   +ecxad 11106   [,]cicc 11322   Basecbs 14193   ↾s cress 14194   +g cplusg 14257   0gc0g 14397    gsumg cgsu 14398   RR*scxrs 14457  SubMndcsubmnd 15482  CMndccmn 16296
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-om 6496  df-1st 6596  df-2nd 6597  df-supp 6710  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fsupp 7640  df-oi 7743  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-xadd 11109  df-icc 11326  df-fz 11457  df-fzo 11568  df-seq 11826  df-hash 12123  df-struct 14195  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-mulr 14271  df-tset 14276  df-ple 14277  df-ds 14279  df-0g 14399  df-gsum 14400  df-xrs 14459  df-mre 14543  df-mrc 14544  df-acs 14546  df-mnd 15434  df-submnd 15484  df-cntz 15854  df-cmn 16298
This theorem is referenced by:  xrge0tsms  20430  xrge0tsmsd  26272
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