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Theorem esumpfinvallem 24417
Description: Lemma for esumpfinval 24418 (Contributed by Thierry Arnoux, 28-Jun-2017.)
Assertion
Ref Expression
esumpfinvallem  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
(fld  gsumg  F )  =  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  F ) )

Proof of Theorem esumpfinvallem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fex 5928 . . . 4  |-  ( ( F : A --> ( 0 [,)  +oo )  /\  A  e.  V )  ->  F  e.  _V )
21ancoms 440 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  F  e.  _V )
3 ovex 6065 . . . 4  |-  (flds  ( 0 [,) 
+oo ) )  e. 
_V
43a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
(flds  (
0 [,)  +oo ) )  e.  _V )
5 ovex 6065 . . . 4  |-  ( RR* ss  ( 0 [,)  +oo ) )  e.  _V
65a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
( RR* ss  ( 0 [,) 
+oo ) )  e. 
_V )
7 mnfxr 10670 . . . . . . . . 9  |-  -oo  e.  RR*
8 pnfxr 10669 . . . . . . . . 9  |-  +oo  e.  RR*
9 0re 9047 . . . . . . . . . 10  |-  0  e.  RR
10 mnflt 10678 . . . . . . . . . 10  |-  ( 0  e.  RR  ->  -oo  <  0 )
119, 10ax-mp 8 . . . . . . . . 9  |-  -oo  <  0
12 pnfge 10683 . . . . . . . . . 10  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
138, 12ax-mp 8 . . . . . . . . 9  |-  +oo  <_  +oo
14 icossioo 24086 . . . . . . . . 9  |-  ( ( (  -oo  e.  RR*  /\ 
+oo  e.  RR* )  /\  (  -oo  <  0  /\  +oo 
<_  +oo ) )  -> 
( 0 [,)  +oo )  C_  (  -oo (,)  +oo ) )
157, 8, 11, 13, 14mp4an 655 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  (  -oo (,)  +oo )
16 ioomax 10941 . . . . . . . 8  |-  (  -oo (,) 
+oo )  =  RR
1715, 16sseqtri 3340 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  RR
18 ax-resscn 9003 . . . . . . 7  |-  RR  C_  CC
1917, 18sstri 3317 . . . . . 6  |-  ( 0 [,)  +oo )  C_  CC
20 eqid 2404 . . . . . . 7  |-  (flds  ( 0 [,) 
+oo ) )  =  (flds  ( 0 [,)  +oo )
)
21 cnfldbas 16662 . . . . . . 7  |-  CC  =  ( Base ` fld )
2220, 21ressbas2 13475 . . . . . 6  |-  ( ( 0 [,)  +oo )  C_  CC  ->  ( 0 [,)  +oo )  =  (
Base `  (flds  ( 0 [,) 
+oo ) ) ) )
2319, 22ax-mp 8 . . . . 5  |-  ( 0 [,)  +oo )  =  (
Base `  (flds  ( 0 [,) 
+oo ) ) )
24 icossxr 10951 . . . . . 6  |-  ( 0 [,)  +oo )  C_  RR*
25 eqid 2404 . . . . . . 7  |-  ( RR* ss  ( 0 [,)  +oo ) )  =  (
RR* ss  ( 0 [,) 
+oo ) )
26 xrsbas 16672 . . . . . . 7  |-  RR*  =  ( Base `  RR* s )
2725, 26ressbas2 13475 . . . . . 6  |-  ( ( 0 [,)  +oo )  C_ 
RR*  ->  ( 0 [,) 
+oo )  =  (
Base `  ( RR* ss  ( 0 [,)  +oo ) ) ) )
2824, 27ax-mp 8 . . . . 5  |-  ( 0 [,)  +oo )  =  (
Base `  ( RR* ss  ( 0 [,)  +oo ) ) )
2923, 28eqtr3i 2426 . . . 4  |-  ( Base `  (flds  ( 0 [,)  +oo )
) )  =  (
Base `  ( RR* ss  ( 0 [,)  +oo ) ) )
3029a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
( Base `  (flds  ( 0 [,) 
+oo ) ) )  =  ( Base `  ( RR* ss  ( 0 [,)  +oo ) ) ) )
31 simprl 733 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,)  +oo )
) )  /\  y  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) ) )  ->  x  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) )
3231, 23syl6eleqr 2495 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,)  +oo )
) )  /\  y  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) ) )  ->  x  e.  ( 0 [,)  +oo ) )
33 simprr 734 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,)  +oo )
) )  /\  y  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) ) )  ->  y  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) )
3433, 23syl6eleqr 2495 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,)  +oo )
) )  /\  y  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) ) )  ->  y  e.  ( 0 [,)  +oo ) )
35 ge0addcl 10965 . . . . 5  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x  +  y )  e.  ( 0 [,) 
+oo ) )
36 ovex 6065 . . . . . . 7  |-  ( 0 [,)  +oo )  e.  _V
37 cnfldadd 16663 . . . . . . . 8  |-  +  =  ( +g  ` fld )
3820, 37ressplusg 13526 . . . . . . 7  |-  ( ( 0 [,)  +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) 
+oo ) ) ) )
3936, 38ax-mp 8 . . . . . 6  |-  +  =  ( +g  `  (flds  ( 0 [,) 
+oo ) ) )
4039oveqi 6053 . . . . 5  |-  ( x  +  y )  =  ( x ( +g  `  (flds  ( 0 [,)  +oo )
) ) y )
4135, 40, 233eltr3g 2486 . . . 4  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x ( +g  `  (flds  ( 0 [,)  +oo ) ) ) y )  e.  (
Base `  (flds  ( 0 [,) 
+oo ) ) ) )
4232, 34, 41syl2anc 643 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,)  +oo )
) )  /\  y  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) ) )  ->  (
x ( +g  `  (flds  ( 0 [,)  +oo ) ) ) y )  e.  (
Base `  (flds  ( 0 [,) 
+oo ) ) ) )
43 simpl 444 . . . . . . 7  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  x  e.  ( 0 [,)  +oo ) )
4417, 43sseldi 3306 . . . . . 6  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  x  e.  RR )
45 simpr 448 . . . . . . 7  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  y  e.  ( 0 [,)  +oo ) )
4617, 45sseldi 3306 . . . . . 6  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  y  e.  RR )
47 rexadd 10774 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x + e
y )  =  ( x  +  y ) )
4847eqcomd 2409 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  =  ( x + e y ) )
4944, 46, 48syl2anc 643 . . . . 5  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x  +  y )  =  ( x + e y ) )
50 xrsadd 16673 . . . . . . . 8  |-  + e  =  ( +g  `  RR* s
)
5125, 50ressplusg 13526 . . . . . . 7  |-  ( ( 0 [,)  +oo )  e.  _V  ->  + e  =  ( +g  `  ( RR* ss  ( 0 [,)  +oo ) ) ) )
5236, 51ax-mp 8 . . . . . 6  |-  + e  =  ( +g  `  ( RR* ss  ( 0 [,)  +oo ) ) )
5352oveqi 6053 . . . . 5  |-  ( x + e y )  =  ( x ( +g  `  ( RR* ss  ( 0 [,)  +oo ) ) ) y )
5449, 40, 533eqtr3g 2459 . . . 4  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x ( +g  `  (flds  ( 0 [,)  +oo ) ) ) y )  =  ( x ( +g  `  ( RR* ss  ( 0 [,)  +oo ) ) ) y ) )
5532, 34, 54syl2anc 643 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,)  +oo )
) )  /\  y  e.  ( Base `  (flds  ( 0 [,)  +oo ) ) ) ) )  ->  (
x ( +g  `  (flds  ( 0 [,)  +oo ) ) ) y )  =  ( x ( +g  `  ( RR* ss  ( 0 [,)  +oo ) ) ) y ) )
56 simpr 448 . . . 4  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  F : A --> ( 0 [,)  +oo ) )
57 ffun 5552 . . . 4  |-  ( F : A --> ( 0 [,)  +oo )  ->  Fun  F )
5856, 57syl 16 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  Fun  F )
59 frn 5556 . . . . 5  |-  ( F : A --> ( 0 [,)  +oo )  ->  ran  F 
C_  ( 0 [,) 
+oo ) )
6056, 59syl 16 . . . 4  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  ran  F  C_  ( 0 [,)  +oo ) )
6160, 23syl6sseq 3354 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  ran  F  C_  ( Base `  (flds  ( 0 [,)  +oo )
) ) )
622, 4, 6, 30, 42, 55, 58, 61gsumpropd2 24174 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
( (flds  ( 0 [,)  +oo )
)  gsumg  F )  =  ( ( RR* ss  ( 0 [,)  +oo ) )  gsumg  F ) )
63 cnfldex 16661 . . . 4  |-fld  e.  _V
6463a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->fld  e.  _V )
65 simpl 444 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  A  e.  V )
6619a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
( 0 [,)  +oo )  C_  CC )
67 0xr 9087 . . . . 5  |-  0  e.  RR*
68 ltpnf 10677 . . . . . 6  |-  ( 0  e.  RR  ->  0  <  +oo )
699, 68ax-mp 8 . . . . 5  |-  0  <  +oo
70 lbico1 10922 . . . . 5  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <  +oo )  ->  0  e.  ( 0 [,)  +oo ) )
7167, 8, 69, 70mp3an 1279 . . . 4  |-  0  e.  ( 0 [,)  +oo )
7271a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
0  e.  ( 0 [,)  +oo ) )
73 simpr 448 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  CC )  ->  x  e.  CC )
7473addid2d 9223 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  CC )  ->  ( 0  +  x
)  =  x )
7573addid1d 9222 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  CC )  ->  ( x  +  0 )  =  x )
7674, 75jca 519 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  CC )  ->  ( ( 0  +  x )  =  x  /\  ( x  + 
0 )  =  x ) )
7721, 37, 20, 64, 65, 66, 56, 72, 76gsumress 14732 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
(fld  gsumg  F )  =  ( (flds  ( 0 [,)  +oo ) )  gsumg  F ) )
78 xrge0base 24160 . . 3  |-  ( 0 [,]  +oo )  =  (
Base `  ( RR* ss  ( 0 [,]  +oo ) ) )
79 xrge0plusg 24162 . . 3  |-  + e  =  ( +g  `  ( RR* ss  ( 0 [,]  +oo ) ) )
80 ovex 6065 . . . . 5  |-  ( 0 [,]  +oo )  e.  _V
81 ressress 13481 . . . . 5  |-  ( ( ( 0 [,]  +oo )  e.  _V  /\  (
0 [,)  +oo )  e. 
_V )  ->  (
( RR* ss  ( 0 [,] 
+oo ) )s  ( 0 [,)  +oo ) )  =  ( RR* ss  ( ( 0 [,]  +oo )  i^i  ( 0 [,)  +oo ) ) ) )
8280, 36, 81mp2an 654 . . . 4  |-  ( (
RR* ss  ( 0 [,] 
+oo ) )s  ( 0 [,)  +oo ) )  =  ( RR* ss  ( ( 0 [,]  +oo )  i^i  ( 0 [,)  +oo ) ) )
83 incom 3493 . . . . . 6  |-  ( ( 0 [,]  +oo )  i^i  ( 0 [,)  +oo ) )  =  ( ( 0 [,)  +oo )  i^i  ( 0 [,] 
+oo ) )
84 icossicc 24082 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
85 dfss 3295 . . . . . . 7  |-  ( ( 0 [,)  +oo )  C_  ( 0 [,]  +oo ) 
<->  ( 0 [,)  +oo )  =  ( (
0 [,)  +oo )  i^i  ( 0 [,]  +oo ) ) )
8684, 85mpbi 200 . . . . . 6  |-  ( 0 [,)  +oo )  =  ( ( 0 [,)  +oo )  i^i  ( 0 [,] 
+oo ) )
8783, 86eqtr4i 2427 . . . . 5  |-  ( ( 0 [,]  +oo )  i^i  ( 0 [,)  +oo ) )  =  ( 0 [,)  +oo )
8887oveq2i 6051 . . . 4  |-  ( RR* ss  ( ( 0 [,] 
+oo )  i^i  (
0 [,)  +oo ) ) )  =  ( RR* ss  ( 0 [,)  +oo ) )
8982, 88eqtr2i 2425 . . 3  |-  ( RR* ss  ( 0 [,)  +oo ) )  =  ( ( RR* ss  ( 0 [,]  +oo ) )s  ( 0 [,)  +oo ) )
90 ovex 6065 . . . 4  |-  ( RR* ss  ( 0 [,]  +oo ) )  e.  _V
9190a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
( RR* ss  ( 0 [,] 
+oo ) )  e. 
_V )
9284a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
( 0 [,)  +oo )  C_  ( 0 [,] 
+oo ) )
93 iccssxr 10949 . . . . . 6  |-  ( 0 [,]  +oo )  C_  RR*
94 simpr 448 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  ( 0 [,]  +oo ) )
9593, 94sseldi 3306 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  RR* )
96 xaddid2 10782 . . . . 5  |-  ( x  e.  RR*  ->  ( 0 + e x )  =  x )
9795, 96syl 16 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( 0 + e
x )  =  x )
98 xaddid1 10781 . . . . 5  |-  ( x  e.  RR*  ->  ( x + e 0 )  =  x )
9995, 98syl 16 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( x + e
0 )  =  x )
10097, 99jca 519 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( ( 0 + e x )  =  x  /\  ( x + e 0 )  =  x ) )
10178, 79, 89, 91, 65, 92, 56, 72, 100gsumress 14732 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
( ( RR* ss  (
0 [,]  +oo ) ) 
gsumg  F )  =  ( ( RR* ss  ( 0 [,)  +oo ) )  gsumg  F ) )
10262, 77, 1013eqtr4d 2446 1  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  -> 
(fld  gsumg  F )  =  ( (
RR* ss  ( 0 [,] 
+oo ) )  gsumg  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280   class class class wbr 4172   ran crn 4838   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075    < clt 9076    <_ cle 9077   + ecxad 10664   (,)cioo 10872   [,)cico 10874   [,]cicc 10875   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   RR* scxrs 13677    gsumg cgsu 13679  ℂfldccnfld 16658
This theorem is referenced by:  esumpfinval  24418  esumpfinvalf  24419  esumpcvgval  24421  esumcvg  24429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-xadd 10667  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-seq 11279  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-xrs 13681  df-0g 13682  df-gsum 13683  df-cnfld 16659
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