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Theorem tsmsadd 18176
Description: The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
tsmsadd.b  |-  B  =  ( Base `  G
)
tsmsadd.p  |-  .+  =  ( +g  `  G )
tsmsadd.1  |-  ( ph  ->  G  e. CMnd )
tsmsadd.2  |-  ( ph  ->  G  e. TopMnd )
tsmsadd.a  |-  ( ph  ->  A  e.  V )
tsmsadd.f  |-  ( ph  ->  F : A --> B )
tsmsadd.h  |-  ( ph  ->  H : A --> B )
tsmsadd.x  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
tsmsadd.y  |-  ( ph  ->  Y  e.  ( G tsums 
H ) )
Assertion
Ref Expression
tsmsadd  |-  ( ph  ->  ( X  .+  Y
)  e.  ( G tsums 
( F  o F 
.+  H ) ) )

Proof of Theorem tsmsadd
Dummy variables  y 
z  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmsadd.b . . . . . 6  |-  B  =  ( Base `  G
)
2 tsmsadd.1 . . . . . 6  |-  ( ph  ->  G  e. CMnd )
3 tsmsadd.2 . . . . . . 7  |-  ( ph  ->  G  e. TopMnd )
4 tmdtps 18106 . . . . . . 7  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  G  e.  TopSp )
6 tsmsadd.a . . . . . 6  |-  ( ph  ->  A  e.  V )
7 tsmsadd.f . . . . . 6  |-  ( ph  ->  F : A --> B )
81, 2, 5, 6, 7tsmscl 18164 . . . . 5  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
9 tsmsadd.x . . . . 5  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
108, 9sseldd 3349 . . . 4  |-  ( ph  ->  X  e.  B )
11 tsmsadd.h . . . . . 6  |-  ( ph  ->  H : A --> B )
121, 2, 5, 6, 11tsmscl 18164 . . . . 5  |-  ( ph  ->  ( G tsums  H ) 
C_  B )
13 tsmsadd.y . . . . 5  |-  ( ph  ->  Y  e.  ( G tsums 
H ) )
1412, 13sseldd 3349 . . . 4  |-  ( ph  ->  Y  e.  B )
15 tsmsadd.p . . . . 5  |-  .+  =  ( +g  `  G )
16 eqid 2436 . . . . 5  |-  ( + f `  G )  =  ( + f `  G )
171, 15, 16plusfval 14703 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X ( + f `  G ) Y )  =  ( X  .+  Y ) )
1810, 14, 17syl2anc 643 . . 3  |-  ( ph  ->  ( X ( + f `  G ) Y )  =  ( X  .+  Y ) )
19 eqid 2436 . . . . . 6  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
201, 19istps 17001 . . . . 5  |-  ( G  e.  TopSp 
<->  ( TopOpen `  G )  e.  (TopOn `  B )
)
215, 20sylib 189 . . . 4  |-  ( ph  ->  ( TopOpen `  G )  e.  (TopOn `  B )
)
22 eqid 2436 . . . . . 6  |-  ( ~P A  i^i  Fin )  =  ( ~P A  i^i  Fin )
23 eqid 2436 . . . . . 6  |-  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y 
C_  z } )  =  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )
24 eqid 2436 . . . . . 6  |-  ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )  =  ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )
2522, 23, 24, 6tsmsfbas 18157 . . . . 5  |-  ( ph  ->  ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )  e.  (
fBas `  ( ~P A  i^i  Fin ) ) )
26 fgcl 17910 . . . . 5  |-  ( ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )  e.  (
fBas `  ( ~P A  i^i  Fin ) )  ->  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) )  e.  ( Fil `  ( ~P A  i^i  Fin )
) )
2725, 26syl 16 . . . 4  |-  ( ph  ->  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) )  e.  ( Fil `  ( ~P A  i^i  Fin )
) )
281, 22, 2, 6, 7tsmslem1 18158 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  z
) )  e.  B
)
291, 22, 2, 6, 11tsmslem1 18158 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( H  |`  z
) )  e.  B
)
301, 19, 22, 24, 2, 6, 7tsmsval 18160 . . . . 5  |-  ( ph  ->  ( G tsums  F )  =  ( ( (
TopOpen `  G )  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) )
319, 30eleqtrd 2512 . . . 4  |-  ( ph  ->  X  e.  ( ( ( TopOpen `  G )  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) )
321, 19, 22, 24, 2, 6, 11tsmsval 18160 . . . . 5  |-  ( ph  ->  ( G tsums  H )  =  ( ( (
TopOpen `  G )  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( H  |`  z ) ) ) ) )
3313, 32eleqtrd 2512 . . . 4  |-  ( ph  ->  Y  e.  ( ( ( TopOpen `  G )  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( H  |`  z ) ) ) ) )
3419, 16tmdcn 18113 . . . . . 6  |-  ( G  e. TopMnd  ->  ( + f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
353, 34syl 16 . . . . 5  |-  ( ph  ->  ( + f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
36 opelxpi 4910 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
3710, 14, 36syl2anc 643 . . . . . 6  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
38 txtopon 17623 . . . . . . . 8  |-  ( ( ( TopOpen `  G )  e.  (TopOn `  B )  /\  ( TopOpen `  G )  e.  (TopOn `  B )
)  ->  ( ( TopOpen
`  G )  tX  ( TopOpen `  G )
)  e.  (TopOn `  ( B  X.  B
) ) )
3921, 21, 38syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  e.  (TopOn `  ( B  X.  B ) ) )
40 toponuni 16992 . . . . . . 7  |-  ( ( ( TopOpen `  G )  tX  ( TopOpen `  G )
)  e.  (TopOn `  ( B  X.  B
) )  ->  ( B  X.  B )  = 
U. ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) ) )
4139, 40syl 16 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  U. (
( TopOpen `  G )  tX  ( TopOpen `  G )
) )
4237, 41eleqtrd 2512 . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e. 
U. ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) ) )
43 eqid 2436 . . . . . 6  |-  U. (
( TopOpen `  G )  tX  ( TopOpen `  G )
)  =  U. (
( TopOpen `  G )  tX  ( TopOpen `  G )
)
4443cncnpi 17342 . . . . 5  |-  ( ( ( + f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
)  /\  <. X ,  Y >.  e.  U. (
( TopOpen `  G )  tX  ( TopOpen `  G )
) )  ->  ( + f `  G )  e.  ( ( ( ( TopOpen `  G )  tX  ( TopOpen `  G )
)  CnP  ( TopOpen `  G ) ) `  <. X ,  Y >. ) )
4535, 42, 44syl2anc 643 . . . 4  |-  ( ph  ->  ( + f `  G )  e.  ( ( ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) )  CnP  ( TopOpen `  G
) ) `  <. X ,  Y >. )
)
4621, 21, 27, 28, 29, 31, 33, 45flfcnp2 18039 . . 3  |-  ( ph  ->  ( X ( + f `  G ) Y )  e.  ( ( ( TopOpen `  G
)  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( ( G 
gsumg  ( F  |`  z ) ) ( + f `  G ) ( G 
gsumg  ( H  |`  z ) ) ) ) ) )
4718, 46eqeltrrd 2511 . 2  |-  ( ph  ->  ( X  .+  Y
)  e.  ( ( ( TopOpen `  G )  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( ( G 
gsumg  ( F  |`  z ) ) ( + f `  G ) ( G 
gsumg  ( H  |`  z ) ) ) ) ) )
48 cmnmnd 15427 . . . . . . 7  |-  ( G  e. CMnd  ->  G  e.  Mnd )
492, 48syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Mnd )
501, 15mndcl 14695 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
51503expb 1154 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
5249, 51sylan 458 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  .+  y
)  e.  B )
53 inidm 3550 . . . . 5  |-  ( A  i^i  A )  =  A
5452, 7, 11, 6, 6, 53off 6320 . . . 4  |-  ( ph  ->  ( F  o F 
.+  H ) : A --> B )
551, 19, 22, 24, 2, 6, 54tsmsval 18160 . . 3  |-  ( ph  ->  ( G tsums  ( F  o F  .+  H
) )  =  ( ( ( TopOpen `  G
)  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( ( F  o F  .+  H )  |`  z
) ) ) ) )
56 eqid 2436 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
572adantr 452 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
58 elfpw 7408 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  <->  ( z  C_  A  /\  z  e. 
Fin ) )
5958simprbi 451 . . . . . . . 8  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  e.  Fin )
6059adantl 453 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  z  e.  Fin )
6158simplbi 447 . . . . . . . 8  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  C_  A )
62 fssres 5610 . . . . . . . 8  |-  ( ( F : A --> B  /\  z  C_  A )  -> 
( F  |`  z
) : z --> B )
637, 61, 62syl2an 464 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) : z --> B )
64 fssres 5610 . . . . . . . 8  |-  ( ( H : A --> B  /\  z  C_  A )  -> 
( H  |`  z
) : z --> B )
6511, 61, 64syl2an 464 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  |`  z ) : z --> B )
6660, 63fisuppfi 14773 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( F  |`  z ) " ( _V  \  { ( 0g
`  G ) } ) )  e.  Fin )
6760, 65fisuppfi 14773 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( H  |`  z ) " ( _V  \  { ( 0g
`  G ) } ) )  e.  Fin )
681, 56, 15, 57, 60, 63, 65, 66, 67gsumadd 15528 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( ( F  |`  z )  o F 
.+  ( H  |`  z ) ) )  =  ( ( G 
gsumg  ( F  |`  z ) )  .+  ( G 
gsumg  ( H  |`  z ) ) ) )
69 fvex 5742 . . . . . . . . . . . 12  |-  ( Base `  G )  e.  _V
701, 69eqeltri 2506 . . . . . . . . . . 11  |-  B  e. 
_V
7170a1i 11 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
72 fex2 5603 . . . . . . . . . 10  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  _V )  ->  F  e.  _V )
737, 6, 71, 72syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  F  e.  _V )
74 fex2 5603 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  A  e.  V  /\  B  e.  _V )  ->  H  e.  _V )
7511, 6, 71, 74syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  H  e.  _V )
76 offres 6319 . . . . . . . . 9  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( F  o F  .+  H )  |`  z )  =  ( ( F  |`  z
)  o F  .+  ( H  |`  z ) ) )
7773, 75, 76syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( F  o F  .+  H )  |`  z )  =  ( ( F  |`  z
)  o F  .+  ( H  |`  z ) ) )
7877adantr 452 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  (
( F  o F 
.+  H )  |`  z )  =  ( ( F  |`  z
)  o F  .+  ( H  |`  z ) ) )
7978oveq2d 6097 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( ( F  o F  .+  H )  |`  z ) )  =  ( G  gsumg  ( ( F  |`  z )  o F 
.+  ( H  |`  z ) ) ) )
801, 15, 16plusfval 14703 . . . . . . 7  |-  ( ( ( G  gsumg  ( F  |`  z
) )  e.  B  /\  ( G  gsumg  ( H  |`  z
) )  e.  B
)  ->  ( ( G  gsumg  ( F  |`  z
) ) ( + f `  G ) ( G  gsumg  ( H  |`  z
) ) )  =  ( ( G  gsumg  ( F  |`  z ) )  .+  ( G  gsumg  ( H  |`  z
) ) ) )
8128, 29, 80syl2anc 643 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  (
( G  gsumg  ( F  |`  z
) ) ( + f `  G ) ( G  gsumg  ( H  |`  z
) ) )  =  ( ( G  gsumg  ( F  |`  z ) )  .+  ( G  gsumg  ( H  |`  z
) ) ) )
8268, 79, 813eqtr4d 2478 . . . . 5  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( ( F  o F  .+  H )  |`  z ) )  =  ( ( G  gsumg  ( F  |`  z ) ) ( + f `  G
) ( G  gsumg  ( H  |`  z ) ) ) )
8382mpteq2dva 4295 . . . 4  |-  ( ph  ->  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( ( F  o F  .+  H )  |`  z
) ) )  =  ( z  e.  ( ~P A  i^i  Fin )  |->  ( ( G 
gsumg  ( F  |`  z ) ) ( + f `  G ) ( G 
gsumg  ( H  |`  z ) ) ) ) )
8483fveq2d 5732 . . 3  |-  ( ph  ->  ( ( ( TopOpen `  G )  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( ( F  o F  .+  H )  |`  z
) ) ) )  =  ( ( (
TopOpen `  G )  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( ( G 
gsumg  ( F  |`  z ) ) ( + f `  G ) ( G 
gsumg  ( H  |`  z ) ) ) ) ) )
8555, 84eqtrd 2468 . 2  |-  ( ph  ->  ( G tsums  ( F  o F  .+  H
) )  =  ( ( ( TopOpen `  G
)  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( ( G 
gsumg  ( F  |`  z ) ) ( + f `  G ) ( G 
gsumg  ( H  |`  z ) ) ) ) ) )
8647, 85eleqtrrd 2513 1  |-  ( ph  ->  ( X  .+  Y
)  e.  ( G tsums 
( F  o F 
.+  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   {csn 3814   <.cop 3817   U.cuni 4015    e. cmpt 4266    X. cxp 4876   ran crn 4879    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Fincfn 7109   Basecbs 13469   +g cplusg 13529   TopOpenctopn 13649   0gc0g 13723    gsumg cgsu 13724   Mndcmnd 14684   + fcplusf 14687  CMndccmn 15412   fBascfbas 16689   filGencfg 16690  TopOnctopon 16959   TopSpctps 16961    Cn ccn 17288    CnP ccnp 17289    tX ctx 17592   Filcfil 17877    fLimf cflf 17967  TopMndctmd 18100   tsums ctsu 18155
This theorem is referenced by:  tsmssub  18178  tsmssplit  18181  esumadd  24448  esumaddf  24453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-topgen 13667  df-0g 13727  df-gsum 13728  df-mnd 14690  df-plusf 14691  df-submnd 14739  df-cntz 15116  df-cmn 15414  df-fbas 16699  df-fg 16700  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-ntr 17084  df-nei 17162  df-cn 17291  df-cnp 17292  df-tx 17594  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-tmd 18102  df-tsms 18156
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