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Theorem tsmsadd 21239
Description: The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
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  oF  .+  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 21169 . . . . . . 7  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
53, 4syl 17 . . . . . 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 21227 . . . . 5  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
9 tsmsadd.x . . . . 5  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
108, 9sseldd 3419 . . . 4  |-  ( ph  ->  X  e.  B )
11 tsmsadd.h . . . . . 6  |-  ( ph  ->  H : A --> B )
121, 2, 5, 6, 11tsmscl 21227 . . . . 5  |-  ( ph  ->  ( G tsums  H ) 
C_  B )
13 tsmsadd.y . . . . 5  |-  ( ph  ->  Y  e.  ( G tsums 
H ) )
1412, 13sseldd 3419 . . . 4  |-  ( ph  ->  Y  e.  B )
15 tsmsadd.p . . . . 5  |-  .+  =  ( +g  `  G )
16 eqid 2471 . . . . 5  |-  ( +f `  G )  =  ( +f `  G )
171, 15, 16plusfval 16572 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X ( +f `  G ) Y )  =  ( X  .+  Y ) )
1810, 14, 17syl2anc 673 . . 3  |-  ( ph  ->  ( X ( +f `  G ) Y )  =  ( X  .+  Y ) )
19 eqid 2471 . . . . . 6  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
201, 19istps 20028 . . . . 5  |-  ( G  e.  TopSp 
<->  ( TopOpen `  G )  e.  (TopOn `  B )
)
215, 20sylib 201 . . . 4  |-  ( ph  ->  ( TopOpen `  G )  e.  (TopOn `  B )
)
22 eqid 2471 . . . . . 6  |-  ( ~P A  i^i  Fin )  =  ( ~P A  i^i  Fin )
23 eqid 2471 . . . . . 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 2471 . . . . . 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 21220 . . . . 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 20971 . . . . 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 17 . . . 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 21221 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  z
) )  e.  B
)
291, 22, 2, 6, 11tsmslem1 21221 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( H  |`  z
) )  e.  B
)
301, 19, 22, 24, 2, 6, 7tsmsval 21223 . . . . 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 2551 . . . 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 21223 . . . . 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 2551 . . . 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 21176 . . . . . 6  |-  ( G  e. TopMnd  ->  ( +f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
353, 34syl 17 . . . . 5  |-  ( ph  ->  ( +f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
36 opelxpi 4871 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
3710, 14, 36syl2anc 673 . . . . . 6  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
38 txtopon 20683 . . . . . . . 8  |-  ( ( ( TopOpen `  G )  e.  (TopOn `  B )  /\  ( TopOpen `  G )  e.  (TopOn `  B )
)  ->  ( ( TopOpen
`  G )  tX  ( TopOpen `  G )
)  e.  (TopOn `  ( B  X.  B
) ) )
3921, 21, 38syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  e.  (TopOn `  ( B  X.  B ) ) )
40 toponuni 20019 . . . . . . 7  |-  ( ( ( TopOpen `  G )  tX  ( TopOpen `  G )
)  e.  (TopOn `  ( B  X.  B
) )  ->  ( B  X.  B )  = 
U. ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) ) )
4139, 40syl 17 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  U. (
( TopOpen `  G )  tX  ( TopOpen `  G )
) )
4237, 41eleqtrd 2551 . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e. 
U. ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) ) )
43 eqid 2471 . . . . . 6  |-  U. (
( TopOpen `  G )  tX  ( TopOpen `  G )
)  =  U. (
( TopOpen `  G )  tX  ( TopOpen `  G )
)
4443cncnpi 20371 . . . . 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 673 . . . 4  |-  ( ph  ->  ( +f `  G )  e.  ( ( ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) )  CnP  ( TopOpen `  G
) ) `  <. X ,  Y >. )
)
4621, 21, 27, 28, 29, 31, 33, 45flfcnp2 21100 . . 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 2550 . 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 17523 . . . . . . 7  |-  ( G  e. CMnd  ->  G  e.  Mnd )
492, 48syl 17 . . . . . 6  |-  ( ph  ->  G  e.  Mnd )
501, 15mndcl 16623 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
51503expb 1232 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
5249, 51sylan 479 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  .+  y
)  e.  B )
53 inidm 3632 . . . . 5  |-  ( A  i^i  A )  =  A
5452, 7, 11, 6, 6, 53off 6565 . . . 4  |-  ( ph  ->  ( F  oF  .+  H ) : A --> B )
551, 19, 22, 24, 2, 6, 54tsmsval 21223 . . 3  |-  ( ph  ->  ( G tsums  ( F  oF  .+  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  oF  .+  H )  |`  z
) ) ) ) )
56 eqid 2471 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
572adantr 472 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
58 elfpw 7894 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  <->  ( z  C_  A  /\  z  e. 
Fin ) )
5958simprbi 471 . . . . . . . 8  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  e.  Fin )
6059adantl 473 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  z  e.  Fin )
6158simplbi 467 . . . . . . . 8  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  C_  A )
62 fssres 5761 . . . . . . . 8  |-  ( ( F : A --> B  /\  z  C_  A )  -> 
( F  |`  z
) : z --> B )
637, 61, 62syl2an 485 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) : z --> B )
64 fssres 5761 . . . . . . . 8  |-  ( ( H : A --> B  /\  z  C_  A )  -> 
( H  |`  z
) : z --> B )
6511, 61, 64syl2an 485 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  |`  z ) : z --> B )
66 fvex 5889 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
6766a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( 0g `  G )  e. 
_V )
6863, 60, 67fdmfifsupp 7911 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) finSupp  ( 0g `  G ) )
6965, 60, 67fdmfifsupp 7911 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  |`  z ) finSupp  ( 0g `  G ) )
701, 56, 15, 57, 60, 63, 65, 68, 69gsumadd 17634 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( ( F  |`  z )  oF  .+  ( H  |`  z ) ) )  =  ( ( G 
gsumg  ( F  |`  z ) )  .+  ( G 
gsumg  ( H  |`  z ) ) ) )
71 fvex 5889 . . . . . . . . . . . 12  |-  ( Base `  G )  e.  _V
721, 71eqeltri 2545 . . . . . . . . . . 11  |-  B  e. 
_V
7372a1i 11 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
74 fex2 6767 . . . . . . . . . 10  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  _V )  ->  F  e.  _V )
757, 6, 73, 74syl3anc 1292 . . . . . . . . 9  |-  ( ph  ->  F  e.  _V )
76 fex2 6767 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  A  e.  V  /\  B  e.  _V )  ->  H  e.  _V )
7711, 6, 73, 76syl3anc 1292 . . . . . . . . 9  |-  ( ph  ->  H  e.  _V )
78 offres 6807 . . . . . . . . 9  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( F  oF  .+  H )  |`  z )  =  ( ( F  |`  z
)  oF  .+  ( H  |`  z ) ) )
7975, 77, 78syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( ( F  oF  .+  H )  |`  z )  =  ( ( F  |`  z
)  oF  .+  ( H  |`  z ) ) )
8079adantr 472 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  (
( F  oF  .+  H )  |`  z )  =  ( ( F  |`  z
)  oF  .+  ( H  |`  z ) ) )
8180oveq2d 6324 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( ( F  oF  .+  H )  |`  z ) )  =  ( G  gsumg  ( ( F  |`  z )  oF  .+  ( H  |`  z ) ) ) )
821, 15, 16plusfval 16572 . . . . . . 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
) ) ) )
8328, 29, 82syl2anc 673 . . . . . 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
) ) ) )
8470, 81, 833eqtr4d 2515 . . . . 5  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( ( F  oF  .+  H )  |`  z ) )  =  ( ( G  gsumg  ( F  |`  z ) ) ( +f `  G
) ( G  gsumg  ( H  |`  z ) ) ) )
8584mpteq2dva 4482 . . . 4  |-  ( ph  ->  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( ( F  oF  .+  H )  |`  z
) ) )  =  ( z  e.  ( ~P A  i^i  Fin )  |->  ( ( G 
gsumg  ( F  |`  z ) ) ( +f `  G ) ( G 
gsumg  ( H  |`  z ) ) ) ) )
8685fveq2d 5883 . . 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  oF  .+  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 ) ) ) ) ) )
8755, 86eqtrd 2505 . 2  |-  ( ph  ->  ( G tsums  ( F  oF  .+  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 ) ) ) ) ) )
8847, 87eleqtrrd 2552 1  |-  ( ph  ->  ( X  .+  Y
)  e.  ( G tsums 
( F  oF  .+  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   <.cop 3965   U.cuni 4190    |-> cmpt 4454    X. cxp 4837   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   Fincfn 7587   Basecbs 15199   +g cplusg 15268   TopOpenctopn 15398   0gc0g 15416    gsumg cgsu 15417   +fcplusf 16563   Mndcmnd 16613  CMndccmn 17508   fBascfbas 19035   filGencfg 19036  TopOnctopon 19995   TopSpctps 19996    Cn ccn 20317    CnP ccnp 20318    tX ctx 20652   Filcfil 20938    fLimf cflf 21028  TopMndctmd 21163   tsums ctsu 21218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-topgen 15420  df-plusf 16565  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-cntz 17049  df-cmn 17510  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-ntr 20112  df-nei 20191  df-cn 20320  df-cnp 20321  df-tx 20654  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tmd 21165  df-tsms 21219
This theorem is referenced by:  tsmssub  21241  tsmssplit  21244  esumadd  28952  esumaddf  28956
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