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Theorem tsmsval 17829
Description: Definition of the topological group sum(s) of a collection  F ( x ) of values in the group with index set  A. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tsmsval.b  |-  B  =  ( Base `  G
)
tsmsval.j  |-  J  =  ( TopOpen `  G )
tsmsval.s  |-  S  =  ( ~P A  i^i  Fin )
tsmsval.l  |-  L  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )
tsmsval.g  |-  ( ph  ->  G  e.  V )
tsmsval.a  |-  ( ph  ->  A  e.  W )
tsmsval.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
tsmsval  |-  ( ph  ->  ( G tsums  F )  =  ( ( J 
fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G 
gsumg  ( F  |`  y ) ) ) ) )
Distinct variable groups:    y, z, F    y, G, z    ph, y,
z    y, S
Allowed substitution hints:    A( y, z)    B( y, z)    S( z)    J( y, z)    L( y, z)    V( y, z)    W( y, z)

Proof of Theorem tsmsval
StepHypRef Expression
1 tsmsval.b . 2  |-  B  =  ( Base `  G
)
2 tsmsval.j . 2  |-  J  =  ( TopOpen `  G )
3 tsmsval.s . 2  |-  S  =  ( ~P A  i^i  Fin )
4 tsmsval.l . 2  |-  L  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )
5 tsmsval.g . 2  |-  ( ph  ->  G  e.  V )
6 tsmsval.f . . 3  |-  ( ph  ->  F : A --> B )
7 tsmsval.a . . 3  |-  ( ph  ->  A  e.  W )
8 fvex 5555 . . . . 5  |-  ( Base `  G )  e.  _V
91, 8eqeltri 2366 . . . 4  |-  B  e. 
_V
109a1i 10 . . 3  |-  ( ph  ->  B  e.  _V )
11 fex2 5417 . . 3  |-  ( ( F : A --> B  /\  A  e.  W  /\  B  e.  _V )  ->  F  e.  _V )
126, 7, 10, 11syl3anc 1182 . 2  |-  ( ph  ->  F  e.  _V )
13 fdm 5409 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
146, 13syl 15 . 2  |-  ( ph  ->  dom  F  =  A )
151, 2, 3, 4, 5, 12, 14tsmsval2 17828 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( J 
fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G 
gsumg  ( F  |`  y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638    e. cmpt 4093   dom cdm 4705   ran crn 4706    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   TopOpenctopn 13342    gsumg cgsu 13417   filGencfg 17535    fLimf cflf 17646   tsums ctsu 17824
This theorem is referenced by:  eltsms  17831  haustsms  17834  tsmscls  17836  tsmsmhm  17844  tsmsadd  17845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tsms 17825
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