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Theorem tsmsval2 20753
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 )
tsmsval2.f  |-  ( ph  ->  F  e.  W )
tsmsval2.a  |-  ( ph  ->  dom  F  =  A )
Assertion
Ref Expression
tsmsval2  |-  ( 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 tsmsval2
Dummy variables  f 
s  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-tsms 20750 . . 3  |- tsums  =  ( w  e.  _V , 
f  e.  _V  |->  [_ ( ~P dom  f  i^i 
Fin )  /  s ]_ ( ( ( TopOpen `  w )  fLimf  ( s
filGen ran  ( z  e.  s  |->  { y  e.  s  |  z  C_  y } ) ) ) `
 ( y  e.  s  |->  ( w  gsumg  ( f  |`  y ) ) ) ) )
21a1i 11 . 2  |-  ( ph  -> tsums  =  ( w  e. 
_V ,  f  e. 
_V  |->  [_ ( ~P dom  f  i^i  Fin )  / 
s ]_ ( ( (
TopOpen `  w )  fLimf  ( s filGen ran  ( z  e.  s  |->  { y  e.  s  |  z 
C_  y } ) ) ) `  (
y  e.  s  |->  ( w  gsumg  ( f  |`  y
) ) ) ) ) )
3 vex 3112 . . . . . . 7  |-  f  e. 
_V
43dmex 6732 . . . . . 6  |-  dom  f  e.  _V
54pwex 4639 . . . . 5  |-  ~P dom  f  e.  _V
65inex1 4597 . . . 4  |-  ( ~P
dom  f  i^i  Fin )  e.  _V
76a1i 11 . . 3  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  e.  _V )
8 simplrl 761 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  ->  w  =  G )
98fveq2d 5876 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( TopOpen `  w )  =  ( TopOpen `  G
) )
10 tsmsval.j . . . . . 6  |-  J  =  ( TopOpen `  G )
119, 10syl6eqr 2516 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( TopOpen `  w )  =  J )
12 id 22 . . . . . . 7  |-  ( s  =  ( ~P dom  f  i^i  Fin )  -> 
s  =  ( ~P
dom  f  i^i  Fin ) )
13 simprr 757 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
f  =  F )
1413dmeqd 5215 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  dom  F )
15 tsmsval2.a . . . . . . . . . . . 12  |-  ( ph  ->  dom  F  =  A )
1615adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  F  =  A )
1714, 16eqtrd 2498 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  A
)
1817pweqd 4020 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  ~P dom  f  =  ~P A )
1918ineq1d 3695 . . . . . . . 8  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  =  ( ~P A  i^i  Fin ) )
20 tsmsval.s . . . . . . . 8  |-  S  =  ( ~P A  i^i  Fin )
2119, 20syl6eqr 2516 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  =  S )
2212, 21sylan9eqr 2520 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
s  =  S )
23 rabeq 3103 . . . . . . . . . 10  |-  ( s  =  S  ->  { y  e.  s  |  z 
C_  y }  =  { y  e.  S  |  z  C_  y } )
2422, 23syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  ->  { y  e.  s  |  z  C_  y }  =  { y  e.  S  |  z  C_  y } )
2522, 24mpteq12dv 4535 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( z  e.  s 
|->  { y  e.  s  |  z  C_  y } )  =  ( z  e.  S  |->  { y  e.  S  | 
z  C_  y }
) )
2625rneqd 5240 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  ->  ran  ( z  e.  s 
|->  { y  e.  s  |  z  C_  y } )  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } ) )
27 tsmsval.l . . . . . . 7  |-  L  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )
2826, 27syl6eqr 2516 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  ->  ran  ( z  e.  s 
|->  { y  e.  s  |  z  C_  y } )  =  L )
2922, 28oveq12d 6314 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( s filGen ran  (
z  e.  s  |->  { y  e.  s  |  z  C_  y }
) )  =  ( S filGen L ) )
3011, 29oveq12d 6314 . . . 4  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( ( TopOpen `  w
)  fLimf  ( s filGen ran  ( z  e.  s 
|->  { y  e.  s  |  z  C_  y } ) ) )  =  ( J  fLimf  ( S filGen L ) ) )
31 simplrr 762 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
f  =  F )
3231reseq1d 5282 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( f  |`  y
)  =  ( F  |`  y ) )
338, 32oveq12d 6314 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( w  gsumg  ( f  |`  y
) )  =  ( G  gsumg  ( F  |`  y
) ) )
3422, 33mpteq12dv 4535 . . . 4  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( y  e.  s 
|->  ( w  gsumg  ( f  |`  y
) ) )  =  ( y  e.  S  |->  ( G  gsumg  ( F  |`  y
) ) ) )
3530, 34fveq12d 5878 . . 3  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( ( ( TopOpen `  w )  fLimf  ( s
filGen ran  ( z  e.  s  |->  { y  e.  s  |  z  C_  y } ) ) ) `
 ( y  e.  s  |->  ( w  gsumg  ( f  |`  y ) ) ) )  =  ( ( J  fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G  gsumg  ( F  |`  y
) ) ) ) )
367, 35csbied 3457 . 2  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  [_ ( ~P dom  f  i^i  Fin )  /  s ]_ ( ( ( TopOpen `  w )  fLimf  ( s
filGen ran  ( z  e.  s  |->  { y  e.  s  |  z  C_  y } ) ) ) `
 ( y  e.  s  |->  ( w  gsumg  ( f  |`  y ) ) ) )  =  ( ( J  fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G  gsumg  ( F  |`  y
) ) ) ) )
37 tsmsval.g . . 3  |-  ( ph  ->  G  e.  V )
38 elex 3118 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3937, 38syl 16 . 2  |-  ( ph  ->  G  e.  _V )
40 tsmsval2.f . . 3  |-  ( ph  ->  F  e.  W )
41 elex 3118 . . 3  |-  ( F  e.  W  ->  F  e.  _V )
4240, 41syl 16 . 2  |-  ( ph  ->  F  e.  _V )
43 fvex 5882 . . 3  |-  ( ( J  fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G  gsumg  ( F  |`  y
) ) ) )  e.  _V
4443a1i 11 . 2  |-  ( ph  ->  ( ( J  fLimf  ( S filGen L ) ) `
 ( y  e.  S  |->  ( G  gsumg  ( F  |`  y ) ) ) )  e.  _V )
452, 36, 39, 42, 44ovmpt2d 6429 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( J 
fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G 
gsumg  ( F  |`  y ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   [_csb 3430    i^i cin 3470    C_ wss 3471   ~Pcpw 4015    |-> cmpt 4515   dom cdm 5008   ran crn 5009    |` cres 5010   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Fincfn 7535   Basecbs 14643   TopOpenctopn 14838    gsumg cgsu 14857   filGencfg 18533    fLimf cflf 20561   tsums ctsu 20749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-tsms 20750
This theorem is referenced by:  tsmsval  20754  tsmspropd  20755
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