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Theorem tsmsval2 17812
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 17809 . . 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 10 . 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 2791 . . . . . . 7  |-  f  e. 
_V
43dmex 4941 . . . . . 6  |-  dom  f  e.  _V
54pwex 4193 . . . . 5  |-  ~P dom  f  e.  _V
65inex1 4155 . . . 4  |-  ( ~P
dom  f  i^i  Fin )  e.  _V
76a1i 10 . . 3  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  e.  _V )
8 simplrl 736 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  ->  w  =  G )
98fveq2d 5529 . . . . . 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 2333 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( TopOpen `  w )  =  J )
12 id 19 . . . . . . 7  |-  ( s  =  ( ~P dom  f  i^i  Fin )  -> 
s  =  ( ~P
dom  f  i^i  Fin ) )
13 simprr 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
f  =  F )
1413dmeqd 4881 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  dom  F )
15 tsmsval2.a . . . . . . . . . . . 12  |-  ( ph  ->  dom  F  =  A )
1615adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  F  =  A )
1714, 16eqtrd 2315 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  A
)
1817pweqd 3630 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  ~P dom  f  =  ~P A )
1918ineq1d 3369 . . . . . . . 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 2333 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  =  S )
2212, 21sylan9eqr 2337 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
s  =  S )
23 rabeq 2782 . . . . . . . . . 10  |-  ( s  =  S  ->  { y  e.  s  |  z 
C_  y }  =  { y  e.  S  |  z  C_  y } )
2422, 23syl 15 . . . . . . . . 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 4098 . . . . . . . 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 4906 . . . . . . 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 2333 . . . . . 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 5876 . . . . 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 5876 . . . 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 737 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
f  =  F )
3231reseq1d 4954 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( f  |`  y
)  =  ( F  |`  y ) )
338, 32oveq12d 5876 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( w  gsumg  ( f  |`  y
) )  =  ( G  gsumg  ( F  |`  y
) ) )
3422, 33mpteq12dv 4098 . . . 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 5531 . . 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 3123 . 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 2796 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3937, 38syl 15 . 2  |-  ( ph  ->  G  e.  _V )
40 tsmsval2.f . . 3  |-  ( ph  ->  F  e.  W )
41 elex 2796 . . 3  |-  ( F  e.  W  ->  F  e.  _V )
4240, 41syl 15 . 2  |-  ( ph  ->  F  e.  _V )
43 fvex 5539 . . 3  |-  ( ( J  fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G  gsumg  ( F  |`  y
) ) ) )  e.  _V
4443a1i 10 . 2  |-  ( ph  ->  ( ( J  fLimf  ( S filGen L ) ) `
 ( y  e.  S  |->  ( G  gsumg  ( F  |`  y ) ) ) )  e.  _V )
452, 36, 39, 42, 44ovmpt2d 5975 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    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [_csb 3081    i^i cin 3151    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Fincfn 6863   Basecbs 13148   TopOpenctopn 13326    gsumg cgsu 13401   filGencfg 17519    fLimf cflf 17630   tsums ctsu 17808
This theorem is referenced by:  tsmsval  17813  tsmspropd  17814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tsms 17809
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