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Theorem tsmsval2 18164
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 18161 . . 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 2961 . . . . . . 7  |-  f  e. 
_V
43dmex 5135 . . . . . 6  |-  dom  f  e.  _V
54pwex 4385 . . . . 5  |-  ~P dom  f  e.  _V
65inex1 4347 . . . 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 738 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  ->  w  =  G )
98fveq2d 5735 . . . . . 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 2488 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( TopOpen `  w )  =  J )
12 id 21 . . . . . . 7  |-  ( s  =  ( ~P dom  f  i^i  Fin )  -> 
s  =  ( ~P
dom  f  i^i  Fin ) )
13 simprr 735 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
f  =  F )
1413dmeqd 5075 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  dom  F )
15 tsmsval2.a . . . . . . . . . . . 12  |-  ( ph  ->  dom  F  =  A )
1615adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  F  =  A )
1714, 16eqtrd 2470 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  A
)
1817pweqd 3806 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  ~P dom  f  =  ~P A )
1918ineq1d 3543 . . . . . . . 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 2488 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  =  S )
2212, 21sylan9eqr 2492 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
s  =  S )
23 rabeq 2952 . . . . . . . . . 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 4290 . . . . . . . 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 5100 . . . . . . 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 2488 . . . . . 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 6102 . . . . 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 6102 . . . 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 739 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
f  =  F )
3231reseq1d 5148 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( f  |`  y
)  =  ( F  |`  y ) )
338, 32oveq12d 6102 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( w  gsumg  ( f  |`  y
) )  =  ( G  gsumg  ( F  |`  y
) ) )
3422, 33mpteq12dv 4290 . . . 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 5737 . . 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 3295 . 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 2966 . . 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 2966 . . 3  |-  ( F  e.  W  ->  F  e.  _V )
4240, 41syl 16 . 2  |-  ( ph  ->  F  e.  _V )
43 fvex 5745 . . 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 6204 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 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   [_csb 3253    i^i cin 3321    C_ wss 3322   ~Pcpw 3801    e. cmpt 4269   dom cdm 4881   ran crn 4882    |` cres 4883   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   Fincfn 7112   Basecbs 13474   TopOpenctopn 13654    gsumg cgsu 13729   filGencfg 16695    fLimf cflf 17972   tsums ctsu 18160
This theorem is referenced by:  tsmsval  18165  tsmspropd  18166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tsms 18161
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