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Theorem tsmsval2 17828
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 17825 . . 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 2804 . . . . . . 7  |-  f  e. 
_V
43dmex 4957 . . . . . 6  |-  dom  f  e.  _V
54pwex 4209 . . . . 5  |-  ~P dom  f  e.  _V
65inex1 4171 . . . 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 5545 . . . . . 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 2346 . . . . 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 4897 . . . . . . . . . . 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 2328 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  A
)
1817pweqd 3643 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  ~P dom  f  =  ~P A )
1918ineq1d 3382 . . . . . . . 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 2346 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  =  S )
2212, 21sylan9eqr 2350 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
s  =  S )
23 rabeq 2795 . . . . . . . . . 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 4114 . . . . . . . 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 4922 . . . . . . 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 2346 . . . . . 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 5892 . . . . 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 5892 . . . 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 4970 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( f  |`  y
)  =  ( F  |`  y ) )
338, 32oveq12d 5892 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( w  gsumg  ( f  |`  y
) )  =  ( G  gsumg  ( F  |`  y
) ) )
3422, 33mpteq12dv 4114 . . . 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 5547 . . 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 3136 . 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 2809 . . 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 2809 . . 3  |-  ( F  e.  W  ->  F  e.  _V )
4240, 41syl 15 . 2  |-  ( ph  ->  F  e.  _V )
43 fvex 5555 . . 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 5991 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   [_csb 3094    i^i cin 3164    C_ wss 3165   ~Pcpw 3638    e. cmpt 4093   dom cdm 4705   ran crn 4706    |` cres 4707   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Fincfn 6879   Basecbs 13164   TopOpenctopn 13342    gsumg cgsu 13417   filGencfg 17535    fLimf cflf 17646   tsums ctsu 17824
This theorem is referenced by:  tsmsval  17829  tsmspropd  17830
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-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tsms 17825
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