MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsmsval2 Unicode version

Theorem tsmsval2 18120
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 18117 . . 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 2927 . . . . . . 7  |-  f  e. 
_V
43dmex 5099 . . . . . 6  |-  dom  f  e.  _V
54pwex 4350 . . . . 5  |-  ~P dom  f  e.  _V
65inex1 4312 . . . 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 737 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  ->  w  =  G )
98fveq2d 5699 . . . . . 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 2462 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( TopOpen `  w )  =  J )
12 id 20 . . . . . . 7  |-  ( s  =  ( ~P dom  f  i^i  Fin )  -> 
s  =  ( ~P
dom  f  i^i  Fin ) )
13 simprr 734 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
f  =  F )
1413dmeqd 5039 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  dom  F )
15 tsmsval2.a . . . . . . . . . . . 12  |-  ( ph  ->  dom  F  =  A )
1615adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  F  =  A )
1714, 16eqtrd 2444 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  dom  f  =  A
)
1817pweqd 3772 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  ->  ~P dom  f  =  ~P A )
1918ineq1d 3509 . . . . . . . 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 2462 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  f  =  F ) )  -> 
( ~P dom  f  i^i  Fin )  =  S )
2212, 21sylan9eqr 2466 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
s  =  S )
23 rabeq 2918 . . . . . . . . . 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 4255 . . . . . . . 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 5064 . . . . . . 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 2462 . . . . . 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 6066 . . . . 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 6066 . . . 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 738 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
f  =  F )
3231reseq1d 5112 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( f  |`  y
)  =  ( F  |`  y ) )
338, 32oveq12d 6066 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  f  =  F )
)  /\  s  =  ( ~P dom  f  i^i 
Fin ) )  -> 
( w  gsumg  ( f  |`  y
) )  =  ( G  gsumg  ( F  |`  y
) ) )
3422, 33mpteq12dv 4255 . . . 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 5701 . . 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 3261 . 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 2932 . . 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 2932 . . 3  |-  ( F  e.  W  ->  F  e.  _V )
4240, 41syl 16 . 2  |-  ( ph  ->  F  e.  _V )
43 fvex 5709 . . 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 6168 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 359    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924   [_csb 3219    i^i cin 3287    C_ wss 3288   ~Pcpw 3767    e. cmpt 4234   dom cdm 4845   ran crn 4846    |` cres 4847   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   Fincfn 7076   Basecbs 13432   TopOpenctopn 13612    gsumg cgsu 13687   filGencfg 16653    fLimf cflf 17928   tsums ctsu 18116
This theorem is referenced by:  tsmsval  18121  tsmspropd  18122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-tsms 18117
  Copyright terms: Public domain W3C validator