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Theorem tsmspropd 19700
Description: The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 15444 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
tsmspropd.f  |-  ( ph  ->  F  e.  V )
tsmspropd.g  |-  ( ph  ->  G  e.  W )
tsmspropd.h  |-  ( ph  ->  H  e.  X )
tsmspropd.b  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
tsmspropd.p  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
tsmspropd.j  |-  ( ph  ->  ( TopOpen `  G )  =  ( TopOpen `  H
) )
Assertion
Ref Expression
tsmspropd  |-  ( ph  ->  ( G tsums  F )  =  ( H tsums  F
) )

Proof of Theorem tsmspropd
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmspropd.j . . . 4  |-  ( ph  ->  ( TopOpen `  G )  =  ( TopOpen `  H
) )
21oveq1d 6104 . . 3  |-  ( ph  ->  ( ( TopOpen `  G
)  fLimf  ( ( ~P
dom  F  i^i  Fin ) filGen ran  ( z  e.  ( ~P dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z  C_  y }
) ) )  =  ( ( TopOpen `  H
)  fLimf  ( ( ~P
dom  F  i^i  Fin ) filGen ran  ( z  e.  ( ~P dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z  C_  y }
) ) ) )
3 tsmspropd.f . . . . . 6  |-  ( ph  ->  F  e.  V )
4 resexg 5147 . . . . . 6  |-  ( F  e.  V  ->  ( F  |`  y )  e. 
_V )
53, 4syl 16 . . . . 5  |-  ( ph  ->  ( F  |`  y
)  e.  _V )
6 tsmspropd.g . . . . 5  |-  ( ph  ->  G  e.  W )
7 tsmspropd.h . . . . 5  |-  ( ph  ->  H  e.  X )
8 tsmspropd.b . . . . 5  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
9 tsmspropd.p . . . . 5  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
105, 6, 7, 8, 9gsumpropd 15502 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  y
) )  =  ( H  gsumg  ( F  |`  y
) ) )
1110mpteq2dv 4377 . . 3  |-  ( ph  ->  ( y  e.  ( ~P dom  F  i^i  Fin )  |->  ( G  gsumg  ( F  |`  y ) ) )  =  ( y  e.  ( ~P dom  F  i^i  Fin )  |->  ( H 
gsumg  ( F  |`  y ) ) ) )
122, 11fveq12d 5695 . 2  |-  ( ph  ->  ( ( ( TopOpen `  G )  fLimf  ( ( ~P dom  F  i^i  Fin ) filGen ran  ( z  e.  ( ~P dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z  C_  y }
) ) ) `  ( y  e.  ( ~P dom  F  i^i  Fin )  |->  ( G  gsumg  ( F  |`  y ) ) ) )  =  ( ( ( TopOpen `  H )  fLimf  ( ( ~P dom  F  i^i  Fin ) filGen ran  ( z  e.  ( ~P dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z 
C_  y } ) ) ) `  (
y  e.  ( ~P
dom  F  i^i  Fin )  |->  ( H  gsumg  ( F  |`  y
) ) ) ) )
13 eqid 2441 . . 3  |-  ( Base `  G )  =  (
Base `  G )
14 eqid 2441 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
15 eqid 2441 . . 3  |-  ( ~P
dom  F  i^i  Fin )  =  ( ~P dom  F  i^i  Fin )
16 eqid 2441 . . 3  |-  ran  (
z  e.  ( ~P
dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z  C_  y } )  =  ran  ( z  e.  ( ~P dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z 
C_  y } )
17 eqidd 2442 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
1813, 14, 15, 16, 6, 3, 17tsmsval2 19698 . 2  |-  ( ph  ->  ( G tsums  F )  =  ( ( (
TopOpen `  G )  fLimf  ( ( ~P dom  F  i^i  Fin ) filGen ran  (
z  e.  ( ~P
dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z  C_  y } ) ) ) `
 ( y  e.  ( ~P dom  F  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  y ) ) ) ) )
19 eqid 2441 . . 3  |-  ( Base `  H )  =  (
Base `  H )
20 eqid 2441 . . 3  |-  ( TopOpen `  H )  =  (
TopOpen `  H )
2119, 20, 15, 16, 7, 3, 17tsmsval2 19698 . 2  |-  ( ph  ->  ( H tsums  F )  =  ( ( (
TopOpen `  H )  fLimf  ( ( ~P dom  F  i^i  Fin ) filGen ran  (
z  e.  ( ~P
dom  F  i^i  Fin )  |->  { y  e.  ( ~P dom  F  i^i  Fin )  |  z  C_  y } ) ) ) `
 ( y  e.  ( ~P dom  F  i^i  Fin )  |->  ( H 
gsumg  ( F  |`  y ) ) ) ) )
2212, 18, 213eqtr4d 2483 1  |-  ( ph  ->  ( G tsums  F )  =  ( H tsums  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2717   _Vcvv 2970    i^i cin 3325    C_ wss 3326   ~Pcpw 3858    e. cmpt 4348   dom cdm 4838   ran crn 4839    |` cres 4840   ` cfv 5416  (class class class)co 6089   Fincfn 7308   Basecbs 14172   +g cplusg 14236   TopOpenctopn 14358    gsumg cgsu 14377   filGencfg 17803    fLimf cflf 19506   tsums ctsu 19694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-recs 6830  df-rdg 6864  df-seq 11805  df-0g 14378  df-gsum 14379  df-tsms 19695
This theorem is referenced by: (None)
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