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Theorem tsmspropd 20755
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 16072 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 6311 . . 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 5326 . . . . . 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 16025 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  y
) )  =  ( H  gsumg  ( F  |`  y
) ) )
1110mpteq2dv 4544 . . 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 5878 . 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 2457 . . 3  |-  ( Base `  G )  =  (
Base `  G )
14 eqid 2457 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
15 eqid 2457 . . 3  |-  ( ~P
dom  F  i^i  Fin )  =  ( ~P dom  F  i^i  Fin )
16 eqid 2457 . . 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 2458 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
1813, 14, 15, 16, 6, 3, 17tsmsval2 20753 . 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 2457 . . 3  |-  ( Base `  H )  =  (
Base `  H )
20 eqid 2457 . . 3  |-  ( TopOpen `  H )  =  (
TopOpen `  H )
2119, 20, 15, 16, 7, 3, 17tsmsval2 20753 . 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 2508 1  |-  ( ph  ->  ( G tsums  F )  =  ( H tsums  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    i^i cin 3470    C_ wss 3471   ~Pcpw 4015    |-> cmpt 4515   dom cdm 5008   ran crn 5009    |` cres 5010   ` cfv 5594  (class class class)co 6296   Fincfn 7535   Basecbs 14643   +g cplusg 14711   TopOpenctopn 14838    gsumg cgsu 14857   filGencfg 18533    fLimf cflf 20561   tsums ctsu 20749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-recs 7060  df-rdg 7094  df-seq 12110  df-0g 14858  df-gsum 14859  df-tsms 20750
This theorem is referenced by: (None)
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