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Theorem tsmspropd 17830
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 14414 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 5889 . . 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 5010 . . . . . 6  |-  ( F  e.  V  ->  ( F  |`  y )  e. 
_V )
53, 4syl 15 . . . . 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 14469 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  y
) )  =  ( H  gsumg  ( F  |`  y
) ) )
1110mpteq2dv 4123 . . 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 5547 . 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 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
14 eqid 2296 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
15 eqid 2296 . . 3  |-  ( ~P
dom  F  i^i  Fin )  =  ( ~P dom  F  i^i  Fin )
16 eqid 2296 . . 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 2297 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
1813, 14, 15, 16, 6, 3, 17tsmsval2 17828 . 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 2296 . . 3  |-  ( Base `  H )  =  (
Base `  H )
20 eqid 2296 . . 3  |-  ( TopOpen `  H )  =  (
TopOpen `  H )
2119, 20, 15, 16, 7, 3, 17tsmsval2 17828 . 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 2338 1  |-  ( ph  ->  ( G tsums  F )  =  ( H tsums  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    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   Fincfn 6879   Basecbs 13164   +g cplusg 13224   TopOpenctopn 13342    gsumg cgsu 13417   filGencfg 17535    fLimf cflf 17646   tsums ctsu 17824
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-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-0g 13420  df-gsum 13421  df-tsms 17825
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