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Theorem tsmspropd 18162
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 14722 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 6097 . . 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 5186 . . . . . 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 14777 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  y
) )  =  ( H  gsumg  ( F  |`  y
) ) )
1110mpteq2dv 4297 . . 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 5735 . 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 2437 . . 3  |-  ( Base `  G )  =  (
Base `  G )
14 eqid 2437 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
15 eqid 2437 . . 3  |-  ( ~P
dom  F  i^i  Fin )  =  ( ~P dom  F  i^i  Fin )
16 eqid 2437 . . 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 2438 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
1813, 14, 15, 16, 6, 3, 17tsmsval2 18160 . 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 2437 . . 3  |-  ( Base `  H )  =  (
Base `  H )
20 eqid 2437 . . 3  |-  ( TopOpen `  H )  =  (
TopOpen `  H )
2119, 20, 15, 16, 7, 3, 17tsmsval2 18160 . 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 2479 1  |-  ( ph  ->  ( G tsums  F )  =  ( H tsums  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957    i^i cin 3320    C_ wss 3321   ~Pcpw 3800    e. cmpt 4267   dom cdm 4879   ran crn 4880    |` cres 4881   ` cfv 5455  (class class class)co 6082   Fincfn 7110   Basecbs 13470   +g cplusg 13530   TopOpenctopn 13650    gsumg cgsu 13725   filGencfg 16691    fLimf cflf 17968   tsums ctsu 18156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-recs 6634  df-rdg 6669  df-seq 11325  df-0g 13728  df-gsum 13729  df-tsms 18157
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