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Theorem tsmspropd 17814
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 14398 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 5873 . . 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 4994 . . . . . 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 14453 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  y
) )  =  ( H  gsumg  ( F  |`  y
) ) )
1110mpteq2dv 4107 . . 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 5531 . 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 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
14 eqid 2283 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
15 eqid 2283 . . 3  |-  ( ~P
dom  F  i^i  Fin )  =  ( ~P dom  F  i^i  Fin )
16 eqid 2283 . . 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 2284 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
1813, 14, 15, 16, 6, 3, 17tsmsval2 17812 . 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 2283 . . 3  |-  ( Base `  H )  =  (
Base `  H )
20 eqid 2283 . . 3  |-  ( TopOpen `  H )  =  (
TopOpen `  H )
2119, 20, 15, 16, 7, 3, 17tsmsval2 17812 . 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 2325 1  |-  ( ph  ->  ( G tsums  F )  =  ( H tsums  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   +g cplusg 13208   TopOpenctopn 13326    gsumg cgsu 13401   filGencfg 17519    fLimf cflf 17630   tsums ctsu 17808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-0g 13404  df-gsum 13405  df-tsms 17809
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