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Theorem tsmsmhm 20383
Description: Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
tsmsmhm.b  |-  B  =  ( Base `  G
)
tsmsmhm.j  |-  J  =  ( TopOpen `  G )
tsmsmhm.k  |-  K  =  ( TopOpen `  H )
tsmsmhm.1  |-  ( ph  ->  G  e. CMnd )
tsmsmhm.2  |-  ( ph  ->  G  e.  TopSp )
tsmsmhm.3  |-  ( ph  ->  H  e. CMnd )
tsmsmhm.4  |-  ( ph  ->  H  e.  TopSp )
tsmsmhm.5  |-  ( ph  ->  C  e.  ( G MndHom  H ) )
tsmsmhm.6  |-  ( ph  ->  C  e.  ( J  Cn  K ) )
tsmsmhm.a  |-  ( ph  ->  A  e.  V )
tsmsmhm.f  |-  ( ph  ->  F : A --> B )
tsmsmhm.x  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
Assertion
Ref Expression
tsmsmhm  |-  ( ph  ->  ( C `  X
)  e.  ( H tsums 
( C  o.  F
) ) )

Proof of Theorem tsmsmhm
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmsmhm.2 . . . 4  |-  ( ph  ->  G  e.  TopSp )
2 tsmsmhm.b . . . . 5  |-  B  =  ( Base `  G
)
3 tsmsmhm.j . . . . 5  |-  J  =  ( TopOpen `  G )
42, 3istps 19204 . . . 4  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
51, 4sylib 196 . . 3  |-  ( ph  ->  J  e.  (TopOn `  B ) )
6 eqid 2467 . . . . 5  |-  ( ~P A  i^i  Fin )  =  ( ~P A  i^i  Fin )
7 eqid 2467 . . . . 5  |-  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y 
C_  z } )  =  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )
8 eqid 2467 . . . . 5  |-  ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )  =  ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )
9 tsmsmhm.a . . . . 5  |-  ( ph  ->  A  e.  V )
106, 7, 8, 9tsmsfbas 20361 . . . 4  |-  ( ph  ->  ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )  e.  (
fBas `  ( ~P A  i^i  Fin ) ) )
11 fgcl 20114 . . . 4  |-  ( ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } )  e.  (
fBas `  ( ~P A  i^i  Fin ) )  ->  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) )  e.  ( Fil `  ( ~P A  i^i  Fin )
) )
1210, 11syl 16 . . 3  |-  ( ph  ->  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) )  e.  ( Fil `  ( ~P A  i^i  Fin )
) )
13 tsmsmhm.1 . . . . 5  |-  ( ph  ->  G  e. CMnd )
14 tsmsmhm.f . . . . 5  |-  ( ph  ->  F : A --> B )
152, 6, 13, 9, 14tsmslem1 20362 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  z
) )  e.  B
)
16 eqid 2467 . . . 4  |-  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G 
gsumg  ( F  |`  z ) ) )  =  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z
) ) )
1715, 16fmptd 6043 . . 3  |-  ( ph  ->  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) : ( ~P A  i^i  Fin ) --> B )
18 tsmsmhm.x . . . 4  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
192, 3, 6, 8, 1, 9, 14tsmsval 20364 . . . 4  |-  ( ph  ->  ( G tsums  F )  =  ( ( J 
fLimf  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) )
2018, 19eleqtrd 2557 . . 3  |-  ( ph  ->  X  e.  ( ( J  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) )
21 tsmsmhm.6 . . . 4  |-  ( ph  ->  C  e.  ( J  Cn  K ) )
222, 13, 1, 9, 14tsmscl 20368 . . . . . 6  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2322, 18sseldd 3505 . . . . 5  |-  ( ph  ->  X  e.  B )
24 toponuni 19195 . . . . . 6  |-  ( J  e.  (TopOn `  B
)  ->  B  =  U. J )
255, 24syl 16 . . . . 5  |-  ( ph  ->  B  =  U. J
)
2623, 25eleqtrd 2557 . . . 4  |-  ( ph  ->  X  e.  U. J
)
27 eqid 2467 . . . . 5  |-  U. J  =  U. J
2827cncnpi 19545 . . . 4  |-  ( ( C  e.  ( J  Cn  K )  /\  X  e.  U. J )  ->  C  e.  ( ( J  CnP  K
) `  X )
)
2921, 26, 28syl2anc 661 . . 3  |-  ( ph  ->  C  e.  ( ( J  CnP  K ) `
 X ) )
30 flfcnp 20240 . . 3  |-  ( ( ( J  e.  (TopOn `  B )  /\  (
( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) )  e.  ( Fil `  ( ~P A  i^i  Fin )
)  /\  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) : ( ~P A  i^i  Fin ) --> B )  /\  ( X  e.  ( ( J  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) )  /\  C  e.  ( ( J  CnP  K ) `  X ) ) )  ->  ( C `  X )  e.  ( ( K  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( C  o.  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) ) )
315, 12, 17, 20, 29, 30syl32anc 1236 . 2  |-  ( ph  ->  ( C `  X
)  e.  ( ( K  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( C  o.  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) ) )
32 eqid 2467 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
33 tsmsmhm.k . . . 4  |-  K  =  ( TopOpen `  H )
34 tsmsmhm.3 . . . 4  |-  ( ph  ->  H  e. CMnd )
35 tsmsmhm.4 . . . . . . 7  |-  ( ph  ->  H  e.  TopSp )
3632, 33istps 19204 . . . . . . 7  |-  ( H  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  H )
) )
3735, 36sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  H )
) )
38 cnf2 19516 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  K  e.  (TopOn `  ( Base `  H ) )  /\  C  e.  ( J  Cn  K ) )  ->  C : B --> ( Base `  H ) )
395, 37, 21, 38syl3anc 1228 . . . . 5  |-  ( ph  ->  C : B --> ( Base `  H ) )
40 fco 5739 . . . . 5  |-  ( ( C : B --> ( Base `  H )  /\  F : A --> B )  -> 
( C  o.  F
) : A --> ( Base `  H ) )
4139, 14, 40syl2anc 661 . . . 4  |-  ( ph  ->  ( C  o.  F
) : A --> ( Base `  H ) )
4232, 33, 6, 8, 34, 9, 41tsmsval 20364 . . 3  |-  ( ph  ->  ( H tsums  ( C  o.  F ) )  =  ( ( K 
fLimf  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( H  gsumg  ( ( C  o.  F )  |`  z ) ) ) ) )
43 eqidd 2468 . . . . . 6  |-  ( ph  ->  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) )  =  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) )
4439feqmptd 5918 . . . . . 6  |-  ( ph  ->  C  =  ( w  e.  B  |->  ( C `
 w ) ) )
45 fveq2 5864 . . . . . 6  |-  ( w  =  ( G  gsumg  ( F  |`  z ) )  -> 
( C `  w
)  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
4615, 43, 44, 45fmptco 6052 . . . . 5  |-  ( ph  ->  ( C  o.  (
z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z
) ) ) )  =  ( z  e.  ( ~P A  i^i  Fin )  |->  ( C `  ( G  gsumg  ( F  |`  z
) ) ) ) )
47 resco 5509 . . . . . . . 8  |-  ( ( C  o.  F )  |`  z )  =  ( C  o.  ( F  |`  z ) )
4847oveq2i 6293 . . . . . . 7  |-  ( H 
gsumg  ( ( C  o.  F )  |`  z
) )  =  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )
49 eqid 2467 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
5013adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
5134adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e. CMnd )
52 cmnmnd 16609 . . . . . . . . 9  |-  ( H  e. CMnd  ->  H  e.  Mnd )
5351, 52syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e.  Mnd )
54 elfpw 7818 . . . . . . . . . 10  |-  ( z  e.  ( ~P A  i^i  Fin )  <->  ( z  C_  A  /\  z  e. 
Fin ) )
5554simprbi 464 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  e.  Fin )
5655adantl 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  z  e.  Fin )
57 tsmsmhm.5 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( G MndHom  H ) )
5857adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  C  e.  ( G MndHom  H ) )
5954simplbi 460 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  C_  A )
60 fssres 5749 . . . . . . . . 9  |-  ( ( F : A --> B  /\  z  C_  A )  -> 
( F  |`  z
) : z --> B )
6114, 59, 60syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) : z --> B )
62 fvex 5874 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
6362a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( 0g `  G )  e. 
_V )
6461, 56, 63fdmfifsupp 7835 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) finSupp  ( 0g `  G ) )
652, 49, 50, 53, 56, 58, 61, 64gsummhm 16750 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
6648, 65syl5eq 2520 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( ( C  o.  F )  |`  z
) )  =  ( C `  ( G 
gsumg  ( F  |`  z ) ) ) )
6766mpteq2dva 4533 . . . . 5  |-  ( ph  ->  ( z  e.  ( ~P A  i^i  Fin )  |->  ( H  gsumg  ( ( C  o.  F )  |`  z ) ) )  =  ( z  e.  ( ~P A  i^i  Fin )  |->  ( C `  ( G  gsumg  ( F  |`  z
) ) ) ) )
6846, 67eqtr4d 2511 . . . 4  |-  ( ph  ->  ( C  o.  (
z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z
) ) ) )  =  ( z  e.  ( ~P A  i^i  Fin )  |->  ( H  gsumg  ( ( C  o.  F )  |`  z ) ) ) )
6968fveq2d 5868 . . 3  |-  ( ph  ->  ( ( K  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( C  o.  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) )  =  ( ( K  fLimf  ( ( ~P A  i^i  Fin ) filGen ran  ( y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( z  e.  ( ~P A  i^i  Fin )  |->  ( H  gsumg  ( ( C  o.  F )  |`  z ) ) ) ) )
7042, 69eqtr4d 2511 . 2  |-  ( ph  ->  ( H tsums  ( C  o.  F ) )  =  ( ( K 
fLimf  ( ( ~P A  i^i  Fin ) filGen ran  (
y  e.  ( ~P A  i^i  Fin )  |->  { z  e.  ( ~P A  i^i  Fin )  |  y  C_  z } ) ) ) `
 ( C  o.  ( z  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  z ) ) ) ) ) )
7131, 70eleqtrrd 2558 1  |-  ( ph  ->  ( C `  X
)  e.  ( H tsums 
( C  o.  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    |-> cmpt 4505   ran crn 5000    |` cres 5001    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282   Fincfn 7513   Basecbs 14486   TopOpenctopn 14673   0gc0g 14691    gsumg cgsu 14692   Mndcmnd 15722   MndHom cmhm 15775  CMndccmn 16594   fBascfbas 18177   filGencfg 18178  TopOnctopon 19162   TopSpctps 19164    Cn ccn 19491    CnP ccnp 19492   Filcfil 20081    fLimf cflf 20171   tsums ctsu 20359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-0g 14693  df-gsum 14694  df-mnd 15728  df-mhm 15777  df-cntz 16150  df-cmn 16596  df-fbas 18187  df-fg 18188  df-top 19166  df-topon 19169  df-topsp 19170  df-ntr 19287  df-nei 19365  df-cn 19494  df-cnp 19495  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-tsms 20360
This theorem is referenced by:  tsmsinv  20385  esumcocn  27726
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