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Theorem tsmsmhm 17828
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 16674 . . . 4  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
51, 4sylib 188 . . 3  |-  ( ph  ->  J  e.  (TopOn `  B ) )
6 eqid 2283 . . . . 5  |-  ( ~P A  i^i  Fin )  =  ( ~P A  i^i  Fin )
7 eqid 2283 . . . . 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 2283 . . . . 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 17810 . . . 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 17573 . . . 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 15 . . 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 17811 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  z
) )  e.  B
)
16 eqid 2283 . . . 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 5684 . . 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 17813 . . . 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 2359 . . 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 17817 . . . . . 6  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2322, 18sseldd 3181 . . . . 5  |-  ( ph  ->  X  e.  B )
24 toponuni 16665 . . . . . 6  |-  ( J  e.  (TopOn `  B
)  ->  B  =  U. J )
255, 24syl 15 . . . . 5  |-  ( ph  ->  B  =  U. J
)
2623, 25eleqtrd 2359 . . . 4  |-  ( ph  ->  X  e.  U. J
)
27 eqid 2283 . . . . 5  |-  U. J  =  U. J
2827cncnpi 17007 . . . 4  |-  ( ( C  e.  ( J  Cn  K )  /\  X  e.  U. J )  ->  C  e.  ( ( J  CnP  K
) `  X )
)
2921, 26, 28syl2anc 642 . . 3  |-  ( ph  ->  C  e.  ( ( J  CnP  K ) `
 X ) )
30 flfcnp 17699 . . 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 1190 . 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 2283 . . . 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 16674 . . . . . . 7  |-  ( H  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  H )
) )
3735, 36sylib 188 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  H )
) )
38 cnf2 16979 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  K  e.  (TopOn `  ( Base `  H ) )  /\  C  e.  ( J  Cn  K ) )  ->  C : B --> ( Base `  H ) )
395, 37, 21, 38syl3anc 1182 . . . . 5  |-  ( ph  ->  C : B --> ( Base `  H ) )
40 fco 5398 . . . . 5  |-  ( ( C : B --> ( Base `  H )  /\  F : A --> B )  -> 
( C  o.  F
) : A --> ( Base `  H ) )
4139, 14, 40syl2anc 642 . . . 4  |-  ( ph  ->  ( C  o.  F
) : A --> ( Base `  H ) )
4232, 33, 6, 8, 34, 9, 41tsmsval 17813 . . 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 2284 . . . . . 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 5575 . . . . . 6  |-  ( ph  ->  C  =  ( w  e.  B  |->  ( C `
 w ) ) )
45 fveq2 5525 . . . . . 6  |-  ( w  =  ( G  gsumg  ( F  |`  z ) )  -> 
( C `  w
)  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
4615, 43, 44, 45fmptco 5691 . . . . 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 5177 . . . . . . . 8  |-  ( ( C  o.  F )  |`  z )  =  ( C  o.  ( F  |`  z ) )
4847oveq2i 5869 . . . . . . 7  |-  ( H 
gsumg  ( ( C  o.  F )  |`  z
) )  =  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )
49 eqid 2283 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
5013adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
5134adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e. CMnd )
52 cmnmnd 15104 . . . . . . . . 9  |-  ( H  e. CMnd  ->  H  e.  Mnd )
5351, 52syl 15 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e.  Mnd )
54 elfpw 7157 . . . . . . . . . 10  |-  ( z  e.  ( ~P A  i^i  Fin )  <->  ( z  C_  A  /\  z  e. 
Fin ) )
5554simprbi 450 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  e.  Fin )
5655adantl 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  z  e.  Fin )
57 tsmsmhm.5 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( G MndHom  H ) )
5857adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  C  e.  ( G MndHom  H ) )
5954simplbi 446 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  C_  A )
60 fssres 5408 . . . . . . . . 9  |-  ( ( F : A --> B  /\  z  C_  A )  -> 
( F  |`  z
) : z --> B )
6114, 59, 60syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) : z --> B )
6256, 61fisuppfi 14450 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( F  |`  z ) " ( _V  \  { ( 0g
`  G ) } ) )  e.  Fin )
632, 49, 50, 53, 56, 58, 61, 62gsummhm 15211 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
6448, 63syl5eq 2327 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( ( C  o.  F )  |`  z
) )  =  ( C `  ( G 
gsumg  ( F  |`  z ) ) ) )
6564mpteq2dva 4106 . . . . 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
) ) ) ) )
6646, 65eqtr4d 2318 . . . 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 ) ) ) )
6766fveq2d 5529 . . 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 ) ) ) ) )
6842, 67eqtr4d 2318 . 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 ) ) ) ) ) )
6931, 68eleqtrrd 2360 1  |-  ( ph  ->  ( C `  X
)  e.  ( H tsums 
( C  o.  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827    e. cmpt 4077   ran crn 4690    |` cres 4691    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   TopOpenctopn 13326   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   MndHom cmhm 14413  CMndccmn 15089  TopOnctopon 16632   TopSpctps 16634    Cn ccn 16954    CnP ccnp 16955   fBascfbas 17518   filGencfg 17519   Filcfil 17540    fLimf cflf 17630   tsums ctsu 17808
This theorem is referenced by:  tsmsinv  17830  esumcocn  23448
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-0g 13404  df-gsum 13405  df-mnd 14367  df-mhm 14415  df-cntz 14793  df-cmn 15091  df-top 16636  df-topon 16639  df-topsp 16640  df-ntr 16757  df-nei 16835  df-cn 16957  df-cnp 16958  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-tsms 17809
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