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Theorem tsmsmhm 18167
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 16993 . . . 4  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
51, 4sylib 189 . . 3  |-  ( ph  ->  J  e.  (TopOn `  B ) )
6 eqid 2435 . . . . 5  |-  ( ~P A  i^i  Fin )  =  ( ~P A  i^i  Fin )
7 eqid 2435 . . . . 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 2435 . . . . 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 18149 . . . 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 17902 . . . 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 18150 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  z
) )  e.  B
)
16 eqid 2435 . . . 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 5885 . . 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 18152 . . . 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 2511 . . 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 18156 . . . . . 6  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2322, 18sseldd 3341 . . . . 5  |-  ( ph  ->  X  e.  B )
24 toponuni 16984 . . . . . 6  |-  ( J  e.  (TopOn `  B
)  ->  B  =  U. J )
255, 24syl 16 . . . . 5  |-  ( ph  ->  B  =  U. J
)
2623, 25eleqtrd 2511 . . . 4  |-  ( ph  ->  X  e.  U. J
)
27 eqid 2435 . . . . 5  |-  U. J  =  U. J
2827cncnpi 17334 . . . 4  |-  ( ( C  e.  ( J  Cn  K )  /\  X  e.  U. J )  ->  C  e.  ( ( J  CnP  K
) `  X )
)
2921, 26, 28syl2anc 643 . . 3  |-  ( ph  ->  C  e.  ( ( J  CnP  K ) `
 X ) )
30 flfcnp 18028 . . 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 1192 . 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 2435 . . . 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 16993 . . . . . . 7  |-  ( H  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  H )
) )
3735, 36sylib 189 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  H )
) )
38 cnf2 17305 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  K  e.  (TopOn `  ( Base `  H ) )  /\  C  e.  ( J  Cn  K ) )  ->  C : B --> ( Base `  H ) )
395, 37, 21, 38syl3anc 1184 . . . . 5  |-  ( ph  ->  C : B --> ( Base `  H ) )
40 fco 5592 . . . . 5  |-  ( ( C : B --> ( Base `  H )  /\  F : A --> B )  -> 
( C  o.  F
) : A --> ( Base `  H ) )
4139, 14, 40syl2anc 643 . . . 4  |-  ( ph  ->  ( C  o.  F
) : A --> ( Base `  H ) )
4232, 33, 6, 8, 34, 9, 41tsmsval 18152 . . 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 2436 . . . . . 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 5771 . . . . . 6  |-  ( ph  ->  C  =  ( w  e.  B  |->  ( C `
 w ) ) )
45 fveq2 5720 . . . . . 6  |-  ( w  =  ( G  gsumg  ( F  |`  z ) )  -> 
( C `  w
)  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
4615, 43, 44, 45fmptco 5893 . . . . 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 5366 . . . . . . . 8  |-  ( ( C  o.  F )  |`  z )  =  ( C  o.  ( F  |`  z ) )
4847oveq2i 6084 . . . . . . 7  |-  ( H 
gsumg  ( ( C  o.  F )  |`  z
) )  =  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )
49 eqid 2435 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
5013adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  G  e. CMnd )
5134adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e. CMnd )
52 cmnmnd 15419 . . . . . . . . 9  |-  ( H  e. CMnd  ->  H  e.  Mnd )
5351, 52syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e.  Mnd )
54 elfpw 7400 . . . . . . . . . 10  |-  ( z  e.  ( ~P A  i^i  Fin )  <->  ( z  C_  A  /\  z  e. 
Fin ) )
5554simprbi 451 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  e.  Fin )
5655adantl 453 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  z  e.  Fin )
57 tsmsmhm.5 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( G MndHom  H ) )
5857adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  C  e.  ( G MndHom  H ) )
5954simplbi 447 . . . . . . . . 9  |-  ( z  e.  ( ~P A  i^i  Fin )  ->  z  C_  A )
60 fssres 5602 . . . . . . . . 9  |-  ( ( F : A --> B  /\  z  C_  A )  -> 
( F  |`  z
) : z --> B )
6114, 59, 60syl2an 464 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) : z --> B )
6256, 61fisuppfi 14765 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( F  |`  z ) " ( _V  \  { ( 0g
`  G ) } ) )  e.  Fin )
632, 49, 50, 53, 56, 58, 61, 62gsummhm 15526 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
6448, 63syl5eq 2479 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( ( C  o.  F )  |`  z
) )  =  ( C `  ( G 
gsumg  ( F  |`  z ) ) ) )
6564mpteq2dva 4287 . . . . 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 2470 . . . 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 5724 . . 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 2470 . 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 2512 1  |-  ( ph  ->  ( C `  X
)  e.  ( H tsums 
( C  o.  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007    e. cmpt 4258   ran crn 4871    |` cres 4872    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   Basecbs 13461   TopOpenctopn 13641   0gc0g 13715    gsumg cgsu 13716   Mndcmnd 14676   MndHom cmhm 14728  CMndccmn 15404   fBascfbas 16681   filGencfg 16682  TopOnctopon 16951   TopSpctps 16953    Cn ccn 17280    CnP ccnp 17281   Filcfil 17869    fLimf cflf 17959   tsums ctsu 18147
This theorem is referenced by:  tsmsinv  18169  esumcocn  24462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-0g 13719  df-gsum 13720  df-mnd 14682  df-mhm 14730  df-cntz 15108  df-cmn 15406  df-fbas 16691  df-fg 16692  df-top 16955  df-topon 16958  df-topsp 16959  df-ntr 17076  df-nei 17154  df-cn 17283  df-cnp 17284  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-tsms 18148
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