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Theorem tsmsmhm 17844
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 16690 . . . 4  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
51, 4sylib 188 . . 3  |-  ( ph  ->  J  e.  (TopOn `  B ) )
6 eqid 2296 . . . . 5  |-  ( ~P A  i^i  Fin )  =  ( ~P A  i^i  Fin )
7 eqid 2296 . . . . 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 2296 . . . . 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 17826 . . . 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 17589 . . . 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 17827 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  z
) )  e.  B
)
16 eqid 2296 . . . 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 5700 . . 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 17829 . . . 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 2372 . . 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 17833 . . . . . 6  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2322, 18sseldd 3194 . . . . 5  |-  ( ph  ->  X  e.  B )
24 toponuni 16681 . . . . . 6  |-  ( J  e.  (TopOn `  B
)  ->  B  =  U. J )
255, 24syl 15 . . . . 5  |-  ( ph  ->  B  =  U. J
)
2623, 25eleqtrd 2372 . . . 4  |-  ( ph  ->  X  e.  U. J
)
27 eqid 2296 . . . . 5  |-  U. J  =  U. J
2827cncnpi 17023 . . . 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 17715 . . 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 2296 . . . 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 16690 . . . . . . 7  |-  ( H  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  H )
) )
3735, 36sylib 188 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  H )
) )
38 cnf2 16995 . . . . . 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 5414 . . . . 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 17829 . . 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 2297 . . . . . 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 5591 . . . . . 6  |-  ( ph  ->  C  =  ( w  e.  B  |->  ( C `
 w ) ) )
45 fveq2 5541 . . . . . 6  |-  ( w  =  ( G  gsumg  ( F  |`  z ) )  -> 
( C `  w
)  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
4615, 43, 44, 45fmptco 5707 . . . . 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 5193 . . . . . . . 8  |-  ( ( C  o.  F )  |`  z )  =  ( C  o.  ( F  |`  z ) )
4847oveq2i 5885 . . . . . . 7  |-  ( H 
gsumg  ( ( C  o.  F )  |`  z
) )  =  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )
49 eqid 2296 . . . . . . . 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 15120 . . . . . . . . 9  |-  ( H  e. CMnd  ->  H  e.  Mnd )
5351, 52syl 15 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e.  Mnd )
54 elfpw 7173 . . . . . . . . . 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 5424 . . . . . . . . 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 14466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( `' ( F  |`  z ) " ( _V  \  { ( 0g
`  G ) } ) )  e.  Fin )
632, 49, 50, 53, 56, 58, 61, 62gsummhm 15227 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
6448, 63syl5eq 2340 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( ( C  o.  F )  |`  z
) )  =  ( C `  ( G 
gsumg  ( F  |`  z ) ) ) )
6564mpteq2dva 4122 . . . . 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 2331 . . . 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 5545 . . 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 2331 . 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 2373 1  |-  ( ph  ->  ( C `  X
)  e.  ( H tsums 
( C  o.  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   {csn 3653   U.cuni 3843    e. cmpt 4093   ran crn 4706    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   TopOpenctopn 13342   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   MndHom cmhm 14429  CMndccmn 15105  TopOnctopon 16648   TopSpctps 16650    Cn ccn 16970    CnP ccnp 16971   fBascfbas 17534   filGencfg 17535   Filcfil 17556    fLimf cflf 17646   tsums ctsu 17824
This theorem is referenced by:  tsmsinv  17846  esumcocn  23463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-0g 13420  df-gsum 13421  df-mnd 14383  df-mhm 14431  df-cntz 14809  df-cmn 15107  df-top 16652  df-topon 16655  df-topsp 16656  df-ntr 16773  df-nei 16851  df-cn 16973  df-cnp 16974  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-tsms 17825
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