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Theorem tsmsmhm 19847
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 18668 . . . 4  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
51, 4sylib 196 . . 3  |-  ( ph  ->  J  e.  (TopOn `  B ) )
6 eqid 2452 . . . . 5  |-  ( ~P A  i^i  Fin )  =  ( ~P A  i^i  Fin )
7 eqid 2452 . . . . 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 2452 . . . . 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 19825 . . . 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 19578 . . . 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 19826 . . . 4  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( G  gsumg  ( F  |`  z
) )  e.  B
)
16 eqid 2452 . . . 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 5971 . . 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 19828 . . . 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 2542 . . 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 19832 . . . . . 6  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2322, 18sseldd 3460 . . . . 5  |-  ( ph  ->  X  e.  B )
24 toponuni 18659 . . . . . 6  |-  ( J  e.  (TopOn `  B
)  ->  B  =  U. J )
255, 24syl 16 . . . . 5  |-  ( ph  ->  B  =  U. J
)
2623, 25eleqtrd 2542 . . . 4  |-  ( ph  ->  X  e.  U. J
)
27 eqid 2452 . . . . 5  |-  U. J  =  U. J
2827cncnpi 19009 . . . 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 19704 . . 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 1227 . 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 2452 . . . 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 18668 . . . . . . 7  |-  ( H  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  H )
) )
3735, 36sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  H )
) )
38 cnf2 18980 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  K  e.  (TopOn `  ( Base `  H ) )  /\  C  e.  ( J  Cn  K ) )  ->  C : B --> ( Base `  H ) )
395, 37, 21, 38syl3anc 1219 . . . . 5  |-  ( ph  ->  C : B --> ( Base `  H ) )
40 fco 5671 . . . . 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 19828 . . 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 2453 . . . . . 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 5848 . . . . . 6  |-  ( ph  ->  C  =  ( w  e.  B  |->  ( C `
 w ) ) )
45 fveq2 5794 . . . . . 6  |-  ( w  =  ( G  gsumg  ( F  |`  z ) )  -> 
( C `  w
)  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
4615, 43, 44, 45fmptco 5980 . . . . 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 5445 . . . . . . . 8  |-  ( ( C  o.  F )  |`  z )  =  ( C  o.  ( F  |`  z ) )
4847oveq2i 6206 . . . . . . 7  |-  ( H 
gsumg  ( ( C  o.  F )  |`  z
) )  =  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )
49 eqid 2452 . . . . . . . 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 16408 . . . . . . . . 9  |-  ( H  e. CMnd  ->  H  e.  Mnd )
5351, 52syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  H  e.  Mnd )
54 elfpw 7719 . . . . . . . . . 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 5681 . . . . . . . . 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 5804 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
6362a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( 0g `  G )  e. 
_V )
6461, 56, 63fdmfifsupp 7736 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  z ) finSupp  ( 0g `  G ) )
652, 49, 50, 53, 56, 58, 61, 64gsummhm 16549 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( C  o.  ( F  |`  z ) ) )  =  ( C `
 ( G  gsumg  ( F  |`  z ) ) ) )
6648, 65syl5eq 2505 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ~P A  i^i  Fin ) )  ->  ( H  gsumg  ( ( C  o.  F )  |`  z
) )  =  ( C `  ( G 
gsumg  ( F  |`  z ) ) ) )
6766mpteq2dva 4481 . . . . 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 2496 . . . 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 5798 . . 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 2496 . 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 2543 1  |-  ( ph  ->  ( C `  X
)  e.  ( H tsums 
( C  o.  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2800   _Vcvv 3072    i^i cin 3430    C_ wss 3431   ~Pcpw 3963   U.cuni 4194    |-> cmpt 4453   ran crn 4944    |` cres 4945    o. ccom 4947   -->wf 5517   ` cfv 5521  (class class class)co 6195   Fincfn 7415   Basecbs 14287   TopOpenctopn 14474   0gc0g 14492    gsumg cgsu 14493   Mndcmnd 15523   MndHom cmhm 15576  CMndccmn 16393   fBascfbas 17924   filGencfg 17925  TopOnctopon 18626   TopSpctps 18628    Cn ccn 18955    CnP ccnp 18956   Filcfil 19545    fLimf cflf 19635   tsums ctsu 19823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-seq 11919  df-hash 12216  df-0g 14494  df-gsum 14495  df-mnd 15529  df-mhm 15578  df-cntz 15949  df-cmn 16395  df-fbas 17934  df-fg 17935  df-top 18630  df-topon 18633  df-topsp 18634  df-ntr 18751  df-nei 18829  df-cn 18958  df-cnp 18959  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-tsms 19824
This theorem is referenced by:  tsmsinv  19849  esumcocn  26669
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