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Theorem tgptsmscls 20760
Description: A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 20716, 0nsg 16386. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
Hypotheses
Ref Expression
tgptsmscls.b  |-  B  =  ( Base `  G
)
tgptsmscls.j  |-  J  =  ( TopOpen `  G )
tgptsmscls.1  |-  ( ph  ->  G  e. CMnd )
tgptsmscls.2  |-  ( ph  ->  G  e.  TopGrp )
tgptsmscls.a  |-  ( ph  ->  A  e.  V )
tgptsmscls.f  |-  ( ph  ->  F : A --> B )
tgptsmscls.x  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
Assertion
Ref Expression
tgptsmscls  |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } ) )

Proof of Theorem tgptsmscls
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgptsmscls.2 . . . . . . . . . 10  |-  ( ph  ->  G  e.  TopGrp )
21adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopGrp )
3 tgpgrp 20685 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
42, 3syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Grp )
5 eqid 2396 . . . . . . . . . . 11  |-  ( 0g
`  G )  =  ( 0g `  G
)
650subg 16366 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
74, 6syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( 0g
`  G ) }  e.  (SubGrp `  G
) )
8 tgptsmscls.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  G )
98clssubg 20715 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  {
( 0g `  G
) }  e.  (SubGrp `  G ) )  -> 
( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G ) )
102, 7, 9syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  e.  (SubGrp `  G )
)
11 tgptsmscls.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
12 eqid 2396 . . . . . . . . 9  |-  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
1311, 12eqger 16391 . . . . . . . 8  |-  ( ( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G )  ->  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  Er  B
)
1410, 13syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )  Er  B
)
15 tgptsmscls.1 . . . . . . . . . 10  |-  ( ph  ->  G  e. CMnd )
16 tgptps 20687 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  TopSp )
171, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  e.  TopSp )
18 tgptsmscls.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  V )
19 tgptsmscls.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
2011, 15, 17, 18, 19tsmscl 20741 . . . . . . . . 9  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2120sselda 3434 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  B
)
22 tgptsmscls.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
2320, 22sseldd 3435 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
2423adantr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  B
)
25 eqid 2396 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
2615adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e. CMnd )
2718adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  A  e.  V
)
2819adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F : A --> B )
2922adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  ( G tsums  F ) )
30 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( G tsums  F ) )
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 20759 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( G tsums  ( F  oF ( -g `  G
) F ) ) )
3228ffvelrnda 5950 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( F `  k )  e.  B )
3328feqmptd 5844 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F  =  ( k  e.  A  |->  ( F `  k ) ) )
3427, 32, 32, 33, 33offval2 6477 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  oF ( -g `  G
) F )  =  ( k  e.  A  |->  ( ( F `  k ) ( -g `  G ) ( F `
 k ) ) ) )
354adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  G  e.  Grp )
3611, 5, 25grpsubid 16262 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( F `  k )  e.  B )  -> 
( ( F `  k ) ( -g `  G ) ( F `
 k ) )  =  ( 0g `  G ) )
3735, 32, 36syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  (
( F `  k
) ( -g `  G
) ( F `  k ) )  =  ( 0g `  G
) )
3837mpteq2dva 4470 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( ( F `
 k ) (
-g `  G )
( F `  k
) ) )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
3934, 38eqtrd 2437 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  oF ( -g `  G
) F )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
4039oveq2d 6234 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  oF ( -g `  G ) F ) )  =  ( G tsums 
( k  e.  A  |->  ( 0g `  G
) ) ) )
412, 16syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopSp )
4211, 5grpidcl 16218 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
434, 42syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( 0g `  G )  e.  B
)
4443adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( 0g `  G )  e.  B )
45 eqid 2396 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  ( 0g
`  G ) )  =  ( k  e.  A  |->  ( 0g `  G ) )
4644, 45fmptd 5974 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) : A --> B )
47 fconstmpt 4974 . . . . . . . . . . . 12  |-  ( A  X.  { ( 0g
`  G ) } )  =  ( k  e.  A  |->  ( 0g
`  G ) )
48 fvex 5801 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
4948a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 0g `  G
)  e.  _V )
5018, 49fczfsuppd 7784 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  X.  {
( 0g `  G
) } ) finSupp  ( 0g `  G ) )
5150adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( A  X.  { ( 0g `  G ) } ) finSupp 
( 0g `  G
) )
5247, 51syl5eqbrr 4418 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) finSupp  ( 0g `  G ) )
5311, 5, 26, 41, 27, 46, 52, 8tsmsgsum 20745 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  (
k  e.  A  |->  ( 0g `  G ) ) )  =  ( ( cls `  J
) `  { ( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) } ) )
54 cmnmnd 16953 . . . . . . . . . . . . . 14  |-  ( G  e. CMnd  ->  G  e.  Mnd )
5526, 54syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Mnd )
565gsumz 16145 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) )  =  ( 0g `  G
) )
5755, 27, 56syl2anc 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g
`  G ) ) )  =  ( 0g
`  G ) )
5857sneqd 3973 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( G 
gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) }  =  { ( 0g
`  G ) } )
5958fveq2d 5795 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6040, 53, 593eqtrd 2441 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  oF ( -g `  G ) F ) )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6131, 60eleqtrd 2486 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
62 isabl 16942 . . . . . . . . . 10  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
634, 26, 62sylanbrc 662 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Abel )
6411subgss 16342 . . . . . . . . . 10  |-  ( ( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G )  ->  (
( cls `  J
) `  { ( 0g `  G ) } )  C_  B )
6510, 64syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  C_  B )
6611, 25, 12eqgabl 16983 . . . . . . . . 9  |-  ( ( G  e.  Abel  /\  (
( cls `  J
) `  { ( 0g `  G ) } )  C_  B )  ->  ( x ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) X  <->  ( x  e.  B  /\  X  e.  B  /\  ( X ( -g `  G
) x )  e.  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) ) )
6763, 65, 66syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( x ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) X  <->  ( x  e.  B  /\  X  e.  B  /\  ( X ( -g `  G
) x )  e.  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) ) )
6821, 24, 61, 67mpbir3and 1177 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) X )
6914, 68ersym 7263 . . . . . 6  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
7012releqg 16388 . . . . . . 7  |-  Rel  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
71 relelec 7292 . . . . . . 7  |-  ( Rel  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  ->  (
x  e.  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )  <->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x ) )
7270, 71ax-mp 5 . . . . . 6  |-  ( x  e.  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  <->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
7369, 72sylibr 212 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) ) )
74 eqid 2396 . . . . . . 7  |-  ( ( cls `  J ) `
 { ( 0g
`  G ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } )
7511, 8, 5, 12, 74snclseqg 20722 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  X  e.  B )  ->  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
762, 24, 75syl2anc 659 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
7773, 76eleqtrd 2486 . . . 4  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( ( cls `  J
) `  { X } ) )
7877ex 432 . . 3  |-  ( ph  ->  ( x  e.  ( G tsums  F )  ->  x  e.  ( ( cls `  J ) `  { X } ) ) )
7978ssrdv 3440 . 2  |-  ( ph  ->  ( G tsums  F ) 
C_  ( ( cls `  J ) `  { X } ) )
8011, 8, 15, 17, 18, 19, 22tsmscls 20744 . 2  |-  ( ph  ->  ( ( cls `  J
) `  { X } )  C_  ( G tsums  F ) )
8179, 80eqssd 3451 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   _Vcvv 3051    C_ wss 3406   {csn 3961   class class class wbr 4384    |-> cmpt 4442    X. cxp 4928   Rel wrel 4935   -->wf 5509   ` cfv 5513  (class class class)co 6218    oFcof 6459    Er wer 7248   [cec 7249   finSupp cfsupp 7766   Basecbs 14657   TopOpenctopn 14852   0gc0g 14870    gsumg cgsu 14871   Mndcmnd 16059   Grpcgrp 16193   -gcsg 16195  SubGrpcsubg 16335   ~QG cqg 16337  CMndccmn 16938   Abelcabl 16939   TopSpctps 19505   clsccl 19627   TopGrpctgp 20678   tsums ctsu 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-se 4770  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-of 6461  df-om 6622  df-1st 6721  df-2nd 6722  df-supp 6840  df-recs 6982  df-rdg 7016  df-1o 7070  df-oadd 7074  df-er 7251  df-ec 7253  df-map 7362  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-fsupp 7767  df-oi 7872  df-card 8255  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-2 10533  df-n0 10735  df-z 10804  df-uz 11024  df-fz 11616  df-fzo 11740  df-seq 12034  df-hash 12331  df-ndx 14660  df-slot 14661  df-base 14662  df-sets 14663  df-ress 14664  df-plusg 14738  df-0g 14872  df-gsum 14873  df-topgen 14874  df-plusf 16011  df-mgm 16012  df-sgrp 16051  df-mnd 16061  df-mhm 16106  df-submnd 16107  df-grp 16197  df-minusg 16198  df-sbg 16199  df-subg 16338  df-eqg 16340  df-ghm 16405  df-cntz 16495  df-cmn 16940  df-abl 16941  df-fbas 18552  df-fg 18553  df-top 19507  df-bases 19509  df-topon 19510  df-topsp 19511  df-cld 19628  df-ntr 19629  df-cls 19630  df-nei 19708  df-cn 19837  df-cnp 19838  df-tx 20171  df-hmeo 20364  df-fil 20455  df-fm 20547  df-flim 20548  df-flf 20549  df-tmd 20679  df-tgp 20680  df-tsms 20733
This theorem is referenced by:  tgptsmscld  20761
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