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Theorem tgptsmscls 21212
Description: A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 21172, 0nsg 16910. (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 471 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopGrp )
3 tgpgrp 21141 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
42, 3syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Grp )
5 eqid 2461 . . . . . . . . . . 11  |-  ( 0g
`  G )  =  ( 0g `  G
)
650subg 16890 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
74, 6syl 17 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( 0g
`  G ) }  e.  (SubGrp `  G
) )
8 tgptsmscls.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  G )
98clssubg 21171 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  {
( 0g `  G
) }  e.  (SubGrp `  G ) )  -> 
( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G ) )
102, 7, 9syl2anc 671 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  e.  (SubGrp `  G )
)
11 tgptsmscls.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
12 eqid 2461 . . . . . . . . 9  |-  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
1311, 12eqger 16915 . . . . . . . 8  |-  ( ( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G )  ->  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  Er  B
)
1410, 13syl 17 . . . . . . 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 21143 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  TopSp )
171, 16syl 17 . . . . . . . . . 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 21197 . . . . . . . . 9  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2120sselda 3443 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  B
)
22 tgptsmscls.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
2320, 22sseldd 3444 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
2423adantr 471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  B
)
25 eqid 2461 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
2615adantr 471 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e. CMnd )
2718adantr 471 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  A  e.  V
)
2819adantr 471 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F : A --> B )
2922adantr 471 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  ( G tsums  F ) )
30 simpr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( G tsums  F ) )
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 21211 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( G tsums  ( F  oF ( -g `  G
) F ) ) )
3228ffvelrnda 6044 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( F `  k )  e.  B )
3328feqmptd 5940 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F  =  ( k  e.  A  |->  ( F `  k ) ) )
3427, 32, 32, 33, 33offval2 6574 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  oF ( -g `  G
) F )  =  ( k  e.  A  |->  ( ( F `  k ) ( -g `  G ) ( F `
 k ) ) ) )
354adantr 471 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  G  e.  Grp )
3611, 5, 25grpsubid 16786 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( F `  k )  e.  B )  -> 
( ( F `  k ) ( -g `  G ) ( F `
 k ) )  =  ( 0g `  G ) )
3735, 32, 36syl2anc 671 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  (
( F `  k
) ( -g `  G
) ( F `  k ) )  =  ( 0g `  G
) )
3837mpteq2dva 4502 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( ( F `
 k ) (
-g `  G )
( F `  k
) ) )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
3934, 38eqtrd 2495 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  oF ( -g `  G
) F )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
4039oveq2d 6330 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  oF ( -g `  G ) F ) )  =  ( G tsums 
( k  e.  A  |->  ( 0g `  G
) ) ) )
412, 16syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopSp )
4211, 5grpidcl 16742 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
434, 42syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( 0g `  G )  e.  B
)
4443adantr 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( 0g `  G )  e.  B )
45 eqid 2461 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  ( 0g
`  G ) )  =  ( k  e.  A  |->  ( 0g `  G ) )
4644, 45fmptd 6068 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) : A --> B )
47 fconstmpt 4896 . . . . . . . . . . . 12  |-  ( A  X.  { ( 0g
`  G ) } )  =  ( k  e.  A  |->  ( 0g
`  G ) )
48 fvex 5897 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
4948a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 0g `  G
)  e.  _V )
5018, 49fczfsuppd 7926 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  X.  {
( 0g `  G
) } ) finSupp  ( 0g `  G ) )
5150adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( A  X.  { ( 0g `  G ) } ) finSupp 
( 0g `  G
) )
5247, 51syl5eqbrr 4450 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) finSupp  ( 0g `  G ) )
5311, 5, 26, 41, 27, 46, 52, 8tsmsgsum 21201 . . . . . . . . . 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 17493 . . . . . . . . . . . . . 14  |-  ( G  e. CMnd  ->  G  e.  Mnd )
5526, 54syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Mnd )
565gsumz 16669 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) )  =  ( 0g `  G
) )
5755, 27, 56syl2anc 671 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g
`  G ) ) )  =  ( 0g
`  G ) )
5857sneqd 3991 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( G 
gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) }  =  { ( 0g
`  G ) } )
5958fveq2d 5891 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6040, 53, 593eqtrd 2499 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  oF ( -g `  G ) F ) )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6131, 60eleqtrd 2541 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
62 isabl 17482 . . . . . . . . . 10  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
634, 26, 62sylanbrc 675 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Abel )
6411subgss 16866 . . . . . . . . . 10  |-  ( ( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G )  ->  (
( cls `  J
) `  { ( 0g `  G ) } )  C_  B )
6510, 64syl 17 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  C_  B )
6611, 25, 12eqgabl 17523 . . . . . . . . 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 671 . . . . . . . 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 1197 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) X )
6914, 68ersym 7400 . . . . . 6  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
7012releqg 16912 . . . . . . 7  |-  Rel  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
71 relelec 7429 . . . . . . 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 217 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) ) )
74 eqid 2461 . . . . . . 7  |-  ( ( cls `  J ) `
 { ( 0g
`  G ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } )
7511, 8, 5, 12, 74snclseqg 21178 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  X  e.  B )  ->  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
762, 24, 75syl2anc 671 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
7773, 76eleqtrd 2541 . . . 4  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( ( cls `  J
) `  { X } ) )
7877ex 440 . . 3  |-  ( ph  ->  ( x  e.  ( G tsums  F )  ->  x  e.  ( ( cls `  J ) `  { X } ) ) )
7978ssrdv 3449 . 2  |-  ( ph  ->  ( G tsums  F ) 
C_  ( ( cls `  J ) `  { X } ) )
8011, 8, 15, 17, 18, 19, 22tsmscls 21200 . 2  |-  ( ph  ->  ( ( cls `  J
) `  { X } )  C_  ( G tsums  F ) )
8179, 80eqssd 3460 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   _Vcvv 3056    C_ wss 3415   {csn 3979   class class class wbr 4415    |-> cmpt 4474    X. cxp 4850   Rel wrel 4857   -->wf 5596   ` cfv 5600  (class class class)co 6314    oFcof 6555    Er wer 7385   [cec 7386   finSupp cfsupp 7908   Basecbs 15169   TopOpenctopn 15368   0gc0g 15386    gsumg cgsu 15387   Mndcmnd 16583   Grpcgrp 16717   -gcsg 16719  SubGrpcsubg 16859   ~QG cqg 16861  CMndccmn 17478   Abelcabl 17479   TopSpctps 19967   clsccl 20081   TopGrpctgp 21134   tsums ctsu 21188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-ec 7390  df-map 7499  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-seq 12245  df-hash 12547  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-gsum 15389  df-topgen 15390  df-plusf 16535  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-mhm 16630  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-subg 16862  df-eqg 16864  df-ghm 16929  df-cntz 17019  df-cmn 17480  df-abl 17481  df-fbas 19015  df-fg 19016  df-top 19969  df-bases 19970  df-topon 19971  df-topsp 19972  df-cld 20082  df-ntr 20083  df-cls 20084  df-nei 20162  df-cn 20291  df-cnp 20292  df-tx 20625  df-hmeo 20818  df-fil 20909  df-fm 21001  df-flim 21002  df-flf 21003  df-tmd 21135  df-tgp 21136  df-tsms 21189
This theorem is referenced by:  tgptsmscld  21213
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