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Theorem tgptsmscls 21156
Description: A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 21116, 0nsg 16855. (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 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopGrp )
3 tgpgrp 21085 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
42, 3syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Grp )
5 eqid 2423 . . . . . . . . . . 11  |-  ( 0g
`  G )  =  ( 0g `  G
)
650subg 16835 . . . . . . . . . 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 21115 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  {
( 0g `  G
) }  e.  (SubGrp `  G ) )  -> 
( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G ) )
102, 7, 9syl2anc 666 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  e.  (SubGrp `  G )
)
11 tgptsmscls.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
12 eqid 2423 . . . . . . . . 9  |-  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
1311, 12eqger 16860 . . . . . . . 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 21087 . . . . . . . . . . 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 21141 . . . . . . . . 9  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2120sselda 3465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  B
)
22 tgptsmscls.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
2320, 22sseldd 3466 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
2423adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  B
)
25 eqid 2423 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
2615adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e. CMnd )
2718adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  A  e.  V
)
2819adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F : A --> B )
2922adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  ( G tsums  F ) )
30 simpr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( G tsums  F ) )
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 21155 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( G tsums  ( F  oF ( -g `  G
) F ) ) )
3228ffvelrnda 6035 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( F `  k )  e.  B )
3328feqmptd 5932 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F  =  ( k  e.  A  |->  ( F `  k ) ) )
3427, 32, 32, 33, 33offval2 6560 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  oF ( -g `  G
) F )  =  ( k  e.  A  |->  ( ( F `  k ) ( -g `  G ) ( F `
 k ) ) ) )
354adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  G  e.  Grp )
3611, 5, 25grpsubid 16731 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( F `  k )  e.  B )  -> 
( ( F `  k ) ( -g `  G ) ( F `
 k ) )  =  ( 0g `  G ) )
3735, 32, 36syl2anc 666 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  (
( F `  k
) ( -g `  G
) ( F `  k ) )  =  ( 0g `  G
) )
3837mpteq2dva 4508 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( ( F `
 k ) (
-g `  G )
( F `  k
) ) )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
3934, 38eqtrd 2464 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  oF ( -g `  G
) F )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
4039oveq2d 6319 . . . . . . . . . 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 16687 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
434, 42syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( 0g `  G )  e.  B
)
4443adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( 0g `  G )  e.  B )
45 eqid 2423 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  ( 0g
`  G ) )  =  ( k  e.  A  |->  ( 0g `  G ) )
4644, 45fmptd 6059 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) : A --> B )
47 fconstmpt 4895 . . . . . . . . . . . 12  |-  ( A  X.  { ( 0g
`  G ) } )  =  ( k  e.  A  |->  ( 0g
`  G ) )
48 fvex 5889 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
4948a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 0g `  G
)  e.  _V )
5018, 49fczfsuppd 7905 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  X.  {
( 0g `  G
) } ) finSupp  ( 0g `  G ) )
5150adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( A  X.  { ( 0g `  G ) } ) finSupp 
( 0g `  G
) )
5247, 51syl5eqbrr 4456 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) finSupp  ( 0g `  G ) )
5311, 5, 26, 41, 27, 46, 52, 8tsmsgsum 21145 . . . . . . . . . 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 17438 . . . . . . . . . . . . . 14  |-  ( G  e. CMnd  ->  G  e.  Mnd )
5526, 54syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Mnd )
565gsumz 16614 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) )  =  ( 0g `  G
) )
5755, 27, 56syl2anc 666 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g
`  G ) ) )  =  ( 0g
`  G ) )
5857sneqd 4009 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( G 
gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) }  =  { ( 0g
`  G ) } )
5958fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6040, 53, 593eqtrd 2468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  oF ( -g `  G ) F ) )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6131, 60eleqtrd 2513 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
62 isabl 17427 . . . . . . . . . 10  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
634, 26, 62sylanbrc 669 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Abel )
6411subgss 16811 . . . . . . . . . 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 17468 . . . . . . . . 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 666 . . . . . . . 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 1189 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) X )
6914, 68ersym 7381 . . . . . 6  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
7012releqg 16857 . . . . . . 7  |-  Rel  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
71 relelec 7410 . . . . . . 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 216 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) ) )
74 eqid 2423 . . . . . . 7  |-  ( ( cls `  J ) `
 { ( 0g
`  G ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } )
7511, 8, 5, 12, 74snclseqg 21122 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  X  e.  B )  ->  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
762, 24, 75syl2anc 666 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
7773, 76eleqtrd 2513 . . . 4  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( ( cls `  J
) `  { X } ) )
7877ex 436 . . 3  |-  ( ph  ->  ( x  e.  ( G tsums  F )  ->  x  e.  ( ( cls `  J ) `  { X } ) ) )
7978ssrdv 3471 . 2  |-  ( ph  ->  ( G tsums  F ) 
C_  ( ( cls `  J ) `  { X } ) )
8011, 8, 15, 17, 18, 19, 22tsmscls 21144 . 2  |-  ( ph  ->  ( ( cls `  J
) `  { X } )  C_  ( G tsums  F ) )
8179, 80eqssd 3482 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   _Vcvv 3082    C_ wss 3437   {csn 3997   class class class wbr 4421    |-> cmpt 4480    X. cxp 4849   Rel wrel 4856   -->wf 5595   ` cfv 5599  (class class class)co 6303    oFcof 6541    Er wer 7366   [cec 7367   finSupp cfsupp 7887   Basecbs 15114   TopOpenctopn 15313   0gc0g 15331    gsumg cgsu 15332   Mndcmnd 16528   Grpcgrp 16662   -gcsg 16664  SubGrpcsubg 16804   ~QG cqg 16806  CMndccmn 17423   Abelcabl 17424   TopSpctps 19911   clsccl 20025   TopGrpctgp 21078   tsums ctsu 21132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-ec 7371  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918  df-seq 12215  df-hash 12517  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-gsum 15334  df-topgen 15335  df-plusf 16480  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-subg 16807  df-eqg 16809  df-ghm 16874  df-cntz 16964  df-cmn 17425  df-abl 17426  df-fbas 18960  df-fg 18961  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-cn 20235  df-cnp 20236  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-tmd 21079  df-tgp 21080  df-tsms 21133
This theorem is referenced by:  tgptsmscld  21157
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