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Theorem tgptsmscls 18171
Description: A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 18131, 0nsg 14977. (Contributed by Mario Carneiro, 22-Sep-2015.)
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 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopGrp )
3 tgpgrp 18100 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
42, 3syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Grp )
5 eqid 2435 . . . . . . . . . . 11  |-  ( 0g
`  G )  =  ( 0g `  G
)
650subg 14957 . . . . . . . . . 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 18130 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  {
( 0g `  G
) }  e.  (SubGrp `  G ) )  -> 
( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G ) )
102, 7, 9syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  e.  (SubGrp `  G )
)
11 tgptsmscls.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
12 eqid 2435 . . . . . . . . 9  |-  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
1311, 12eqger 14982 . . . . . . . 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 18102 . . . . . . . . . . 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 18156 . . . . . . . . 9  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2120sselda 3340 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  B
)
22 tgptsmscls.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
2320, 22sseldd 3341 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
2423adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  B
)
25 eqid 2435 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
2615adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e. CMnd )
2718adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  A  e.  V
)
2819adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F : A --> B )
2922adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  ( G tsums  F ) )
30 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( G tsums  F ) )
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 18170 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( G tsums  ( F  o F ( -g `  G
) F ) ) )
3228ffvelrnda 5862 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( F `  k )  e.  B )
3328feqmptd 5771 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F  =  ( k  e.  A  |->  ( F `  k ) ) )
3427, 32, 32, 33, 33offval2 6314 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  o F ( -g `  G
) F )  =  ( k  e.  A  |->  ( ( F `  k ) ( -g `  G ) ( F `
 k ) ) ) )
354adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  G  e.  Grp )
3611, 5, 25grpsubid 14865 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( F `  k )  e.  B )  -> 
( ( F `  k ) ( -g `  G ) ( F `
 k ) )  =  ( 0g `  G ) )
3735, 32, 36syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  (
( F `  k
) ( -g `  G
) ( F `  k ) )  =  ( 0g `  G
) )
3837mpteq2dva 4287 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( ( F `
 k ) (
-g `  G )
( F `  k
) ) )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
3934, 38eqtrd 2467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  o F ( -g `  G
) F )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
4039oveq2d 6089 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  o F ( -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 14825 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
434, 42syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( 0g `  G )  e.  B
)
4443adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( 0g `  G )  e.  B )
45 eqid 2435 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  ( 0g
`  G ) )  =  ( k  e.  A  |->  ( 0g `  G ) )
4644, 45fmptd 5885 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) : A --> B )
47 0fin 7328 . . . . . . . . . . . 12  |-  (/)  e.  Fin
48 eqidd 2436 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  ( A  \  (/) ) )  ->  ( 0g `  G )  =  ( 0g `  G ) )
4948suppss2 6292 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( `' ( k  e.  A  |->  ( 0g `  G ) ) " ( _V 
\  { ( 0g
`  G ) } ) )  C_  (/) )
50 ssfi 7321 . . . . . . . . . . . 12  |-  ( (
(/)  e.  Fin  /\  ( `' ( k  e.  A  |->  ( 0g `  G ) ) "
( _V  \  {
( 0g `  G
) } ) ) 
C_  (/) )  ->  ( `' ( k  e.  A  |->  ( 0g `  G ) ) "
( _V  \  {
( 0g `  G
) } ) )  e.  Fin )
5147, 49, 50sylancr 645 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( `' ( k  e.  A  |->  ( 0g `  G ) ) " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin )
5211, 5, 26, 41, 27, 46, 51, 8tsmsgsum 18160 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  (
k  e.  A  |->  ( 0g `  G ) ) )  =  ( ( cls `  J
) `  { ( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) } ) )
53 cmnmnd 15419 . . . . . . . . . . . . . 14  |-  ( G  e. CMnd  ->  G  e.  Mnd )
5426, 53syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Mnd )
555gsumz 14773 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) )  =  ( 0g `  G
) )
5654, 27, 55syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g
`  G ) ) )  =  ( 0g
`  G ) )
5756sneqd 3819 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( G 
gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) }  =  { ( 0g
`  G ) } )
5857fveq2d 5724 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
5940, 52, 583eqtrd 2471 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  o F ( -g `  G ) F ) )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6031, 59eleqtrd 2511 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
61 isabl 15408 . . . . . . . . . 10  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
624, 26, 61sylanbrc 646 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Abel )
6311subgss 14937 . . . . . . . . . 10  |-  ( ( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G )  ->  (
( cls `  J
) `  { ( 0g `  G ) } )  C_  B )
6410, 63syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  C_  B )
6511, 25, 12eqgabl 15446 . . . . . . . . 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 ) } ) ) ) )
6662, 64, 65syl2anc 643 . . . . . . . 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 ) } ) ) ) )
6721, 24, 60, 66mpbir3and 1137 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) X )
6814, 67ersym 6909 . . . . . 6  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
6912releqg 14979 . . . . . . 7  |-  Rel  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
70 relelec 6937 . . . . . . 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 ) )
7169, 70ax-mp 8 . . . . . 6  |-  ( x  e.  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  <->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
7268, 71sylibr 204 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) ) )
73 eqid 2435 . . . . . . 7  |-  ( ( cls `  J ) `
 { ( 0g
`  G ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } )
7411, 8, 5, 12, 73snclseqg 18137 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  X  e.  B )  ->  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
752, 24, 74syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
7672, 75eleqtrd 2511 . . . 4  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( ( cls `  J
) `  { X } ) )
7776ex 424 . . 3  |-  ( ph  ->  ( x  e.  ( G tsums  F )  ->  x  e.  ( ( cls `  J ) `  { X } ) ) )
7877ssrdv 3346 . 2  |-  ( ph  ->  ( G tsums  F ) 
C_  ( ( cls `  J ) `  { X } ) )
7911, 8, 15, 17, 18, 19, 22tsmscls 18159 . 2  |-  ( ph  ->  ( ( cls `  J
) `  { X } )  C_  ( G tsums  F ) )
8078, 79eqssd 3357 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   "cima 4873   Rel wrel 4875   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295    Er wer 6894   [cec 6895   Fincfn 7101   Basecbs 13461   TopOpenctopn 13641   0gc0g 13715    gsumg cgsu 13716   Mndcmnd 14676   Grpcgrp 14677   -gcsg 14680  SubGrpcsubg 14930   ~QG cqg 14932  CMndccmn 15404   Abelcabel 15405   TopSpctps 16953   clsccl 17074   TopGrpctgp 18093   tsums ctsu 18147
This theorem is referenced by:  tgptsmscld  18172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ec 6899  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-topgen 13659  df-0g 13719  df-gsum 13720  df-mnd 14682  df-plusf 14683  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-eqg 14935  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-fbas 16691  df-fg 16692  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-cn 17283  df-cnp 17284  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-tmd 18094  df-tgp 18095  df-tsms 18148
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