MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgptsmscls Unicode version

Theorem tgptsmscls 17848
Description: A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 17808, 0nsg 14678. (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 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopGrp )
3 tgpgrp 17777 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
42, 3syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Grp )
5 eqid 2296 . . . . . . . . . . 11  |-  ( 0g
`  G )  =  ( 0g `  G
)
650subg 14658 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
74, 6syl 15 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( 0g
`  G ) }  e.  (SubGrp `  G
) )
8 tgptsmscls.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  G )
98clssubg 17807 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  {
( 0g `  G
) }  e.  (SubGrp `  G ) )  -> 
( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G ) )
102, 7, 9syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  e.  (SubGrp `  G )
)
11 tgptsmscls.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
12 eqid 2296 . . . . . . . . 9  |-  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
1311, 12eqger 14683 . . . . . . . 8  |-  ( ( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G )  ->  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  Er  B
)
1410, 13syl 15 . . . . . . 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 17779 . . . . . . . . . . 11  |-  ( G  e.  TopGrp  ->  G  e.  TopSp )
171, 16syl 15 . . . . . . . . . 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 17833 . . . . . . . . 9  |-  ( ph  ->  ( G tsums  F ) 
C_  B )
2120sselda 3193 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  B
)
22 tgptsmscls.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( G tsums 
F ) )
2320, 22sseldd 3194 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
2423adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  B
)
25 eqid 2296 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
2615adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e. CMnd )
2718adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  A  e.  V
)
2819adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F : A --> B )
2922adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X  e.  ( G tsums  F ) )
30 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( G tsums  F ) )
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 17847 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( G tsums  ( F  o F ( -g `  G
) F ) ) )
32 ffvelrn 5679 . . . . . . . . . . . . . 14  |-  ( ( F : A --> B  /\  k  e.  A )  ->  ( F `  k
)  e.  B )
3328, 32sylan 457 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( F `  k )  e.  B )
3428feqmptd 5591 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  F  =  ( k  e.  A  |->  ( F `  k ) ) )
3527, 33, 33, 34, 34offval2 6111 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  o F ( -g `  G
) F )  =  ( k  e.  A  |->  ( ( F `  k ) ( -g `  G ) ( F `
 k ) ) ) )
364adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  G  e.  Grp )
3711, 5, 25grpsubid 14566 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( F `  k )  e.  B )  -> 
( ( F `  k ) ( -g `  G ) ( F `
 k ) )  =  ( 0g `  G ) )
3836, 33, 37syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  (
( F `  k
) ( -g `  G
) ( F `  k ) )  =  ( 0g `  G
) )
3938mpteq2dva 4122 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( ( F `
 k ) (
-g `  G )
( F `  k
) ) )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
4035, 39eqtrd 2328 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( F  o F ( -g `  G
) F )  =  ( k  e.  A  |->  ( 0g `  G
) ) )
4140oveq2d 5890 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  o F ( -g `  G ) F ) )  =  ( G tsums 
( k  e.  A  |->  ( 0g `  G
) ) ) )
422, 16syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  TopSp )
4311, 5grpidcl 14526 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
444, 43syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( 0g `  G )  e.  B
)
4544adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  A )  ->  ( 0g `  G )  e.  B )
46 eqid 2296 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  ( 0g
`  G ) )  =  ( k  e.  A  |->  ( 0g `  G ) )
4745, 46fmptd 5700 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( k  e.  A  |->  ( 0g `  G ) ) : A --> B )
48 0fin 7103 . . . . . . . . . . . 12  |-  (/)  e.  Fin
49 eqidd 2297 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( G tsums  F ) )  /\  k  e.  ( A  \  (/) ) )  ->  ( 0g `  G )  =  ( 0g `  G ) )
5049suppss2 6089 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( `' ( k  e.  A  |->  ( 0g `  G ) ) " ( _V 
\  { ( 0g
`  G ) } ) )  C_  (/) )
51 ssfi 7099 . . . . . . . . . . . 12  |-  ( (
(/)  e.  Fin  /\  ( `' ( k  e.  A  |->  ( 0g `  G ) ) "
( _V  \  {
( 0g `  G
) } ) ) 
C_  (/) )  ->  ( `' ( k  e.  A  |->  ( 0g `  G ) ) "
( _V  \  {
( 0g `  G
) } ) )  e.  Fin )
5248, 50, 51sylancr 644 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( `' ( k  e.  A  |->  ( 0g `  G ) ) " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin )
5311, 5, 26, 42, 27, 47, 52, 8tsmsgsum 17837 . . . . . . . . . 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 15120 . . . . . . . . . . . . . 14  |-  ( G  e. CMnd  ->  G  e.  Mnd )
5526, 54syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Mnd )
565gsumz 14474 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) )  =  ( 0g `  G
) )
5755, 27, 56syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G  gsumg  ( k  e.  A  |->  ( 0g
`  G ) ) )  =  ( 0g
`  G ) )
5857sneqd 3666 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  { ( G 
gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) }  =  { ( 0g
`  G ) } )
5958fveq2d 5545 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( G  gsumg  ( k  e.  A  |->  ( 0g `  G
) ) ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6041, 53, 593eqtrd 2332 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( G tsums  ( F  o F ( -g `  G ) F ) )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )
6131, 60eleqtrd 2372 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( X (
-g `  G )
x )  e.  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
62 isabl 15109 . . . . . . . . . 10  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
634, 26, 62sylanbrc 645 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  G  e.  Abel )
6411subgss 14638 . . . . . . . . . 10  |-  ( ( ( cls `  J
) `  { ( 0g `  G ) } )  e.  (SubGrp `  G )  ->  (
( cls `  J
) `  { ( 0g `  G ) } )  C_  B )
6510, 64syl 15 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  ( ( cls `  J ) `  {
( 0g `  G
) } )  C_  B )
6611, 25, 12eqgabl 15147 . . . . . . . . 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 642 . . . . . . . 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 1135 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) X )
6914, 68ersym 6688 . . . . . 6  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
7012releqg 14680 . . . . . . 7  |-  Rel  ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )
71 relelec 6716 . . . . . . 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 8 . . . . . 6  |-  ( x  e.  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  <->  X ( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) ) x )
7369, 72sylibr 203 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) ) )
74 eqid 2296 . . . . . . 7  |-  ( ( cls `  J ) `
 { ( 0g
`  G ) } )  =  ( ( cls `  J ) `
 { ( 0g
`  G ) } )
7511, 8, 5, 12, 74snclseqg 17814 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  X  e.  B )  ->  [ X ] ( G ~QG  ( ( cls `  J ) `
 { ( 0g
`  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
762, 24, 75syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  [ X ]
( G ~QG  ( ( cls `  J
) `  { ( 0g `  G ) } ) )  =  ( ( cls `  J
) `  { X } ) )
7773, 76eleqtrd 2372 . . . 4  |-  ( (
ph  /\  x  e.  ( G tsums  F ) )  ->  x  e.  ( ( cls `  J
) `  { X } ) )
7877ex 423 . . 3  |-  ( ph  ->  ( x  e.  ( G tsums  F )  ->  x  e.  ( ( cls `  J ) `  { X } ) ) )
7978ssrdv 3198 . 2  |-  ( ph  ->  ( G tsums  F ) 
C_  ( ( cls `  J ) `  { X } ) )
8011, 8, 15, 17, 18, 19, 22tsmscls 17836 . 2  |-  ( ph  ->  ( ( cls `  J
) `  { X } )  C_  ( G tsums  F ) )
8179, 80eqssd 3209 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092    Er wer 6673   [cec 6674   Fincfn 6879   Basecbs 13164   TopOpenctopn 13342   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631   ~QG cqg 14633  CMndccmn 15105   Abelcabel 15106   TopSpctps 16650   clsccl 16771   TopGrpctgp 17770   tsums ctsu 17824
This theorem is referenced by:  tgptsmscld  17849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-topgen 13360  df-0g 13420  df-gsum 13421  df-mnd 14383  df-plusf 14384  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-eqg 14636  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-tmd 17771  df-tgp 17772  df-tsms 17825
  Copyright terms: Public domain W3C validator