Theorem tgptsmscls 21953

Description: A sum in a topological group is uniquely determined up to a coset of cls ( { 0 } ), which is a normal subgroup by clsnsg 21913, 0nsg 17639. (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.

Step	Hyp	Ref	Expression
1		tgptsmscls.2	. . . . . . . . . 10 |- ( ph -> G e. TopGrp )
2	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. TopGrp )
3		tgpgrp 21882	. . . . . . . . . . 11 |- ( G e. TopGrp -> G e. Grp )
4	2, 3	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Grp )
5		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
6	5	0subg 17619	. . . . . . . . . 10 |- ( G e. Grp -> { ( 0g ` G ) } e. (SubGrp ` G ) )
7	4, 6	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> { ( 0g ` G ) } e. (SubGrp ` G ) )
8		tgptsmscls.j	. . . . . . . . . 10 |- J = ( TopOpen ` G )
9	8	clssubg 21912	. . . . . . . . 9 |- ( ( G e. TopGrp /\ { ( 0g ` G ) } e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) )
10	2, 7, 9	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) )
11		tgptsmscls.b	. . . . . . . . 9 |- B = ( Base ` G )
12		eqid 2622	. . . . . . . . 9 |- ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) )
13	11, 12	eqger 17644	. . . . . . . 8 |- ( ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) -> ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) Er B )
14	10, 13	syl 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 21884	. . . . . . . . . . 11 |- ( G e. TopGrp -> G e. TopSp )
17	1, 16	syl 17	. . . . . . . . . 10 |- ( ph -> G e. TopSp )
18		tgptsmscls.a	. . . . . . . . . 10 |- ( ph -> A e. V )
19		tgptsmscls.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
20	11, 15, 17, 18, 19	tsmscl 21938	. . . . . . . . 9 |- ( ph -> ( G tsums F ) C_ B )
21	20	sselda 3603	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. B )
22		tgptsmscls.x	. . . . . . . . . 10 |- ( ph -> X e. ( G tsums F ) )
23	20, 22	sseldd 3604	. . . . . . . . 9 |- ( ph -> X e. B )
24	23	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> X e. B )
25		eqid 2622	. . . . . . . . . 10 |- ( -g ` G ) = ( -g ` G )
26	15	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. CMnd )
27	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> A e. V )
28	19	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> F : A --> B )
29	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> X e. ( G tsums F ) )
30		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. ( G tsums F ) )
31	11, 25, 26, 2, 27, 28, 28, 29, 30	tsmssub 21952	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( X ( -g ` G ) x ) e. ( G tsums ( F oF ( -g ` G ) F ) ) )
32	28	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( F ` k ) e. B )
33	28	feqmptd 6249	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> F = ( k e. A |-> ( F ` k ) ) )
34	27, 32, 32, 33, 33	offval2 6914	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( F oF ( -g ` G ) F ) = ( k e. A |-> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) ) )
35	4	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> G e. Grp )
36	11, 5, 25	grpsubid 17499	. . . . . . . . . . . . . 14 |- ( ( G e. Grp /\ ( F ` k ) e. B ) -> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) = ( 0g ` G ) )
37	35, 32, 36	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) = ( 0g ` G ) )
38	37	mpteq2dva 4744	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( k e. A |-> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) ) = ( k e. A |-> ( 0g ` G ) ) )
39	34, 38	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( F oF ( -g ` G ) F ) = ( k e. A |-> ( 0g ` G ) ) )
40	39	oveq2d 6666	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( G tsums ( F oF ( -g ` G ) F ) ) = ( G tsums ( k e. A |-> ( 0g ` G ) ) ) )
41	2, 16	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. TopSp )
42	11, 5	grpidcl 17450	. . . . . . . . . . . . . 14 |- ( G e. Grp -> ( 0g ` G ) e. B )
43	4, 42	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( 0g ` G ) e. B )
44	43	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( 0g ` G ) e. B )
45		eqid 2622	. . . . . . . . . . . 12 |- ( k e. A |-> ( 0g ` G ) ) = ( k e. A |-> ( 0g ` G ) )
46	44, 45	fmptd 6385	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( k e. A |-> ( 0g ` G ) ) : A --> B )
47		fconstmpt 5163	. . . . . . . . . . . 12 |- ( A X. { ( 0g ` G ) } ) = ( k e. A |-> ( 0g ` G ) )
48		fvexd 6203	. . . . . . . . . . . . . 14 |- ( ph -> ( 0g ` G ) e. _V )
49	18, 48	fczfsuppd 8293	. . . . . . . . . . . . 13 |- ( ph -> ( A X. { ( 0g ` G ) } ) finSupp ( 0g ` G ) )
50	49	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( A X. { ( 0g ` G ) } ) finSupp ( 0g ` G ) )
51	47, 50	syl5eqbrr 4689	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( k e. A |-> ( 0g ` G ) ) finSupp ( 0g ` G ) )
52	11, 5, 26, 41, 27, 46, 51, 8	tsmsgsum 21942	. . . . . . . . . 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 18208	. . . . . . . . . . . . . 14 |- ( G e. CMnd -> G e. Mnd )
54	26, 53	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Mnd )
55	5	gsumz 17374	. . . . . . . . . . . . 13 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) = ( 0g ` G ) )
56	54, 27, 55	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) = ( 0g ` G ) )
57	56	sneqd 4189	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> { ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) } = { ( 0g ` G ) } )
58	57	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( ( cls ` J ) ` { ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) } ) = ( ( cls ` J ) ` { ( 0g ` G ) } ) )
59	40, 52, 58	3eqtrd 2660	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( G tsums ( F oF ( -g ` G ) F ) ) = ( ( cls ` J ) ` { ( 0g ` G ) } ) )
60	31, 59	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( X ( -g ` G ) x ) e. ( ( cls ` J ) ` { ( 0g ` G ) } ) )
61		isabl 18197	. . . . . . . . . 10 |- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
62	4, 26, 61	sylanbrc 698	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Abel )
63	11	subgss 17595	. . . . . . . . . 10 |- ( ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) C_ B )
64	10, 63	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) C_ B )
65	11, 25, 12	eqgabl 18240	. . . . . . . . 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 ) } ) ) ) )
66	62, 64, 65	syl2anc 693	. . . . . . . 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 ) } ) ) ) )
67	21, 24, 60, 66	mpbir3and 1245	. . . . . . 7 |- ( ( ph /\ x e. ( G tsums F ) ) -> x ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) X )
68	14, 67	ersym 7754	. . . . . 6 |- ( ( ph /\ x e. ( G tsums F ) ) -> X ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) x )
69	12	releqg 17641	. . . . . . 7 |- Rel ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) )
70		relelec 7787	. . . . . . 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 ) )
71	69, 70	ax-mp 5	. . . . . 6 |- ( x e. [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) <-> X ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) x )
72	68, 71	sylibr 224	. . . . 5 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) )
73		eqid 2622	. . . . . . 7 |- ( ( cls ` J ) ` { ( 0g ` G ) } ) = ( ( cls ` J ) ` { ( 0g ` G ) } )
74	11, 8, 5, 12, 73	snclseqg 21919	. . . . . 6 |- ( ( G e. TopGrp /\ X e. B ) -> [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( ( cls ` J ) ` { X } ) )
75	2, 24, 74	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. ( G tsums F ) ) -> [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( ( cls ` J ) ` { X } ) )
76	72, 75	eleqtrd 2703	. . . 4 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. ( ( cls ` J ) ` { X } ) )
77	76	ex 450	. . 3 |- ( ph -> ( x e. ( G tsums F ) -> x e. ( ( cls ` J ) ` { X } ) ) )
78	77	ssrdv 3609	. 2 |- ( ph -> ( G tsums F ) C_ ( ( cls ` J ) ` { X } ) )
79	11, 8, 15, 17, 18, 19, 22	tsmscls 21941	. 2 |- ( ph -> ( ( cls ` J ) ` { X } ) C_ ( G tsums F ) )
80	78, 79	eqssd 3620	1 |- ( ph -> ( G tsums F ) = ( ( cls ` J ) ` { X } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Er wer 7739   [cec 7740   finSupp cfsupp 8275   Basecbs 15857   TopOpenctopn 16082   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588   ~_QG cqg 17590  CMndccmn 18193   Abelcabl 18194   TopSpctps 20736   clsccl 20822   TopGrpctgp 21875   tsums ctsu 21929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-eqg 17593  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930

This theorem is referenced by:  tgptsmscld  21954