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.

Step	Hyp	Ref	Expression
1		tgptsmscls.2	. . . . . . . . . 10 |- ( ph -> G e. TopGrp )
2	1	adantr 471	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. TopGrp )
3		tgpgrp 21141	. . . . . . . . . . 11 |- ( G e. TopGrp -> G e. Grp )
4	2, 3	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Grp )
5		eqid 2461	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
6	5	0subg 16890	. . . . . . . . . 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 21171	. . . . . . . . 9 |- ( ( G e. TopGrp /\ { ( 0g ` G ) } e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) )
10	2, 7, 9	syl2anc 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 ) } ) )
13	11, 12	eqger 16915	. . . . . . . 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 21143	. . . . . . . . . . 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 21197	. . . . . . . . 9 |- ( ph -> ( G tsums F ) C_ B )
21	20	sselda 3443	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. B )
22		tgptsmscls.x	. . . . . . . . . 10 |- ( ph -> X e. ( G tsums F ) )
23	20, 22	sseldd 3444	. . . . . . . . 9 |- ( ph -> X e. B )
24	23	adantr 471	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> X e. B )
25		eqid 2461	. . . . . . . . . 10 |- ( -g ` G ) = ( -g ` G )
26	15	adantr 471	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. CMnd )
27	18	adantr 471	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> A e. V )
28	19	adantr 471	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> F : A --> B )
29	22	adantr 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 ) )
31	11, 25, 26, 2, 27, 28, 28, 29, 30	tsmssub 21211	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( X ( -g ` G ) x ) e. ( G tsums ( F oF ( -g ` G ) F ) ) )
32	28	ffvelrnda 6044	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( F ` k ) e. B )
33	28	feqmptd 5940	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> F = ( k e. A |-> ( F ` k ) ) )
34	27, 32, 32, 33, 33	offval2 6574	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( F oF ( -g ` G ) F ) = ( k e. A |-> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) ) )
35	4	adantr 471	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> G e. Grp )
36	11, 5, 25	grpsubid 16786	. . . . . . . . . . . . . 14 |- ( ( G e. Grp /\ ( F ` k ) e. B ) -> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) = ( 0g ` G ) )
37	35, 32, 36	syl2anc 671	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) = ( 0g ` G ) )
38	37	mpteq2dva 4502	. . . . . . . . . . . 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 2495	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( F oF ( -g ` G ) F ) = ( k e. A |-> ( 0g ` G ) ) )
40	39	oveq2d 6330	. . . . . . . . . 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 16742	. . . . . . . . . . . . . 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 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 ) )
46	44, 45	fmptd 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
49	48	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> ( 0g ` G ) e. _V )
50	18, 49	fczfsuppd 7926	. . . . . . . . . . . . 13 |- ( ph -> ( A X. { ( 0g ` G ) } ) finSupp ( 0g ` G ) )
51	50	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( A X. { ( 0g ` G ) } ) finSupp ( 0g ` G ) )
52	47, 51	syl5eqbrr 4450	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( k e. A |-> ( 0g ` G ) ) finSupp ( 0g ` G ) )
53	11, 5, 26, 41, 27, 46, 52, 8	tsmsgsum 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 )
55	26, 54	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Mnd )
56	5	gsumz 16669	. . . . . . . . . . . . 13 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) = ( 0g ` G ) )
57	55, 27, 56	syl2anc 671	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) = ( 0g ` G ) )
58	57	sneqd 3991	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> { ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) } = { ( 0g ` G ) } )
59	58	fveq2d 5891	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( ( cls ` J ) ` { ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) } ) = ( ( cls ` J ) ` { ( 0g ` G ) } ) )
60	40, 53, 59	3eqtrd 2499	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( G tsums ( F oF ( -g ` G ) F ) ) = ( ( cls ` J ) ` { ( 0g ` G ) } ) )
61	31, 60	eleqtrd 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 ) )
63	4, 26, 62	sylanbrc 675	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Abel )
64	11	subgss 16866	. . . . . . . . . 10 |- ( ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) C_ B )
65	10, 64	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) C_ B )
66	11, 25, 12	eqgabl 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 ) } ) ) ) )
67	63, 65, 66	syl2anc 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 ) } ) ) ) )
68	21, 24, 61, 67	mpbir3and 1197	. . . . . . 7 |- ( ( ph /\ x e. ( G tsums F ) ) -> x ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) X )
69	14, 68	ersym 7400	. . . . . 6 |- ( ( ph /\ x e. ( G tsums F ) ) -> X ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) x )
70	12	releqg 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 ) )
72	70, 71	ax-mp 5	. . . . . 6 |- ( x e. [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) <-> X ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) x )
73	69, 72	sylibr 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 ) } )
75	11, 8, 5, 12, 74	snclseqg 21178	. . . . . 6 |- ( ( G e. TopGrp /\ X e. B ) -> [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( ( cls ` J ) ` { X } ) )
76	2, 24, 75	syl2anc 671	. . . . 5 |- ( ( ph /\ x e. ( G tsums F ) ) -> [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( ( cls ` J ) ` { X } ) )
77	73, 76	eleqtrd 2541	. . . 4 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. ( ( cls ` J ) ` { X } ) )
78	77	ex 440	. . 3 |- ( ph -> ( x e. ( G tsums F ) -> x e. ( ( cls ` J ) ` { X } ) ) )
79	78	ssrdv 3449	. 2 |- ( ph -> ( G tsums F ) C_ ( ( cls ` J ) ` { X } ) )
80	11, 8, 15, 17, 18, 19, 22	tsmscls 21200	. 2 |- ( ph -> ( ( cls ` J ) ` { X } ) C_ ( G tsums F ) )
81	79, 80	eqssd 3460	1 |- ( ph -> ( G tsums F ) = ( ( cls ` J ) ` { X } ) )

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   gsum_g 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