Theorem tgptsmscls 19859

Description: A sum in a topological group is uniquely determined up to a coset of cls ( { 0 } ), which is a normal subgroup by clsnsg 19815, 0nsg 15848. (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 465	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. TopGrp )
3		tgpgrp 19784	. . . . . . . . . . 11 |- ( G e. TopGrp -> G e. Grp )
4	2, 3	syl 16	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Grp )
5		eqid 2454	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
6	5	0subg 15828	. . . . . . . . . 10 |- ( G e. Grp -> { ( 0g ` G ) } e. (SubGrp ` G ) )
7	4, 6	syl 16	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> { ( 0g ` G ) } e. (SubGrp ` G ) )
8		tgptsmscls.j	. . . . . . . . . 10 |- J = ( TopOpen ` G )
9	8	clssubg 19814	. . . . . . . . 9 |- ( ( G e. TopGrp /\ { ( 0g ` G ) } e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) )
10	2, 7, 9	syl2anc 661	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) )
11		tgptsmscls.b	. . . . . . . . 9 |- B = ( Base ` G )
12		eqid 2454	. . . . . . . . 9 |- ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) )
13	11, 12	eqger 15853	. . . . . . . 8 |- ( ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) -> ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) Er B )
14	10, 13	syl 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 19786	. . . . . . . . . . 11 |- ( G e. TopGrp -> G e. TopSp )
17	1, 16	syl 16	. . . . . . . . . 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 19840	. . . . . . . . 9 |- ( ph -> ( G tsums F ) C_ B )
21	20	sselda 3467	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. B )
22		tgptsmscls.x	. . . . . . . . . 10 |- ( ph -> X e. ( G tsums F ) )
23	20, 22	sseldd 3468	. . . . . . . . 9 |- ( ph -> X e. B )
24	23	adantr 465	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> X e. B )
25		eqid 2454	. . . . . . . . . 10 |- ( -g ` G ) = ( -g ` G )
26	15	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. CMnd )
27	18	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> A e. V )
28	19	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> F : A --> B )
29	22	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> X e. ( G tsums F ) )
30		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. ( G tsums F ) )
31	11, 25, 26, 2, 27, 28, 28, 29, 30	tsmssub 19858	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( X ( -g ` G ) x ) e. ( G tsums ( F oF ( -g ` G ) F ) ) )
32	28	ffvelrnda 5955	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( F ` k ) e. B )
33	28	feqmptd 5856	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> F = ( k e. A |-> ( F ` k ) ) )
34	27, 32, 32, 33, 33	offval2 6449	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( F oF ( -g ` G ) F ) = ( k e. A |-> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) ) )
35	4	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> G e. Grp )
36	11, 5, 25	grpsubid 15732	. . . . . . . . . . . . . 14 |- ( ( G e. Grp /\ ( F ` k ) e. B ) -> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) = ( 0g ` G ) )
37	35, 32, 36	syl2anc 661	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( ( F ` k ) ( -g ` G ) ( F ` k ) ) = ( 0g ` G ) )
38	37	mpteq2dva 4489	. . . . . . . . . . . 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 6219	. . . . . . . . . 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 16	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. TopSp )
42	11, 5	grpidcl 15688	. . . . . . . . . . . . . 14 |- ( G e. Grp -> ( 0g ` G ) e. B )
43	4, 42	syl 16	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( 0g ` G ) e. B )
44	43	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( G tsums F ) ) /\ k e. A ) -> ( 0g ` G ) e. B )
45		eqid 2454	. . . . . . . . . . . 12 |- ( k e. A |-> ( 0g ` G ) ) = ( k e. A |-> ( 0g ` G ) )
46	44, 45	fmptd 5979	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( k e. A |-> ( 0g ` G ) ) : A --> B )
47		fconstmpt 4993	. . . . . . . . . . . 12 |- ( A X. { ( 0g ` G ) } ) = ( k e. A |-> ( 0g ` G ) )
48		fvex 5812	. . . . . . . . . . . . . . 15 |- ( 0g ` G ) e. _V
49	48	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> ( 0g ` G ) e. _V )
50	18, 49	fczfsuppd 7752	. . . . . . . . . . . . 13 |- ( ph -> ( A X. { ( 0g ` G ) } ) finSupp ( 0g ` G ) )
51	50	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( A X. { ( 0g ` G ) } ) finSupp ( 0g ` G ) )
52	47, 51	syl5eqbrr 4437	. . . . . . . . . . 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 19844	. . . . . . . . . 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 16416	. . . . . . . . . . . . . 14 |- ( G e. CMnd -> G e. Mnd )
55	26, 54	syl 16	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Mnd )
56	5	gsumz 15633	. . . . . . . . . . . . 13 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) = ( 0g ` G ) )
57	55, 27, 56	syl2anc 661	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) = ( 0g ` G ) )
58	57	sneqd 4000	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( G tsums F ) ) -> { ( G gsumg ( k e. A |-> ( 0g ` G ) ) ) } = { ( 0g ` G ) } )
59	58	fveq2d 5806	. . . . . . . . . 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 2544	. . . . . . . 8 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( X ( -g ` G ) x ) e. ( ( cls ` J ) ` { ( 0g ` G ) } ) )
62		isabl 16405	. . . . . . . . . 10 |- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
63	4, 26, 62	sylanbrc 664	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> G e. Abel )
64	11	subgss 15804	. . . . . . . . . 10 |- ( ( ( cls ` J ) ` { ( 0g ` G ) } ) e. (SubGrp ` G ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) C_ B )
65	10, 64	syl 16	. . . . . . . . 9 |- ( ( ph /\ x e. ( G tsums F ) ) -> ( ( cls ` J ) ` { ( 0g ` G ) } ) C_ B )
66	11, 25, 12	eqgabl 16443	. . . . . . . . 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 661	. . . . . . . 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 1171	. . . . . . 7 |- ( ( ph /\ x e. ( G tsums F ) ) -> x ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) X )
69	14, 68	ersym 7226	. . . . . 6 |- ( ( ph /\ x e. ( G tsums F ) ) -> X ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) x )
70	12	releqg 15850	. . . . . . 7 |- Rel ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) )
71		relelec 7254	. . . . . . 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 212	. . . . 5 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) )
74		eqid 2454	. . . . . . 7 |- ( ( cls ` J ) ` { ( 0g ` G ) } ) = ( ( cls ` J ) ` { ( 0g ` G ) } )
75	11, 8, 5, 12, 74	snclseqg 19821	. . . . . 6 |- ( ( G e. TopGrp /\ X e. B ) -> [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( ( cls ` J ) ` { X } ) )
76	2, 24, 75	syl2anc 661	. . . . 5 |- ( ( ph /\ x e. ( G tsums F ) ) -> [ X ] ( G ~QG ( ( cls ` J ) ` { ( 0g ` G ) } ) ) = ( ( cls ` J ) ` { X } ) )
77	73, 76	eleqtrd 2544	. . . 4 |- ( ( ph /\ x e. ( G tsums F ) ) -> x e. ( ( cls ` J ) ` { X } ) )
78	77	ex 434	. . 3 |- ( ph -> ( x e. ( G tsums F ) -> x e. ( ( cls ` J ) ` { X } ) ) )
79	78	ssrdv 3473	. 2 |- ( ph -> ( G tsums F ) C_ ( ( cls ` J ) ` { X } ) )
80	11, 8, 15, 17, 18, 19, 22	tsmscls 19843	. 2 |- ( ph -> ( ( cls ` J ) ` { X } ) C_ ( G tsums F ) )
81	79, 80	eqssd 3484	1 |- ( ph -> ( G tsums F ) = ( ( cls ` J ) ` { X } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   _Vcvv 3078   C_ wss 3439   {csn 3988   class class class wbr 4403   |-> cmpt 4461   X. cxp 4949   Rel wrel 4956   -->wf 5525   ` cfv 5529  (class class class)co 6203   oFcof 6431   Er wer 7211   [cec 7212   finSupp cfsupp 7734   Basecbs 14295   TopOpenctopn 14482   0gc0g 14500   gsum_g cgsu 14501   Mndcmnd 15531   Grpcgrp 15532   -gcsg 15535  SubGrpcsubg 15797   ~_QG cqg 15799  CMndccmn 16401   Abelcabel 16402   TopSpctps 18636   clsccl 18757   TopGrpctgp 19777   tsums ctsu 19831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-ec 7216  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-0g 14502  df-gsum 14503  df-topgen 14504  df-mnd 15537  df-plusf 15538  df-mhm 15586  df-submnd 15587  df-grp 15667  df-minusg 15668  df-sbg 15669  df-subg 15800  df-eqg 15802  df-ghm 15867  df-cntz 15957  df-cmn 16403  df-abl 16404  df-fbas 17942  df-fg 17943  df-top 18638  df-bases 18640  df-topon 18641  df-topsp 18642  df-cld 18758  df-ntr 18759  df-cls 18760  df-nei 18837  df-cn 18966  df-cnp 18967  df-tx 19270  df-hmeo 19463  df-fil 19554  df-fm 19646  df-flim 19647  df-flf 19648  df-tmd 19778  df-tgp 19779  df-tsms 19832

This theorem is referenced by:  tgptsmscld  19860