Theorem gsumzinv 16948

Description: Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
gsumzinv.b	|- B = ( Base ` G )
gsumzinv.0	|- .0. = ( 0g ` G )
gsumzinv.z	|- Z = (Cntz ` G )
gsumzinv.i	|- I = ( invg ` G )
gsumzinv.g	|- ( ph -> G e. Grp )
gsumzinv.a	|- ( ph -> A e. V )
gsumzinv.f	|- ( ph -> F : A --> B )
gsumzinv.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumzinv.n	|- ( ph -> F finSupp .0. )

Assertion

Ref	Expression
gsumzinv	|- ( ph -> ( G gsumg ( I o. F ) ) = ( I ` ( G gsumg F ) ) )

Proof of Theorem gsumzinv

Step	Hyp	Ref	Expression
1		gsumzinv.b	. . 3 |- B = ( Base ` G )
2		gsumzinv.0	. . 3 |- .0. = ( 0g ` G )
3		gsumzinv.z	. . 3 |- Z = (Cntz ` G )
4		eqid 2443	. . 3 |- (oppg ` G ) = (oppg ` G )
5		gsumzinv.g	. . . 4 |- ( ph -> G e. Grp )
6		grpmnd 16041	. . . 4 |- ( G e. Grp -> G e. Mnd )
7	5, 6	syl 16	. . 3 |- ( ph -> G e. Mnd )
8		gsumzinv.a	. . 3 |- ( ph -> A e. V )
9		gsumzinv.i	. . . . . 6 |- I = ( invg ` G )
10	1, 9	grpinvf 16073	. . . . 5 |- ( G e. Grp -> I : B --> B )
11	5, 10	syl 16	. . . 4 |- ( ph -> I : B --> B )
12		gsumzinv.f	. . . 4 |- ( ph -> F : A --> B )
13		fco 5731	. . . 4 |- ( ( I : B --> B /\ F : A --> B ) -> ( I o. F ) : A --> B )
14	11, 12, 13	syl2anc 661	. . 3 |- ( ph -> ( I o. F ) : A --> B )
15	4, 9	invoppggim 16374	. . . . . 6 |- ( G e. Grp -> I e. ( G GrpIso (oppg ` G ) ) )
16		gimghm 16291	. . . . . 6 |- ( I e. ( G GrpIso (oppg ` G ) ) -> I e. ( G GrpHom (oppg ` G ) ) )
17		ghmmhm 16256	. . . . . 6 |- ( I e. ( G GrpHom (oppg ` G ) ) -> I e. ( G MndHom (oppg ` G ) ) )
18	5, 15, 16, 17	4syl 21	. . . . 5 |- ( ph -> I e. ( G MndHom (oppg ` G ) ) )
19		gsumzinv.c	. . . . 5 |- ( ph -> ran F C_ ( Z ` ran F ) )
20		eqid 2443	. . . . . 6 |- (Cntz ` (oppg ` G ) ) = (Cntz ` (oppg ` G ) )
21	3, 20	cntzmhm2 16356	. . . . 5 |- ( ( I e. ( G MndHom (oppg ` G ) ) /\ ran F C_ ( Z ` ran F ) ) -> ( I " ran F ) C_ ( (Cntz ` (oppg ` G ) ) ` ( I " ran F ) ) )
22	18, 19, 21	syl2anc 661	. . . 4 |- ( ph -> ( I " ran F ) C_ ( (Cntz ` (oppg ` G ) ) ` ( I " ran F ) ) )
23		rnco2 5504	. . . 4 |- ran ( I o. F ) = ( I " ran F )
24	23	fveq2i 5859	. . . . 5 |- ( Z ` ran ( I o. F ) ) = ( Z ` ( I " ran F ) )
25	4, 3	oppgcntz 16378	. . . . 5 |- ( Z ` ( I " ran F ) ) = ( (Cntz ` (oppg ` G ) ) ` ( I " ran F ) )
26	24, 25	eqtri 2472	. . . 4 |- ( Z ` ran ( I o. F ) ) = ( (Cntz ` (oppg ` G ) ) ` ( I " ran F ) )
27	22, 23, 26	3sstr4g 3530	. . 3 |- ( ph -> ran ( I o. F ) C_ ( Z ` ran ( I o. F ) ) )
28		fvex 5866	. . . . . 6 |- ( 0g ` G ) e. _V
29	2, 28	eqeltri 2527	. . . . 5 |- .0. e. _V
30	29	a1i 11	. . . 4 |- ( ph -> .0. e. _V )
31		fvex 5866	. . . . . 6 |- ( Base ` G ) e. _V
32	1, 31	eqeltri 2527	. . . . 5 |- B e. _V
33	32	a1i 11	. . . 4 |- ( ph -> B e. _V )
34		gsumzinv.n	. . . 4 |- ( ph -> F finSupp .0. )
35	2, 9	grpinvid 16080	. . . . 5 |- ( G e. Grp -> ( I ` .0. ) = .0. )
36	5, 35	syl 16	. . . 4 |- ( ph -> ( I ` .0. ) = .0. )
37	30, 12, 11, 8, 33, 34, 36	fsuppco2 7864	. . 3 |- ( ph -> ( I o. F ) finSupp .0. )
38	1, 2, 3, 4, 7, 8, 14, 27, 37	gsumzoppg 16946	. 2 |- ( ph -> ( (oppg ` G ) gsumg ( I o. F ) ) = ( G gsumg ( I o. F ) ) )
39	4	oppgmnd 16368	. . . 4 |- ( G e. Mnd -> (oppg ` G ) e. Mnd )
40	7, 39	syl 16	. . 3 |- ( ph -> (oppg ` G ) e. Mnd )
41	1, 3, 7, 40, 8, 18, 12, 19, 2, 34	gsumzmhm 16936	. 2 |- ( ph -> ( (oppg ` G ) gsumg ( I o. F ) ) = ( I ` ( G gsumg F ) ) )
42	38, 41	eqtr3d 2486	1 |- ( ph -> ( G gsumg ( I o. F ) ) = ( I ` ( G gsumg F ) ) )

Syntax hints:   -> wi 4   = wceq 1383   e. wcel 1804   _Vcvv 3095   C_ wss 3461   class class class wbr 4437   ran crn 4990   "cima 4992   o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281   finSupp cfsupp 7831   Basecbs 14614   0gc0g 14819   gsum_g cgsu 14820   Mndcmnd 15898   MndHom cmhm 15943   Grpcgrp 16032   invgcminusg 16033   GrpHom cghm 16243   GrpIso cgim 16284  Cntzccntz 16332  opp_gcoppg 16359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-seq 12090  df-hash 12388  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-0g 14821  df-gsum 14822  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-mhm 15945  df-submnd 15946  df-grp 16036  df-minusg 16037  df-ghm 16244  df-gim 16286  df-cntz 16334  df-oppg 16360  df-cmn 16779

This theorem is referenced by:  dprdfinv  17038