Theorem gsumzinv 16565

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 2454	. . 3 |- (oppg ` G ) = (oppg ` G )
5		gsumzinv.g	. . . 4 |- ( ph -> G e. Grp )
6		grpmnd 15670	. . . 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 15702	. . . . 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 5677	. . . 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 15995	. . . . . 6 |- ( G e. Grp -> I e. ( G GrpIso (oppg ` G ) ) )
16		gimghm 15912	. . . . . 6 |- ( I e. ( G GrpIso (oppg ` G ) ) -> I e. ( G GrpHom (oppg ` G ) ) )
17		ghmmhm 15877	. . . . . 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 2454	. . . . . 6 |- (Cntz ` (oppg ` G ) ) = (Cntz ` (oppg ` G ) )
21	3, 20	cntzmhm2 15977	. . . . 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 5454	. . . 4 |- ran ( I o. F ) = ( I " ran F )
24	23	fveq2i 5803	. . . . 5 |- ( Z ` ran ( I o. F ) ) = ( Z ` ( I " ran F ) )
25	4, 3	oppgcntz 15999	. . . . 5 |- ( Z ` ( I " ran F ) ) = ( (Cntz ` (oppg ` G ) ) ` ( I " ran F ) )
26	24, 25	eqtri 2483	. . . 4 |- ( Z ` ran ( I o. F ) ) = ( (Cntz ` (oppg ` G ) ) ` ( I " ran F ) )
27	22, 23, 26	3sstr4g 3506	. . 3 |- ( ph -> ran ( I o. F ) C_ ( Z ` ran ( I o. F ) ) )
28		fvex 5810	. . . . . 6 |- ( 0g ` G ) e. _V
29	2, 28	eqeltri 2538	. . . . 5 |- .0. e. _V
30	29	a1i 11	. . . 4 |- ( ph -> .0. e. _V )
31		fvex 5810	. . . . . 6 |- ( Base ` G ) e. _V
32	1, 31	eqeltri 2538	. . . . 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 15709	. . . . 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 7764	. . 3 |- ( ph -> ( I o. F ) finSupp .0. )
38	1, 2, 3, 4, 7, 8, 14, 27, 37	gsumzoppg 16563	. 2 |- ( ph -> ( (oppg ` G ) gsumg ( I o. F ) ) = ( G gsumg ( I o. F ) ) )
39	4	oppgmnd 15989	. . . 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 16553	. 2 |- ( ph -> ( (oppg ` G ) gsumg ( I o. F ) ) = ( I ` ( G gsumg F ) ) )
42	38, 41	eqtr3d 2497	1 |- ( ph -> ( G gsumg ( I o. F ) ) = ( I ` ( G gsumg F ) ) )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   _Vcvv 3078   C_ wss 3437   class class class wbr 4401   ran crn 4950   "cima 4952   o. ccom 4953   -->wf 5523   ` cfv 5527  (class class class)co 6201   finSupp cfsupp 7732   Basecbs 14293   0gc0g 14498   gsum_g cgsu 14499   Mndcmnd 15529   Grpcgrp 15530   invgcminusg 15531   MndHom cmhm 15582   GrpHom cghm 15864   GrpIso cgim 15905  Cntzccntz 15953  opp_gcoppg 15980

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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471

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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-tpos 6856  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-gsum 14501  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-ghm 15865  df-gim 15907  df-cntz 15955  df-oppg 15981  df-cmn 16401

This theorem is referenced by:  dprdfinv  16632