Theorem dprdfinv 18418

Description: Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
eldprdi.0	|- .0. = ( 0g ` G )
eldprdi.w	|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
eldprdi.1	|- ( ph -> G dom DProd S )
eldprdi.2	|- ( ph -> dom S = I )
eldprdi.3	|- ( ph -> F e. W )
dprdfinv.b	|- N = ( invg ` G )

Assertion

Ref	Expression
dprdfinv	|- ( ph -> ( ( N o. F ) e. W /\ ( G gsumg ( N o. F ) ) = ( N ` ( G gsumg F ) ) ) )

Distinct variable groups:   h, F   h, i, G   h, I, i   h, N   .0. , h   S, h, i

Allowed substitution hints:   ph( h, i)   F( i)   N( i)   W( h, i)   .0. ( i)

Proof of Theorem dprdfinv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldprdi.1	. . . . . 6 |- ( ph -> G dom DProd S )
2		dprdgrp 18404	. . . . . 6 |- ( G dom DProd S -> G e. Grp )
3	1, 2	syl 17	. . . . 5 |- ( ph -> G e. Grp )
4		eqid 2622	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
5		dprdfinv.b	. . . . . 6 |- N = ( invg ` G )
6	4, 5	grpinvf 17466	. . . . 5 |- ( G e. Grp -> N : ( Base ` G ) --> ( Base ` G ) )
7	3, 6	syl 17	. . . 4 |- ( ph -> N : ( Base ` G ) --> ( Base ` G ) )
8		eldprdi.w	. . . . 5 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
9		eldprdi.2	. . . . 5 |- ( ph -> dom S = I )
10		eldprdi.3	. . . . 5 |- ( ph -> F e. W )
11	8, 1, 9, 10, 4	dprdff 18411	. . . 4 |- ( ph -> F : I --> ( Base ` G ) )
12		fcompt 6400	. . . 4 |- ( ( N : ( Base ` G ) --> ( Base ` G ) /\ F : I --> ( Base ` G ) ) -> ( N o. F ) = ( x e. I |-> ( N ` ( F ` x ) ) ) )
13	7, 11, 12	syl2anc 693	. . 3 |- ( ph -> ( N o. F ) = ( x e. I |-> ( N ` ( F ` x ) ) ) )
14	1, 9	dprdf2 18406	. . . . . 6 |- ( ph -> S : I --> (SubGrp ` G ) )
15	14	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
16	8, 1, 9, 10	dprdfcl 18412	. . . . 5 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( S ` x ) )
17	5	subginvcl 17603	. . . . 5 |- ( ( ( S ` x ) e. (SubGrp ` G ) /\ ( F ` x ) e. ( S ` x ) ) -> ( N ` ( F ` x ) ) e. ( S ` x ) )
18	15, 16, 17	syl2anc 693	. . . 4 |- ( ( ph /\ x e. I ) -> ( N ` ( F ` x ) ) e. ( S ` x ) )
19	1, 9	dprddomcld 18400	. . . . . 6 |- ( ph -> I e. _V )
20		mptexg 6484	. . . . . 6 |- ( I e. _V -> ( x e. I |-> ( N ` ( F ` x ) ) ) e. _V )
21	19, 20	syl 17	. . . . 5 |- ( ph -> ( x e. I |-> ( N ` ( F ` x ) ) ) e. _V )
22		funmpt 5926	. . . . . 6 |- Fun ( x e. I |-> ( N ` ( F ` x ) ) )
23	22	a1i 11	. . . . 5 |- ( ph -> Fun ( x e. I |-> ( N ` ( F ` x ) ) ) )
24	8, 1, 9, 10	dprdffsupp 18413	. . . . 5 |- ( ph -> F finSupp .0. )
25		ssid 3624	. . . . . . . . . 10 |- ( F supp .0. ) C_ ( F supp .0. )
26	25	a1i 11	. . . . . . . . 9 |- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) )
27		eldprdi.0	. . . . . . . . . . 11 |- .0. = ( 0g ` G )
28		fvex 6201	. . . . . . . . . . 11 |- ( 0g ` G ) e. _V
29	27, 28	eqeltri 2697	. . . . . . . . . 10 |- .0. e. _V
30	29	a1i 11	. . . . . . . . 9 |- ( ph -> .0. e. _V )
31	11, 26, 19, 30	suppssr 7326	. . . . . . . 8 |- ( ( ph /\ x e. ( I \ ( F supp .0. ) ) ) -> ( F ` x ) = .0. )
32	31	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ x e. ( I \ ( F supp .0. ) ) ) -> ( N ` ( F ` x ) ) = ( N ` .0. ) )
33	27, 5	grpinvid 17476	. . . . . . . . 9 |- ( G e. Grp -> ( N ` .0. ) = .0. )
34	3, 33	syl 17	. . . . . . . 8 |- ( ph -> ( N ` .0. ) = .0. )
35	34	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( I \ ( F supp .0. ) ) ) -> ( N ` .0. ) = .0. )
36	32, 35	eqtrd 2656	. . . . . 6 |- ( ( ph /\ x e. ( I \ ( F supp .0. ) ) ) -> ( N ` ( F ` x ) ) = .0. )
37	36, 19	suppss2 7329	. . . . 5 |- ( ph -> ( ( x e. I |-> ( N ` ( F ` x ) ) ) supp .0. ) C_ ( F supp .0. ) )
38		fsuppsssupp 8291	. . . . 5 |- ( ( ( ( x e. I |-> ( N ` ( F ` x ) ) ) e. _V /\ Fun ( x e. I |-> ( N ` ( F ` x ) ) ) ) /\ ( F finSupp .0. /\ ( ( x e. I |-> ( N ` ( F ` x ) ) ) supp .0. ) C_ ( F supp .0. ) ) ) -> ( x e. I |-> ( N ` ( F ` x ) ) ) finSupp .0. )
39	21, 23, 24, 37, 38	syl22anc 1327	. . . 4 |- ( ph -> ( x e. I |-> ( N ` ( F ` x ) ) ) finSupp .0. )
40	8, 1, 9, 18, 39	dprdwd 18410	. . 3 |- ( ph -> ( x e. I |-> ( N ` ( F ` x ) ) ) e. W )
41	13, 40	eqeltrd 2701	. 2 |- ( ph -> ( N o. F ) e. W )
42		eqid 2622	. . 3 |- (Cntz ` G ) = (Cntz ` G )
43	8, 1, 9, 10, 42	dprdfcntz 18414	. . 3 |- ( ph -> ran F C_ ( (Cntz ` G ) ` ran F ) )
44	4, 27, 42, 5, 3, 19, 11, 43, 24	gsumzinv 18345	. 2 |- ( ph -> ( G gsumg ( N o. F ) ) = ( N ` ( G gsumg F ) ) )
45	41, 44	jca 554	1 |- ( ph -> ( ( N o. F ) e. W /\ ( G gsumg ( N o. F ) ) = ( N ` ( G gsumg F ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   X_cixp 7908   finSupp cfsupp 8275   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   Grpcgrp 17422   invgcminusg 17423  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  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-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-cmn 18195  df-dprd 18394

This theorem is referenced by:  dprdfsub  18420