Theorem gsumsub 15501

Description: The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.)

Hypotheses

Ref	Expression
gsumsub.b	|- B = ( Base ` G )
gsumsub.z	|- .0. = ( 0g ` G )
gsumsub.m	|- .- = ( -g ` G )
gsumsub.g	|- ( ph -> G e. Abel )
gsumsub.a	|- ( ph -> A e. V )
gsumsub.f	|- ( ph -> F : A --> B )
gsumsub.h	|- ( ph -> H : A --> B )
gsumsub.fn	|- ( ph -> ( `' F " ( _V \ { .0. } ) ) e. Fin )
gsumsub.hn	|- ( ph -> ( `' H " ( _V \ { .0. } ) ) e. Fin )

Assertion

Ref	Expression
gsumsub	|- ( ph -> ( G gsumg ( F o F .- H ) ) = ( ( G gsumg F ) .- ( G gsumg H ) ) )

Proof of Theorem gsumsub

Dummy variables k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumsub.b	. . . 4 |- B = ( Base ` G )
2		gsumsub.z	. . . 4 |- .0. = ( 0g ` G )
3		eqid 2408	. . . 4 |- ( +g ` G ) = ( +g ` G )
4		gsumsub.g	. . . . 5 |- ( ph -> G e. Abel )
5		ablcmn 15377	. . . . 5 |- ( G e. Abel -> G e. CMnd )
6	4, 5	syl 16	. . . 4 |- ( ph -> G e. CMnd )
7		gsumsub.a	. . . 4 |- ( ph -> A e. V )
8		gsumsub.f	. . . 4 |- ( ph -> F : A --> B )
9		eqid 2408	. . . . . . 7 |- ( inv g ` G ) = ( inv g ` G )
10		ablgrp 15376	. . . . . . . 8 |- ( G e. Abel -> G e. Grp )
11	4, 10	syl 16	. . . . . . 7 |- ( ph -> G e. Grp )
12	1, 9, 11	grpinvf1o 14820	. . . . . 6 |- ( ph -> ( inv g ` G ) : B -1-1-onto-> B )
13		f1of 5637	. . . . . 6 |- ( ( inv g ` G ) : B -1-1-onto-> B -> ( inv g ` G ) : B --> B )
14	12, 13	syl 16	. . . . 5 |- ( ph -> ( inv g ` G ) : B --> B )
15		gsumsub.h	. . . . 5 |- ( ph -> H : A --> B )
16		fco 5563	. . . . 5 |- ( ( ( inv g ` G ) : B --> B /\ H : A --> B ) -> ( ( inv g ` G ) o. H ) : A --> B )
17	14, 15, 16	syl2anc 643	. . . 4 |- ( ph -> ( ( inv g ` G ) o. H ) : A --> B )
18		gsumsub.fn	. . . 4 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) e. Fin )
19		gsumsub.hn	. . . . 5 |- ( ph -> ( `' H " ( _V \ { .0. } ) ) e. Fin )
20		eldifi 3433	. . . . . . . 8 |- ( k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) -> k e. A )
21		fvco3 5763	. . . . . . . 8 |- ( ( H : A --> B /\ k e. A ) -> ( ( ( inv g ` G ) o. H ) ` k ) = ( ( inv g ` G ) ` ( H ` k ) ) )
22	15, 20, 21	syl2an 464	. . . . . . 7 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( ( inv g ` G ) o. H ) ` k ) = ( ( inv g ` G ) ` ( H ` k ) ) )
23		ssid 3331	. . . . . . . . . . 11 |- ( `' H " ( _V \ { .0. } ) ) C_ ( `' H " ( _V \ { .0. } ) )
24	23	a1i 11	. . . . . . . . . 10 |- ( ph -> ( `' H " ( _V \ { .0. } ) ) C_ ( `' H " ( _V \ { .0. } ) ) )
25	15, 24	suppssr 5827	. . . . . . . . 9 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( H ` k ) = .0. )
26	25	fveq2d 5695	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( inv g ` G ) ` ( H ` k ) ) = ( ( inv g ` G ) ` .0. ) )
27	2, 9	grpinvid 14815	. . . . . . . . . 10 |- ( G e. Grp -> ( ( inv g ` G ) ` .0. ) = .0. )
28	11, 27	syl 16	. . . . . . . . 9 |- ( ph -> ( ( inv g ` G ) ` .0. ) = .0. )
29	28	adantr 452	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( inv g ` G ) ` .0. ) = .0. )
30	26, 29	eqtrd 2440	. . . . . . 7 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( inv g ` G ) ` ( H ` k ) ) = .0. )
31	22, 30	eqtrd 2440	. . . . . 6 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( ( inv g ` G ) o. H ) ` k ) = .0. )
32	17, 31	suppss 5826	. . . . 5 |- ( ph -> ( `' ( ( inv g ` G ) o. H ) " ( _V \ { .0. } ) ) C_ ( `' H " ( _V \ { .0. } ) ) )
33		ssfi 7292	. . . . 5 |- ( ( ( `' H " ( _V \ { .0. } ) ) e. Fin /\ ( `' ( ( inv g ` G ) o. H ) " ( _V \ { .0. } ) ) C_ ( `' H " ( _V \ { .0. } ) ) ) -> ( `' ( ( inv g ` G ) o. H ) " ( _V \ { .0. } ) ) e. Fin )
34	19, 32, 33	syl2anc 643	. . . 4 |- ( ph -> ( `' ( ( inv g ` G ) o. H ) " ( _V \ { .0. } ) ) e. Fin )
35	1, 2, 3, 6, 7, 8, 17, 18, 34	gsumadd 15487	. . 3 |- ( ph -> ( G gsumg ( F o F ( +g ` G ) ( ( inv g ` G ) o. H ) ) ) = ( ( G gsumg F ) ( +g ` G ) ( G gsumg ( ( inv g ` G ) o. H ) ) ) )
36	1, 2, 9, 4, 7, 15, 19	gsuminv 15500	. . . 4 |- ( ph -> ( G gsumg ( ( inv g ` G ) o. H ) ) = ( ( inv g ` G ) ` ( G gsumg H ) ) )
37	36	oveq2d 6060	. . 3 |- ( ph -> ( ( G gsumg F ) ( +g ` G ) ( G gsumg ( ( inv g ` G ) o. H ) ) ) = ( ( G gsumg F ) ( +g ` G ) ( ( inv g ` G ) ` ( G gsumg H ) ) ) )
38	35, 37	eqtrd 2440	. 2 |- ( ph -> ( G gsumg ( F o F ( +g ` G ) ( ( inv g ` G ) o. H ) ) ) = ( ( G gsumg F ) ( +g ` G ) ( ( inv g ` G ) ` ( G gsumg H ) ) ) )
39	8	ffvelrnda 5833	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. B )
40	15	ffvelrnda 5833	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( H ` k ) e. B )
41		gsumsub.m	. . . . . . 7 |- .- = ( -g ` G )
42	1, 3, 9, 41	grpsubval 14807	. . . . . 6 |- ( ( ( F ` k ) e. B /\ ( H ` k ) e. B ) -> ( ( F ` k ) .- ( H ` k ) ) = ( ( F ` k ) ( +g ` G ) ( ( inv g ` G ) ` ( H ` k ) ) ) )
43	39, 40, 42	syl2anc 643	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( F ` k ) .- ( H ` k ) ) = ( ( F ` k ) ( +g ` G ) ( ( inv g ` G ) ` ( H ` k ) ) ) )
44	43	mpteq2dva 4259	. . . 4 |- ( ph -> ( k e. A |-> ( ( F ` k ) .- ( H ` k ) ) ) = ( k e. A |-> ( ( F ` k ) ( +g ` G ) ( ( inv g ` G ) ` ( H ` k ) ) ) ) )
45	8	feqmptd 5742	. . . . 5 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
46	15	feqmptd 5742	. . . . 5 |- ( ph -> H = ( k e. A |-> ( H ` k ) ) )
47	7, 39, 40, 45, 46	offval2 6285	. . . 4 |- ( ph -> ( F o F .- H ) = ( k e. A |-> ( ( F ` k ) .- ( H ` k ) ) ) )
48		fvex 5705	. . . . . 6 |- ( ( inv g ` G ) ` ( H ` k ) ) e. _V
49	48	a1i 11	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( inv g ` G ) ` ( H ` k ) ) e. _V )
50	14	feqmptd 5742	. . . . . 6 |- ( ph -> ( inv g ` G ) = ( x e. B |-> ( ( inv g ` G ) ` x ) ) )
51		fveq2 5691	. . . . . 6 |- ( x = ( H ` k ) -> ( ( inv g ` G ) ` x ) = ( ( inv g ` G ) ` ( H ` k ) ) )
52	40, 46, 50, 51	fmptco 5864	. . . . 5 |- ( ph -> ( ( inv g ` G ) o. H ) = ( k e. A |-> ( ( inv g ` G ) ` ( H ` k ) ) ) )
53	7, 39, 49, 45, 52	offval2 6285	. . . 4 |- ( ph -> ( F o F ( +g ` G ) ( ( inv g ` G ) o. H ) ) = ( k e. A |-> ( ( F ` k ) ( +g ` G ) ( ( inv g ` G ) ` ( H ` k ) ) ) ) )
54	44, 47, 53	3eqtr4d 2450	. . 3 |- ( ph -> ( F o F .- H ) = ( F o F ( +g ` G ) ( ( inv g ` G ) o. H ) ) )
55	54	oveq2d 6060	. 2 |- ( ph -> ( G gsumg ( F o F .- H ) ) = ( G gsumg ( F o F ( +g ` G ) ( ( inv g ` G ) o. H ) ) ) )
56	1, 2, 6, 7, 8, 18	gsumcl 15480	. . 3 |- ( ph -> ( G gsumg F ) e. B )
57	1, 2, 6, 7, 15, 19	gsumcl 15480	. . 3 |- ( ph -> ( G gsumg H ) e. B )
58	1, 3, 9, 41	grpsubval 14807	. . 3 |- ( ( ( G gsumg F ) e. B /\ ( G gsumg H ) e. B ) -> ( ( G gsumg F ) .- ( G gsumg H ) ) = ( ( G gsumg F ) ( +g ` G ) ( ( inv g ` G ) ` ( G gsumg H ) ) ) )
59	56, 57, 58	syl2anc 643	. 2 |- ( ph -> ( ( G gsumg F ) .- ( G gsumg H ) ) = ( ( G gsumg F ) ( +g ` G ) ( ( inv g ` G ) ` ( G gsumg H ) ) ) )
60	38, 55, 59	3eqtr4d 2450	1 |- ( ph -> ( G gsumg ( F o F .- H ) ) = ( ( G gsumg F ) .- ( G gsumg H ) ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2920   \ cdif 3281   C_ wss 3284   {csn 3778   e. cmpt 4230   `'ccnv 4840   "cima 4844   o. ccom 4845   -->wf 5413   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6044   o Fcof 6266   Fincfn 7072   Basecbs 13428   +g cplusg 13488   0gc0g 13682   gsum_g cgsu 13683   Grpcgrp 14644   inv gcminusg 14645   -gcsg 14647  CMndccmn 15371   Abelcabel 15372

This theorem is referenced by:  tsmsxplem2  18140

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-fzo 11095  df-seq 11283  df-hash 11578  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-0g 13686  df-gsum 13687  df-mnd 14649  df-mhm 14697  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-ghm 14963  df-cntz 15075  df-cmn 15373  df-abl 15374