Theorem gsumsub 15542

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 2436	. . . 4 |- ( +g ` G ) = ( +g ` G )
4		gsumsub.g	. . . . 5 |- ( ph -> G e. Abel )
5		ablcmn 15418	. . . . 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 2436	. . . . . . 7 |- ( inv g ` G ) = ( inv g ` G )
10		ablgrp 15417	. . . . . . . 8 |- ( G e. Abel -> G e. Grp )
11	4, 10	syl 16	. . . . . . 7 |- ( ph -> G e. Grp )
12	1, 9, 11	grpinvf1o 14861	. . . . . 6 |- ( ph -> ( inv g ` G ) : B -1-1-onto-> B )
13		f1of 5674	. . . . . 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 5600	. . . . 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 3469	. . . . . . . 8 |- ( k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) -> k e. A )
21		fvco3 5800	. . . . . . . 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 3367	. . . . . . . . . . 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 5864	. . . . . . . . 9 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( H ` k ) = .0. )
26	25	fveq2d 5732	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( inv g ` G ) ` ( H ` k ) ) = ( ( inv g ` G ) ` .0. ) )
27	2, 9	grpinvid 14856	. . . . . . . . . 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 2468	. . . . . . 7 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( inv g ` G ) ` ( H ` k ) ) = .0. )
31	22, 30	eqtrd 2468	. . . . . 6 |- ( ( ph /\ k e. ( A \ ( `' H " ( _V \ { .0. } ) ) ) ) -> ( ( ( inv g ` G ) o. H ) ` k ) = .0. )
32	17, 31	suppss 5863	. . . . 5 |- ( ph -> ( `' ( ( inv g ` G ) o. H ) " ( _V \ { .0. } ) ) C_ ( `' H " ( _V \ { .0. } ) ) )
33		ssfi 7329	. . . . 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 15528	. . 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 15541	. . . 4 |- ( ph -> ( G gsumg ( ( inv g ` G ) o. H ) ) = ( ( inv g ` G ) ` ( G gsumg H ) ) )
37	36	oveq2d 6097	. . 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 2468	. 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 5870	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. B )
40	15	ffvelrnda 5870	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( H ` k ) e. B )
41		gsumsub.m	. . . . . . 7 |- .- = ( -g ` G )
42	1, 3, 9, 41	grpsubval 14848	. . . . . 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 4295	. . . 4 |- ( ph -> ( k e. A |-> ( ( F ` k ) .- ( H ` k ) ) ) = ( k e. A |-> ( ( F ` k ) ( +g ` G ) ( ( inv g ` G ) ` ( H ` k ) ) ) ) )
45	8	feqmptd 5779	. . . . 5 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
46	15	feqmptd 5779	. . . . 5 |- ( ph -> H = ( k e. A |-> ( H ` k ) ) )
47	7, 39, 40, 45, 46	offval2 6322	. . . 4 |- ( ph -> ( F o F .- H ) = ( k e. A |-> ( ( F ` k ) .- ( H ` k ) ) ) )
48		fvex 5742	. . . . . 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 5779	. . . . . 6 |- ( ph -> ( inv g ` G ) = ( x e. B |-> ( ( inv g ` G ) ` x ) ) )
51		fveq2 5728	. . . . . 6 |- ( x = ( H ` k ) -> ( ( inv g ` G ) ` x ) = ( ( inv g ` G ) ` ( H ` k ) ) )
52	40, 46, 50, 51	fmptco 5901	. . . . 5 |- ( ph -> ( ( inv g ` G ) o. H ) = ( k e. A |-> ( ( inv g ` G ) ` ( H ` k ) ) ) )
53	7, 39, 49, 45, 52	offval2 6322	. . . 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 2478	. . 3 |- ( ph -> ( F o F .- H ) = ( F o F ( +g ` G ) ( ( inv g ` G ) o. H ) ) )
55	54	oveq2d 6097	. 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 15521	. . 3 |- ( ph -> ( G gsumg F ) e. B )
57	1, 2, 6, 7, 15, 19	gsumcl 15521	. . 3 |- ( ph -> ( G gsumg H ) e. B )
58	1, 3, 9, 41	grpsubval 14848	. . 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 2478	1 |- ( ph -> ( G gsumg ( F o F .- H ) ) = ( ( G gsumg F ) .- ( G gsumg H ) ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1652   e. wcel 1725   _Vcvv 2956   \ cdif 3317   C_ wss 3320   {csn 3814   e. cmpt 4266   `'ccnv 4877   "cima 4881   o. ccom 4882   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   o Fcof 6303   Fincfn 7109   Basecbs 13469   +g cplusg 13529   0gc0g 13723   gsum_g cgsu 13724   Grpcgrp 14685   inv gcminusg 14686   -gcsg 14688  CMndccmn 15412   Abelcabel 15413

This theorem is referenced by:  tsmsxplem2  18183

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415