Theorem gsumress 13097

Description: The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither G nor H need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
gsumress.b	|- B = ( Base ` G )
gsumress.o	|- .+ = ( +g ` G )
gsumress.h	|- H = ( G ↾s S )
gsumress.g	|- ( ph -> G e. V )
gsumress.a	|- ( ph -> A e. X )
gsumress.s	|- ( ph -> S C_ B )
gsumress.f	|- ( ph -> F : A --> S )
gsumress.z	|- ( ph -> .0. e. S )
gsumress.c	|- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )

Assertion

Ref	Expression
gsumress	|- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Distinct variable groups:   x, B   x, G   ph, x   x, S   x, H   x, .+   x, .0.

Allowed substitution hints:   A( x)   F( x)   V( x)   X( x)

Proof of Theorem gsumress

Dummy variables m n y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumress.g	. . . . . . . . . 10 |- ( ph -> G e. V )
2		gsumress.b	. . . . . . . . . . 11 |- B = ( Base ` G )
3		eqid 2196	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
4		gsumress.o	. . . . . . . . . . 11 |- .+ = ( +g ` G )
5		eqid 2196	. . . . . . . . . . 11 |- { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } = { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) }
6	2, 3, 4, 5	mgmidsssn0 13086	. . . . . . . . . 10 |- ( G e. V -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { ( 0g ` G ) } )
7	1, 6	syl 14	. . . . . . . . 9 |- ( ph -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { ( 0g ` G ) } )
8		oveq1 5932	. . . . . . . . . . . 12 |- ( y = .0. -> ( y .+ x ) = ( .0. .+ x ) )
9	8	eqeq1d 2205	. . . . . . . . . . 11 |- ( y = .0. -> ( ( y .+ x ) = x <-> ( .0. .+ x ) = x ) )
10	9	ovanraleqv 5949	. . . . . . . . . 10 |- ( y = .0. -> ( A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
11		gsumress.s	. . . . . . . . . . 11 |- ( ph -> S C_ B )
12		gsumress.z	. . . . . . . . . . 11 |- ( ph -> .0. e. S )
13	11, 12	sseldd 3185	. . . . . . . . . 10 |- ( ph -> .0. e. B )
14		gsumress.c	. . . . . . . . . . 11 |- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
15	14	ralrimiva 2570	. . . . . . . . . 10 |- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
16	10, 13, 15	elrabd 2922	. . . . . . . . 9 |- ( ph -> .0. e. { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
17	7, 16	sseldd 3185	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` G ) } )
18		elsni 3641	. . . . . . . 8 |- ( .0. e. { ( 0g ` G ) } -> .0. = ( 0g ` G ) )
19	17, 18	syl 14	. . . . . . 7 |- ( ph -> .0. = ( 0g ` G ) )
20		gsumress.h	. . . . . . . . . . . . 13 |- H = ( G ↾s S )
21	20	a1i 9	. . . . . . . . . . . 12 |- ( ph -> H = ( G ↾s S ) )
22	2	a1i 9	. . . . . . . . . . . 12 |- ( ph -> B = ( Base ` G ) )
23	21, 22, 1, 11	ressbas2d 12771	. . . . . . . . . . 11 |- ( ph -> S = ( Base ` H ) )
24	23, 12	basmexd 12763	. . . . . . . . . 10 |- ( ph -> H e. _V )
25		eqid 2196	. . . . . . . . . . 11 |- ( Base ` H ) = ( Base ` H )
26		eqid 2196	. . . . . . . . . . 11 |- ( 0g ` H ) = ( 0g ` H )
27		eqid 2196	. . . . . . . . . . 11 |- ( +g ` H ) = ( +g ` H )
28		eqid 2196	. . . . . . . . . . 11 |- { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } = { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) }
29	25, 26, 27, 28	mgmidsssn0 13086	. . . . . . . . . 10 |- ( H e. _V -> { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } C_ { ( 0g ` H ) } )
30	24, 29	syl 14	. . . . . . . . 9 |- ( ph -> { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } C_ { ( 0g ` H ) } )
31	9	ovanraleqv 5949	. . . . . . . . . . 11 |- ( y = .0. -> ( A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
32	11	sselda 3184	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. S ) -> x e. B )
33	32, 14	syldan 282	. . . . . . . . . . . 12 |- ( ( ph /\ x e. S ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
34	33	ralrimiva 2570	. . . . . . . . . . 11 |- ( ph -> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
35	31, 12, 34	elrabd 2922	. . . . . . . . . 10 |- ( ph -> .0. e. { y e. S | A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
36	4	a1i 9	. . . . . . . . . . . . . . . 16 |- ( ph -> .+ = ( +g ` G ) )
37		basfn 12761	. . . . . . . . . . . . . . . . . 18 |- Base Fn _V
38		funfvex 5578	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun Base /\ H e. dom Base ) -> ( Base ` H ) e. _V )
39	38	funfni 5361	. . . . . . . . . . . . . . . . . 18 |- ( ( Base Fn _V /\ H e. _V ) -> ( Base ` H ) e. _V )
40	37, 24, 39	sylancr 414	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( Base ` H ) e. _V )
41	23, 40	eqeltrd 2273	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. _V )
42	21, 36, 41, 1	ressplusgd 12831	. . . . . . . . . . . . . . 15 |- ( ph -> .+ = ( +g ` H ) )
43	42	oveqd 5942	. . . . . . . . . . . . . 14 |- ( ph -> ( y .+ x ) = ( y ( +g ` H ) x ) )
44	43	eqeq1d 2205	. . . . . . . . . . . . 13 |- ( ph -> ( ( y .+ x ) = x <-> ( y ( +g ` H ) x ) = x ) )
45	42	oveqd 5942	. . . . . . . . . . . . . 14 |- ( ph -> ( x .+ y ) = ( x ( +g ` H ) y ) )
46	45	eqeq1d 2205	. . . . . . . . . . . . 13 |- ( ph -> ( ( x .+ y ) = x <-> ( x ( +g ` H ) y ) = x ) )
47	44, 46	anbi12d 473	. . . . . . . . . . . 12 |- ( ph -> ( ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) ) )
48	23, 47	raleqbidv 2709	. . . . . . . . . . 11 |- ( ph -> ( A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) ) )
49	23, 48	rabeqbidv 2758	. . . . . . . . . 10 |- ( ph -> { y e. S | A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } = { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } )
50	35, 49	eleqtrd 2275	. . . . . . . . 9 |- ( ph -> .0. e. { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } )
51	30, 50	sseldd 3185	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` H ) } )
52		elsni 3641	. . . . . . . 8 |- ( .0. e. { ( 0g ` H ) } -> .0. = ( 0g ` H ) )
53	51, 52	syl 14	. . . . . . 7 |- ( ph -> .0. = ( 0g ` H ) )
54	19, 53	eqtr3d 2231	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
55	54	eqeq2d 2208	. . . . 5 |- ( ph -> ( z = ( 0g ` G ) <-> z = ( 0g ` H ) ) )
56	55	anbi2d 464	. . . 4 |- ( ph -> ( ( A = (/) /\ z = ( 0g ` G ) ) <-> ( A = (/) /\ z = ( 0g ` H ) ) ) )
57	42	seqeq2d 10563	. . . . . . . . 9 |- ( ph -> seq m ( .+ , F ) = seq m ( ( +g ` H ) , F ) )
58	57	fveq1d 5563	. . . . . . . 8 |- ( ph -> ( seq m ( .+ , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
59	58	eqeq2d 2208	. . . . . . 7 |- ( ph -> ( z = ( seq m ( .+ , F ) ` n ) <-> z = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
60	59	anbi2d 464	. . . . . 6 |- ( ph -> ( ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
61	60	rexbidv 2498	. . . . 5 |- ( ph -> ( E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
62	61	exbidv 1839	. . . 4 |- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
63	56, 62	orbi12d 794	. . 3 |- ( ph -> ( ( ( A = (/) /\ z = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) <-> ( ( A = (/) /\ z = ( 0g ` H ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) )
64	63	iotabidv 5242	. 2 |- ( ph -> ( iota z ( ( A = (/) /\ z = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) ) = ( iota z ( ( A = (/) /\ z = ( 0g ` H ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) )
65		gsumress.a	. . 3 |- ( ph -> A e. X )
66		gsumress.f	. . . 4 |- ( ph -> F : A --> S )
67	66, 11	fssd 5423	. . 3 |- ( ph -> F : A --> B )
68	2, 3, 4, 1, 65, 67	igsumval 13092	. 2 |- ( ph -> ( G gsumg F ) = ( iota z ( ( A = (/) /\ z = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) ) )
69	23	feq3d 5399	. . . 4 |- ( ph -> ( F : A --> S <-> F : A --> ( Base ` H ) ) )
70	66, 69	mpbid 147	. . 3 |- ( ph -> F : A --> ( Base ` H ) )
71	25, 26, 27, 24, 65, 70	igsumval 13092	. 2 |- ( ph -> ( H gsumg F ) = ( iota z ( ( A = (/) /\ z = ( 0g ` H ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) )
72	64, 68, 71	3eqtr4d 2239	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476   {crab 2479   _Vcvv 2763   C_ wss 3157   (/)c0 3451   {csn 3623   iotacio 5218   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   ZZ>=cuz 9618   ...cfz 10100   seqcseq 10556   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   0gc0g 12958   gsum_g cgsu 12959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-neg 8217  df-inn 9008  df-2 9066  df-z 9344  df-uz 9619  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-igsum 12961

This theorem is referenced by:  gsumsubm  13196