Theorem gsumress 12978

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 2193	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
4		gsumress.o	. . . . . . . . . . 11 |- .+ = ( +g ` G )
5		eqid 2193	. . . . . . . . . . 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 12967	. . . . . . . . . 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 5925	. . . . . . . . . . . 12 |- ( y = .0. -> ( y .+ x ) = ( .0. .+ x ) )
9	8	eqeq1d 2202	. . . . . . . . . . 11 |- ( y = .0. -> ( ( y .+ x ) = x <-> ( .0. .+ x ) = x ) )
10	9	ovanraleqv 5942	. . . . . . . . . 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 3180	. . . . . . . . . 10 |- ( ph -> .0. e. B )
14		gsumress.c	. . . . . . . . . . 11 |- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
15	14	ralrimiva 2567	. . . . . . . . . 10 |- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
16	10, 13, 15	elrabd 2918	. . . . . . . . 9 |- ( ph -> .0. e. { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
17	7, 16	sseldd 3180	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` G ) } )
18		elsni 3636	. . . . . . . 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 12686	. . . . . . . . . . 11 |- ( ph -> S = ( Base ` H ) )
24	23, 12	basmexd 12678	. . . . . . . . . 10 |- ( ph -> H e. _V )
25		eqid 2193	. . . . . . . . . . 11 |- ( Base ` H ) = ( Base ` H )
26		eqid 2193	. . . . . . . . . . 11 |- ( 0g ` H ) = ( 0g ` H )
27		eqid 2193	. . . . . . . . . . 11 |- ( +g ` H ) = ( +g ` H )
28		eqid 2193	. . . . . . . . . . 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 12967	. . . . . . . . . 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 5942	. . . . . . . . . . 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 3179	. . . . . . . . . . . . 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 2567	. . . . . . . . . . 11 |- ( ph -> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
35	31, 12, 34	elrabd 2918	. . . . . . . . . 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 12676	. . . . . . . . . . . . . . . . . 18 |- Base Fn _V
38		funfvex 5571	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun Base /\ H e. dom Base ) -> ( Base ` H ) e. _V )
39	38	funfni 5354	. . . . . . . . . . . . . . . . . 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 2270	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. _V )
42	21, 36, 41, 1	ressplusgd 12746	. . . . . . . . . . . . . . 15 |- ( ph -> .+ = ( +g ` H ) )
43	42	oveqd 5935	. . . . . . . . . . . . . 14 |- ( ph -> ( y .+ x ) = ( y ( +g ` H ) x ) )
44	43	eqeq1d 2202	. . . . . . . . . . . . 13 |- ( ph -> ( ( y .+ x ) = x <-> ( y ( +g ` H ) x ) = x ) )
45	42	oveqd 5935	. . . . . . . . . . . . . 14 |- ( ph -> ( x .+ y ) = ( x ( +g ` H ) y ) )
46	45	eqeq1d 2202	. . . . . . . . . . . . 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 2706	. . . . . . . . . . 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 2755	. . . . . . . . . 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 2272	. . . . . . . . 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 3180	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` H ) } )
52		elsni 3636	. . . . . . . 8 |- ( .0. e. { ( 0g ` H ) } -> .0. = ( 0g ` H ) )
53	51, 52	syl 14	. . . . . . 7 |- ( ph -> .0. = ( 0g ` H ) )
54	19, 53	eqtr3d 2228	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
55	54	eqeq2d 2205	. . . . 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 10525	. . . . . . . . 9 |- ( ph -> seq m ( .+ , F ) = seq m ( ( +g ` H ) , F ) )
58	57	fveq1d 5556	. . . . . . . 8 |- ( ph -> ( seq m ( .+ , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
59	58	eqeq2d 2205	. . . . . . 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 2495	. . . . 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 1836	. . . 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 5237	. 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 5416	. . 3 |- ( ph -> F : A --> B )
68	2, 3, 4, 1, 65, 67	igsumval 12973	. 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 5392	. . . 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 12973	. 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 2236	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   _Vcvv 2760   C_ wss 3153   (/)c0 3446   {csn 3618   iotacio 5213   Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   0gc0g 12867   gsum_g cgsu 12868

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-neg 8193  df-inn 8983  df-2 9041  df-z 9318  df-uz 9593  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-0g 12869  df-igsum 12870

This theorem is referenced by:  gsumsubm  13066