Theorem gsumress 13658

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 2234	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
4		gsumress.o	. . . . . . . . . . 11 |- .+ = ( +g ` G )
5		eqid 2234	. . . . . . . . . . 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 13647	. . . . . . . . . 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 6065	. . . . . . . . . . . 12 |- ( y = .0. -> ( y .+ x ) = ( .0. .+ x ) )
9	8	eqeq1d 2243	. . . . . . . . . . 11 |- ( y = .0. -> ( ( y .+ x ) = x <-> ( .0. .+ x ) = x ) )
10	9	ovanraleqv 6082	. . . . . . . . . 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 3243	. . . . . . . . . 10 |- ( ph -> .0. e. B )
14		gsumress.c	. . . . . . . . . . 11 |- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
15	14	ralrimiva 2617	. . . . . . . . . 10 |- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
16	10, 13, 15	elrabd 2978	. . . . . . . . 9 |- ( ph -> .0. e. { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
17	7, 16	sseldd 3243	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` G ) } )
18		elsni 3712	. . . . . . . 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 13365	. . . . . . . . . . 11 |- ( ph -> S = ( Base ` H ) )
24	23, 12	basmexd 13357	. . . . . . . . . 10 |- ( ph -> H e. _V )
25		eqid 2234	. . . . . . . . . . 11 |- ( Base ` H ) = ( Base ` H )
26		eqid 2234	. . . . . . . . . . 11 |- ( 0g ` H ) = ( 0g ` H )
27		eqid 2234	. . . . . . . . . . 11 |- ( +g ` H ) = ( +g ` H )
28		eqid 2234	. . . . . . . . . . 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 13647	. . . . . . . . . 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 6082	. . . . . . . . . . 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 3242	. . . . . . . . . . . . 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 2617	. . . . . . . . . . 11 |- ( ph -> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
35	31, 12, 34	elrabd 2978	. . . . . . . . . 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 13355	. . . . . . . . . . . . . . . . . 18 |- Base Fn _V
38		funfvex 5692	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun Base /\ H e. dom Base ) -> ( Base ` H ) e. _V )
39	38	funfni 5463	. . . . . . . . . . . . . . . . . 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 2311	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. _V )
42	21, 36, 41, 1	ressplusgd 13426	. . . . . . . . . . . . . . 15 |- ( ph -> .+ = ( +g ` H ) )
43	42	oveqd 6075	. . . . . . . . . . . . . 14 |- ( ph -> ( y .+ x ) = ( y ( +g ` H ) x ) )
44	43	eqeq1d 2243	. . . . . . . . . . . . 13 |- ( ph -> ( ( y .+ x ) = x <-> ( y ( +g ` H ) x ) = x ) )
45	42	oveqd 6075	. . . . . . . . . . . . . 14 |- ( ph -> ( x .+ y ) = ( x ( +g ` H ) y ) )
46	45	eqeq1d 2243	. . . . . . . . . . . . 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 2759	. . . . . . . . . . 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 2810	. . . . . . . . . 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 2313	. . . . . . . . 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 3243	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` H ) } )
52		elsni 3712	. . . . . . . 8 |- ( .0. e. { ( 0g ` H ) } -> .0. = ( 0g ` H ) )
53	51, 52	syl 14	. . . . . . 7 |- ( ph -> .0. = ( 0g ` H ) )
54	19, 53	eqtr3d 2269	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
55	54	eqeq2d 2246	. . . . 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 10840	. . . . . . . . 9 |- ( ph -> seq m ( .+ , F ) = seq m ( ( +g ` H ) , F ) )
58	57	fveq1d 5677	. . . . . . . 8 |- ( ph -> ( seq m ( .+ , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
59	58	eqeq2d 2246	. . . . . . 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 2545	. . . . 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 1874	. . . 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 801	. . 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 5340	. 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 5527	. . 3 |- ( ph -> F : A --> B )
68	2, 3, 4, 1, 65, 67	igsumval 13653	. 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 5502	. . . 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 13653	. 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 2277	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   _Vcvv 2815   C_ wss 3214   (/)c0 3512   {csn 3694   iotacio 5315   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   ↾_s cress 13297   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-neg 8463  df-inn 9255  df-2 9313  df-z 9595  df-uz 9872  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-0g 13555  df-igsum 13556

This theorem is referenced by:  gsumsubm  13749