Theorem gzsumress 13689

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

Assertion

Ref	Expression
gzsumress	|- ( ph -> ( G gzsumgz F ) = ( H gzsumgz 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 gzsumress

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

Step	Hyp	Ref	Expression
1		gzsumress.g	. . . . . . . . . 10 |- ( ph -> G e. V )
2		gzsumress.b	. . . . . . . . . . 11 |- B = ( Base ` G )
3		eqid 2238	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
4		gzsumress.o	. . . . . . . . . . 11 |- .+ = ( +g ` G )
5		eqid 2238	. . . . . . . . . . 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 13681	. . . . . . . . . 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 6082	. . . . . . . . . . . 12 |- ( y = .0. -> ( y .+ x ) = ( .0. .+ x ) )
9	8	eqeq1d 2247	. . . . . . . . . . 11 |- ( y = .0. -> ( ( y .+ x ) = x <-> ( .0. .+ x ) = x ) )
10	9	ovanraleqv 6099	. . . . . . . . . 10 |- ( y = .0. -> ( A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
11		gzsumress.s	. . . . . . . . . . 11 |- ( ph -> S C_ B )
12		gzsumress.z	. . . . . . . . . . 11 |- ( ph -> .0. e. S )
13	11, 12	sseldd 3249	. . . . . . . . . 10 |- ( ph -> .0. e. B )
14		gzsumress.c	. . . . . . . . . . 11 |- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
15	14	ralrimiva 2623	. . . . . . . . . 10 |- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
16	10, 13, 15	elrabd 2984	. . . . . . . . 9 |- ( ph -> .0. e. { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
17	7, 16	sseldd 3249	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` G ) } )
18		elsni 3723	. . . . . . . 8 |- ( .0. e. { ( 0g ` G ) } -> .0. = ( 0g ` G ) )
19	17, 18	syl 14	. . . . . . 7 |- ( ph -> .0. = ( 0g ` G ) )
20		gzsumress.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 13399	. . . . . . . . . . 11 |- ( ph -> S = ( Base ` H ) )
24	23, 12	basmexd 13391	. . . . . . . . . 10 |- ( ph -> H e. _V )
25		eqid 2238	. . . . . . . . . . 11 |- ( Base ` H ) = ( Base ` H )
26		eqid 2238	. . . . . . . . . . 11 |- ( 0g ` H ) = ( 0g ` H )
27		eqid 2238	. . . . . . . . . . 11 |- ( +g ` H ) = ( +g ` H )
28		eqid 2238	. . . . . . . . . . 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 13681	. . . . . . . . . 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 6099	. . . . . . . . . . 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 3248	. . . . . . . . . . . . 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 2623	. . . . . . . . . . 11 |- ( ph -> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
35	31, 12, 34	elrabd 2984	. . . . . . . . . 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 13389	. . . . . . . . . . . . . . . . . 18 |- Base Fn _V
38		funfvex 5707	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun Base /\ H e. dom Base ) -> ( Base ` H ) e. _V )
39	38	funfni 5478	. . . . . . . . . . . . . . . . . 18 |- ( ( Base Fn _V /\ H e. _V ) -> ( Base ` H ) e. _V )
40	37, 24, 39	sylancr 418	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( Base ` H ) e. _V )
41	23, 40	eqeltrd 2315	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. _V )
42	21, 36, 41, 1	ressplusgd 13460	. . . . . . . . . . . . . . 15 |- ( ph -> .+ = ( +g ` H ) )
43	42	oveqd 6092	. . . . . . . . . . . . . 14 |- ( ph -> ( y .+ x ) = ( y ( +g ` H ) x ) )
44	43	eqeq1d 2247	. . . . . . . . . . . . 13 |- ( ph -> ( ( y .+ x ) = x <-> ( y ( +g ` H ) x ) = x ) )
45	42	oveqd 6092	. . . . . . . . . . . . . 14 |- ( ph -> ( x .+ y ) = ( x ( +g ` H ) y ) )
46	45	eqeq1d 2247	. . . . . . . . . . . . 13 |- ( ph -> ( ( x .+ y ) = x <-> ( x ( +g ` H ) y ) = x ) )
47	44, 46	anbi12d 477	. . . . . . . . . . . 12 |- ( ph -> ( ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) ) )
48	23, 47	raleqbidv 2765	. . . . . . . . . . 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 2816	. . . . . . . . . 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 2317	. . . . . . . . 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 3249	. . . . . . . 8 |- ( ph -> .0. e. { ( 0g ` H ) } )
52		elsni 3723	. . . . . . . 8 |- ( .0. e. { ( 0g ` H ) } -> .0. = ( 0g ` H ) )
53	51, 52	syl 14	. . . . . . 7 |- ( ph -> .0. = ( 0g ` H ) )
54	19, 53	eqtr3d 2273	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
55	54	eqeq2d 2250	. . . . 5 |- ( ph -> ( z = ( 0g ` G ) <-> z = ( 0g ` H ) ) )
56	55	anbi2d 468	. . . 4 |- ( ph -> ( ( A = (/) /\ z = ( 0g ` G ) ) <-> ( A = (/) /\ z = ( 0g ` H ) ) ) )
57	42	seqeq2d 10869	. . . . . . . . 9 |- ( ph -> seq m ( .+ , F ) = seq m ( ( +g ` H ) , F ) )
58	57	fveq1d 5692	. . . . . . . 8 |- ( ph -> ( seq m ( .+ , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
59	58	eqeq2d 2250	. . . . . . 7 |- ( ph -> ( z = ( seq m ( .+ , F ) ` n ) <-> z = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
60	59	anbi2d 468	. . . . . 6 |- ( ph -> ( ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
61	60	rexbidv 2551	. . . . 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 1878	. . . 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 805	. . 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 5355	. 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		gzsumress.a	. . 3 |- ( ph -> A e. X )
66		gzsumress.f	. . . 4 |- ( ph -> F : A --> S )
67	66, 11	fssd 5542	. . 3 |- ( ph -> F : A --> B )
68	2, 3, 4, 1, 65, 67	gzsumval 13687	. 2 |- ( ph -> ( G gzsumgz 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 5517	. . . 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	gzsumval 13687	. 2 |- ( ph -> ( H gzsumgz 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 2281	1 |- ( ph -> ( G gzsumgz F ) = ( H gzsumgz F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   C_ wss 3220   (/)c0 3520   {csn 3705   iotacio 5330   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-neg 8490  df-inn 9284  df-2 9342  df-z 9624  df-uz 9901  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-gzsum 13590

This theorem is referenced by:  gsumressfi  14144