Theorem gsumress 17276

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

Step	Hyp	Ref	Expression
1		gsumress.s	. . . . . . . . 9 |- ( ph -> S C_ B )
2		gsumress.z	. . . . . . . . 9 |- ( ph -> .0. e. S )
3	1, 2	sseldd 3604	. . . . . . . 8 |- ( ph -> .0. e. B )
4		gsumress.c	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
5	4	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
6		oveq1 6657	. . . . . . . . . . . 12 |- ( y = .0. -> ( y .+ x ) = ( .0. .+ x ) )
7	6	eqeq1d 2624	. . . . . . . . . . 11 |- ( y = .0. -> ( ( y .+ x ) = x <-> ( .0. .+ x ) = x ) )
8		oveq2 6658	. . . . . . . . . . . 12 |- ( y = .0. -> ( x .+ y ) = ( x .+ .0. ) )
9	8	eqeq1d 2624	. . . . . . . . . . 11 |- ( y = .0. -> ( ( x .+ y ) = x <-> ( x .+ .0. ) = x ) )
10	7, 9	anbi12d 747	. . . . . . . . . 10 |- ( y = .0. -> ( ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
11	10	ralbidv 2986	. . . . . . . . 9 |- ( y = .0. -> ( A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
12	11	elrab 3363	. . . . . . . 8 |- ( .0. e. { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } <-> ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
13	3, 5, 12	sylanbrc 698	. . . . . . 7 |- ( ph -> .0. e. { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
14	13	snssd 4340	. . . . . 6 |- ( ph -> { .0. } C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
15		gsumress.g	. . . . . . . 8 |- ( ph -> G e. V )
16		gsumress.b	. . . . . . . . 9 |- B = ( Base ` G )
17		eqid 2622	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
18		gsumress.o	. . . . . . . . 9 |- .+ = ( +g ` G )
19		eqid 2622	. . . . . . . . 9 |- { 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 ) }
20	16, 17, 18, 19	mgmidsssn0 17269	. . . . . . . 8 |- ( G e. V -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { ( 0g ` G ) } )
21	15, 20	syl 17	. . . . . . 7 |- ( ph -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { ( 0g ` G ) } )
22	21, 13	sseldd 3604	. . . . . . . . 9 |- ( ph -> .0. e. { ( 0g ` G ) } )
23		elsni 4194	. . . . . . . . 9 |- ( .0. e. { ( 0g ` G ) } -> .0. = ( 0g ` G ) )
24	22, 23	syl 17	. . . . . . . 8 |- ( ph -> .0. = ( 0g ` G ) )
25	24	sneqd 4189	. . . . . . 7 |- ( ph -> { .0. } = { ( 0g ` G ) } )
26	21, 25	sseqtr4d 3642	. . . . . 6 |- ( ph -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { .0. } )
27	14, 26	eqssd 3620	. . . . 5 |- ( ph -> { .0. } = { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
28	1	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> x e. B )
29	28, 4	syldan 487	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
30	29	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
31	10	ralbidv 2986	. . . . . . . . . 10 |- ( y = .0. -> ( A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
32	31	elrab 3363	. . . . . . . . 9 |- ( .0. e. { y e. S | A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } <-> ( .0. e. S /\ A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
33	2, 30, 32	sylanbrc 698	. . . . . . . 8 |- ( ph -> .0. e. { y e. S | A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } )
34		gsumress.h	. . . . . . . . . . 11 |- H = ( G ↾s S )
35	34, 16	ressbas2 15931	. . . . . . . . . 10 |- ( S C_ B -> S = ( Base ` H ) )
36	1, 35	syl 17	. . . . . . . . 9 |- ( ph -> S = ( Base ` H ) )
37		fvex 6201	. . . . . . . . . . . . . . 15 |- ( Base ` H ) e. _V
38	36, 37	syl6eqel 2709	. . . . . . . . . . . . . 14 |- ( ph -> S e. _V )
39	34, 18	ressplusg 15993	. . . . . . . . . . . . . 14 |- ( S e. _V -> .+ = ( +g ` H ) )
40	38, 39	syl 17	. . . . . . . . . . . . 13 |- ( ph -> .+ = ( +g ` H ) )
41	40	oveqd 6667	. . . . . . . . . . . 12 |- ( ph -> ( y .+ x ) = ( y ( +g ` H ) x ) )
42	41	eqeq1d 2624	. . . . . . . . . . 11 |- ( ph -> ( ( y .+ x ) = x <-> ( y ( +g ` H ) x ) = x ) )
43	40	oveqd 6667	. . . . . . . . . . . 12 |- ( ph -> ( x .+ y ) = ( x ( +g ` H ) y ) )
44	43	eqeq1d 2624	. . . . . . . . . . 11 |- ( ph -> ( ( x .+ y ) = x <-> ( x ( +g ` H ) y ) = x ) )
45	42, 44	anbi12d 747	. . . . . . . . . 10 |- ( ph -> ( ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) ) )
46	36, 45	raleqbidv 3152	. . . . . . . . 9 |- ( 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 ) ) )
47	36, 46	rabeqbidv 3195	. . . . . . . 8 |- ( 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 ) } )
48	33, 47	eleqtrd 2703	. . . . . . 7 |- ( ph -> .0. e. { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } )
49	48	snssd 4340	. . . . . 6 |- ( ph -> { .0. } C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } )
50		ovex 6678	. . . . . . . . . 10 |- ( G ↾s S ) e. _V
51	34, 50	eqeltri 2697	. . . . . . . . 9 |- H e. _V
52	51	a1i 11	. . . . . . . 8 |- ( ph -> H e. _V )
53		eqid 2622	. . . . . . . . 9 |- ( Base ` H ) = ( Base ` H )
54		eqid 2622	. . . . . . . . 9 |- ( 0g ` H ) = ( 0g ` H )
55		eqid 2622	. . . . . . . . 9 |- ( +g ` H ) = ( +g ` H )
56		eqid 2622	. . . . . . . . 9 |- { 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 ) }
57	53, 54, 55, 56	mgmidsssn0 17269	. . . . . . . 8 |- ( 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 ) } )
58	52, 57	syl 17	. . . . . . 7 |- ( ph -> { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } C_ { ( 0g ` H ) } )
59	58, 48	sseldd 3604	. . . . . . . . 9 |- ( ph -> .0. e. { ( 0g ` H ) } )
60		elsni 4194	. . . . . . . . 9 |- ( .0. e. { ( 0g ` H ) } -> .0. = ( 0g ` H ) )
61	59, 60	syl 17	. . . . . . . 8 |- ( ph -> .0. = ( 0g ` H ) )
62	61	sneqd 4189	. . . . . . 7 |- ( ph -> { .0. } = { ( 0g ` H ) } )
63	58, 62	sseqtr4d 3642	. . . . . 6 |- ( ph -> { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } C_ { .0. } )
64	49, 63	eqssd 3620	. . . . 5 |- ( ph -> { .0. } = { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } )
65	27, 64	eqtr3d 2658	. . . 4 |- ( ph -> { y e. B | A. x e. B ( ( 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 ) } )
66	65	sseq2d 3633	. . 3 |- ( ph -> ( ran F C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } <-> ran F C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) )
67	24, 61	eqtr3d 2658	. . 3 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
68	40	seqeq2d 12808	. . . . . . . . . 10 |- ( ph -> seq m ( .+ , F ) = seq m ( ( +g ` H ) , F ) )
69	68	fveq1d 6193	. . . . . . . . 9 |- ( ph -> ( seq m ( .+ , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
70	69	eqeq2d 2632	. . . . . . . 8 |- ( ph -> ( z = ( seq m ( .+ , F ) ` n ) <-> z = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
71	70	anbi2d 740	. . . . . . 7 |- ( ph -> ( ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
72	71	rexbidv 3052	. . . . . 6 |- ( 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 ) ) ) )
73	72	exbidv 1850	. . . . 5 |- ( 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 ) ) ) )
74	73	iotabidv 5872	. . . 4 |- ( ph -> ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) = ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
75	40	seqeq2d 12808	. . . . . . . . 9 |- ( ph -> seq 1 ( .+ , ( F o. f ) ) = seq 1 ( ( +g ` H ) , ( F o. f ) ) )
76	75	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) )
77	76	eqeq2d 2632	. . . . . . 7 |- ( ph -> ( z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) <-> z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) )
78	77	anbi2d 740	. . . . . 6 |- ( ph -> ( ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) <-> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) )
79	78	exbidv 1850	. . . . 5 |- ( ph -> ( E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) <-> E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) )
80	79	iotabidv 5872	. . . 4 |- ( ph -> ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) = ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) )
81	74, 80	ifeq12d 4106	. . 3 |- ( ph -> if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) = if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) )
82	66, 67, 81	ifbieq12d 4113	. 2 |- ( ph -> if ( ran F C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } , ( 0g ` G ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) = if ( ran F C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } , ( 0g ` H ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) )
83	27	difeq2d 3728	. . . 4 |- ( ph -> ( _V \ { .0. } ) = ( _V \ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) )
84	83	imaeq2d 5466	. . 3 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( `' F " ( _V \ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) ) )
85		gsumress.a	. . 3 |- ( ph -> A e. X )
86		gsumress.f	. . . 4 |- ( ph -> F : A --> S )
87	86, 1	fssd 6057	. . 3 |- ( ph -> F : A --> B )
88	16, 17, 18, 19, 84, 15, 85, 87	gsumval 17271	. 2 |- ( ph -> ( G gsumg F ) = if ( ran F C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } , ( 0g ` G ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) )
89	64	difeq2d 3728	. . . 4 |- ( ph -> ( _V \ { .0. } ) = ( _V \ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) )
90	89	imaeq2d 5466	. . 3 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( `' F " ( _V \ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) ) )
91	36	feq3d 6032	. . . 4 |- ( ph -> ( F : A --> S <-> F : A --> ( Base ` H ) ) )
92	86, 91	mpbid 222	. . 3 |- ( ph -> F : A --> ( Base ` H ) )
93	53, 54, 55, 56, 90, 52, 85, 92	gsumval 17271	. 2 |- ( ph -> ( H gsumg F ) = if ( ran F C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } , ( 0g ` H ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) )
94	82, 88, 93	3eqtr4d 2666	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   `'ccnv 5113   ran crn 5115   "cima 5117   o. ccom 5118   iotacio 5849   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1c1 9937   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-seq 12802  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103

This theorem is referenced by:  gsumsubm  17373  regsumfsum  19814  regsumsupp  19968  frlmgsum  20111  imasdsf1olem  22178  esumpfinvallem  30136  sge0tsms  40597  aacllem  42547