Theorem gsumpropd 16593

Description: The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 16640 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
gsumpropd.f	|- ( ph -> F e. V )
gsumpropd.g	|- ( ph -> G e. W )
gsumpropd.h	|- ( ph -> H e. X )
gsumpropd.b	|- ( ph -> ( Base ` G ) = ( Base ` H ) )
gsumpropd.p	|- ( ph -> ( +g ` G ) = ( +g ` H ) )

Assertion

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

Proof of Theorem gsumpropd

Dummy variables a b f m n s t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumpropd.b	. . . . 5 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
2		gsumpropd.p	. . . . . . . . 9 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
3	2	oveqd 6325	. . . . . . . 8 |- ( ph -> ( s ( +g ` G ) t ) = ( s ( +g ` H ) t ) )
4	3	eqeq1d 2473	. . . . . . 7 |- ( ph -> ( ( s ( +g ` G ) t ) = t <-> ( s ( +g ` H ) t ) = t ) )
5	2	oveqd 6325	. . . . . . . 8 |- ( ph -> ( t ( +g ` G ) s ) = ( t ( +g ` H ) s ) )
6	5	eqeq1d 2473	. . . . . . 7 |- ( ph -> ( ( t ( +g ` G ) s ) = t <-> ( t ( +g ` H ) s ) = t ) )
7	4, 6	anbi12d 725	. . . . . 6 |- ( ph -> ( ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) <-> ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) ) )
8	1, 7	raleqbidv 2987	. . . . 5 |- ( ph -> ( A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) <-> A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) ) )
9	1, 8	rabeqbidv 3026	. . . 4 |- ( ph -> { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } = { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } )
10	9	sseq2d 3446	. . 3 |- ( ph -> ( ran F C_ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } <-> ran F C_ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) )
11		eqidd 2472	. . . 4 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
12	2	oveqdr 6332	. . . 4 |- ( ( ph /\ ( a e. ( Base ` G ) /\ b e. ( Base ` G ) ) ) -> ( a ( +g ` G ) b ) = ( a ( +g ` H ) b ) )
13	11, 1, 12	grpidpropd 16582	. . 3 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
14	2	seqeq2d 12258	. . . . . . . . . 10 |- ( ph -> seq m ( ( +g ` G ) , F ) = seq m ( ( +g ` H ) , F ) )
15	14	fveq1d 5881	. . . . . . . . 9 |- ( ph -> ( seq m ( ( +g ` G ) , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
16	15	eqeq2d 2481	. . . . . . . 8 |- ( ph -> ( x = ( seq m ( ( +g ` G ) , F ) ` n ) <-> x = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
17	16	anbi2d 718	. . . . . . 7 |- ( ph -> ( ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) <-> ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
18	17	rexbidv 2892	. . . . . 6 |- ( ph -> ( E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) <-> E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
19	18	exbidv 1776	. . . . 5 |- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) <-> E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
20	19	iotabidv 5574	. . . 4 |- ( ph -> ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) ) = ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
21	9	difeq2d 3540	. . . . . . . . . . . 12 |- ( ph -> ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) = ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) )
22	21	imaeq2d 5174	. . . . . . . . . . 11 |- ( ph -> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) = ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) )
23	22	fveq2d 5883	. . . . . . . . . 10 |- ( ph -> ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) = ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) )
24	23	oveq2d 6324	. . . . . . . . 9 |- ( ph -> ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) = ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) )
25		f1oeq2 5819	. . . . . . . . 9 |- ( ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) = ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) <-> f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) )
26	24, 25	syl 17	. . . . . . . 8 |- ( ph -> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) <-> f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) )
27		f1oeq3 5820	. . . . . . . . 9 |- ( ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) = ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) -> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) <-> f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) )
28	22, 27	syl 17	. . . . . . . 8 |- ( ph -> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) <-> f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) )
29	26, 28	bitrd 261	. . . . . . 7 |- ( ph -> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) <-> f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) )
30	2	seqeq2d 12258	. . . . . . . . 9 |- ( ph -> seq 1 ( ( +g ` G ) , ( F o. f ) ) = seq 1 ( ( +g ` H ) , ( F o. f ) ) )
31	30, 23	fveq12d 5885	. . . . . . . 8 |- ( ph -> ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) )
32	31	eqeq2d 2481	. . . . . . 7 |- ( ph -> ( x = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) <-> x = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) ) )
33	29, 32	anbi12d 725	. . . . . 6 |- ( ph -> ( ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) ) <-> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) ) ) )
34	33	exbidv 1776	. . . . 5 |- ( ph -> ( E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) ) <-> E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) ) ) )
35	34	iotabidv 5574	. . . 4 |- ( ph -> ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) ) ) = ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) ) ) )
36	20, 35	ifeq12d 3892	. . 3 |- ( ph -> if ( dom F e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) ) ) ) = if ( dom F e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) ) ) ) )
37	10, 13, 36	ifbieq12d 3899	. 2 |- ( ph -> if ( ran F C_ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } , ( 0g ` G ) , if ( dom F e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) ) ) ) ) = if ( ran F C_ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } , ( 0g ` H ) , if ( dom F e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) ) ) ) ) )
38		eqid 2471	. . 3 |- ( Base ` G ) = ( Base ` G )
39		eqid 2471	. . 3 |- ( 0g ` G ) = ( 0g ` G )
40		eqid 2471	. . 3 |- ( +g ` G ) = ( +g ` G )
41		eqid 2471	. . 3 |- { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } = { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) }
42		eqidd 2472	. . 3 |- ( ph -> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) = ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) )
43		gsumpropd.g	. . 3 |- ( ph -> G e. W )
44		gsumpropd.f	. . 3 |- ( ph -> F e. V )
45		eqidd 2472	. . 3 |- ( ph -> dom F = dom F )
46	38, 39, 40, 41, 42, 43, 44, 45	gsumvalx 16591	. 2 |- ( ph -> ( G gsumg F ) = if ( ran F C_ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } , ( 0g ` G ) , if ( dom F e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` G ) | A. t e. ( Base ` G ) ( ( s ( +g ` G ) t ) = t /\ ( t ( +g ` G ) s ) = t ) } ) ) ) ) ) ) ) ) )
47		eqid 2471	. . 3 |- ( Base ` H ) = ( Base ` H )
48		eqid 2471	. . 3 |- ( 0g ` H ) = ( 0g ` H )
49		eqid 2471	. . 3 |- ( +g ` H ) = ( +g ` H )
50		eqid 2471	. . 3 |- { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } = { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) }
51		eqidd 2472	. . 3 |- ( ph -> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) = ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) )
52		gsumpropd.h	. . 3 |- ( ph -> H e. X )
53	47, 48, 49, 50, 51, 52, 44, 45	gsumvalx 16591	. 2 |- ( ph -> ( H gsumg F ) = if ( ran F C_ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } , ( 0g ` H ) , if ( dom F e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) /\ x = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { s e. ( Base ` H ) | A. t e. ( Base ` H ) ( ( s ( +g ` H ) t ) = t /\ ( t ( +g ` H ) s ) = t ) } ) ) ) ) ) ) ) ) )
54	37, 46, 53	3eqtr4d 2515	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   \ cdif 3387   C_ wss 3390   ifcif 3872   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   o. ccom 4843   iotacio 5551   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   1c1 9558   ZZ>=cuz 11182   ...cfz 11810   seqcseq 12251   #chash 12553   Basecbs 15199   +g cplusg 15268   0gc0g 15416   gsum_g cgsu 15417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-seq 12252  df-0g 15418  df-gsum 15419

This theorem is referenced by:  psropprmul  18908  ply1coe  18966  ply1coeOLD  18967  frlmgsum  19407  matgsum  19539  tsmspropd  21224