Theorem gsumpropd 15509

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 15451 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 6113	. . . . . . . 8 |- ( ph -> ( s ( +g ` G ) t ) = ( s ( +g ` H ) t ) )
4	3	eqeq1d 2451	. . . . . . 7 |- ( ph -> ( ( s ( +g ` G ) t ) = t <-> ( s ( +g ` H ) t ) = t ) )
5	2	oveqd 6113	. . . . . . . 8 |- ( ph -> ( t ( +g ` G ) s ) = ( t ( +g ` H ) s ) )
6	5	eqeq1d 2451	. . . . . . 7 |- ( ph -> ( ( t ( +g ` G ) s ) = t <-> ( t ( +g ` H ) s ) = t ) )
7	4, 6	anbi12d 710	. . . . . 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 2936	. . . . 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 2972	. . . 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 3389	. . 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 2444	. . . 4 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
12	2	proplem3 14634	. . . 4 |- ( ( ph /\ ( a e. ( Base ` G ) /\ b e. ( Base ` G ) ) ) -> ( a ( +g ` G ) b ) = ( a ( +g ` H ) b ) )
13	11, 1, 12	grpidpropd 15452	. . 3 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
14	2	seqeq2d 11818	. . . . . . . . . 10 |- ( ph -> seq m ( ( +g ` G ) , F ) = seq m ( ( +g ` H ) , F ) )
15	14	fveq1d 5698	. . . . . . . . 9 |- ( ph -> ( seq m ( ( +g ` G ) , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
16	15	eqeq2d 2454	. . . . . . . 8 |- ( ph -> ( x = ( seq m ( ( +g ` G ) , F ) ` n ) <-> x = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
17	16	anbi2d 703	. . . . . . 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 2741	. . . . . 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 1680	. . . . 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 5407	. . . 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 3479	. . . . . . . . . . . 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 5700	. . . . . . . . . 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 6112	. . . . . . . . 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 5638	. . . . . . . . 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 16	. . . . . . . 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 5639	. . . . . . . . 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 16	. . . . . . . 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 253	. . . . . . 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 11818	. . . . . . . . 9 |- ( ph -> seq 1 ( ( +g ` G ) , ( F o. f ) ) = seq 1 ( ( +g ` H ) , ( F o. f ) ) )
31	30, 23	fveq12d 5702	. . . . . . . 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 2454	. . . . . . 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 710	. . . . . 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 1680	. . . . 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 5407	. . . 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 3814	. . 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 3821	. 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 2443	. . 3 |- ( Base ` G ) = ( Base ` G )
39		eqid 2443	. . 3 |- ( 0g ` G ) = ( 0g ` G )
40		eqid 2443	. . 3 |- ( +g ` G ) = ( +g ` G )
41		eqid 2443	. . 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 2444	. . 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 2444	. . 3 |- ( ph -> dom F = dom F )
46	38, 39, 40, 41, 42, 43, 44, 45	gsumvalx 15507	. 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 2443	. . 3 |- ( Base ` H ) = ( Base ` H )
48		eqid 2443	. . 3 |- ( 0g ` H ) = ( 0g ` H )
49		eqid 2443	. . 3 |- ( +g ` H ) = ( +g ` H )
50		eqid 2443	. . 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 2444	. . 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 15507	. 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 2485	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977   \ cdif 3330   C_ wss 3333   ifcif 3796   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848   o. ccom 4849   iotacio 5384   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   1c1 9288   ZZ>=cuz 10866   ...cfz 11442   seqcseq 11811   #chash 12108   Basecbs 14179   +g cplusg 14243   0gc0g 14383   gsum_g cgsu 14384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-recs 6837  df-rdg 6871  df-seq 11812  df-0g 14385  df-gsum 14386

This theorem is referenced by:  psropprmul  17698  ply1coe  17751  ply1coeOLD  17752  frlmgsumOLD  18200  frlmgsum  18201  tsmspropd  19707  matgsum  30867