Theorem gsumpropd 13094

Description: The group sum depends only on the base set and additive operation. (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 m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2197	. . . . . . 7 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
2		gsumpropd.b	. . . . . . 7 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
3		gsumpropd.g	. . . . . . 7 |- ( ph -> G e. W )
4		gsumpropd.h	. . . . . . 7 |- ( ph -> H e. X )
5		gsumpropd.p	. . . . . . . 8 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
6	5	oveqdr 5953	. . . . . . 7 |- ( ( ph /\ ( a e. ( Base ` G ) /\ b e. ( Base ` G ) ) ) -> ( a ( +g ` G ) b ) = ( a ( +g ` H ) b ) )
7	1, 2, 3, 4, 6	grpidpropdg 13076	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
8	7	eqeq2d 2208	. . . . 5 |- ( ph -> ( x = ( 0g ` G ) <-> x = ( 0g ` H ) ) )
9	8	anbi2d 464	. . . 4 |- ( ph -> ( ( dom F = (/) /\ x = ( 0g ` G ) ) <-> ( dom F = (/) /\ x = ( 0g ` H ) ) ) )
10	5	seqeq2d 10563	. . . . . . . . 9 |- ( ph -> seq m ( ( +g ` G ) , F ) = seq m ( ( +g ` H ) , F ) )
11	10	fveq1d 5563	. . . . . . . 8 |- ( ph -> ( seq m ( ( +g ` G ) , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
12	11	eqeq2d 2208	. . . . . . 7 |- ( ph -> ( x = ( seq m ( ( +g ` G ) , F ) ` n ) <-> x = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
13	12	anbi2d 464	. . . . . 6 |- ( ph -> ( ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) <-> ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
14	13	rexbidv 2498	. . . . 5 |- ( 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 ) ) ) )
15	14	exbidv 1839	. . . 4 |- ( 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 ) ) ) )
16	9, 15	orbi12d 794	. . 3 |- ( ph -> ( ( ( dom F = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) ) <-> ( ( dom F = (/) /\ x = ( 0g ` H ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) )
17	16	iotabidv 5242	. 2 |- ( ph -> ( iota x ( ( dom F = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) ) ) = ( iota x ( ( dom F = (/) /\ x = ( 0g ` H ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) )
18		eqid 2196	. . 3 |- ( Base ` G ) = ( Base ` G )
19		eqid 2196	. . 3 |- ( 0g ` G ) = ( 0g ` G )
20		eqid 2196	. . 3 |- ( +g ` G ) = ( +g ` G )
21		gsumpropd.f	. . 3 |- ( ph -> F e. V )
22		eqidd 2197	. . 3 |- ( ph -> dom F = dom F )
23	18, 19, 20, 3, 21, 22	igsumvalx 13091	. 2 |- ( ph -> ( G gsumg F ) = ( iota x ( ( dom F = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) ) ) )
24		eqid 2196	. . 3 |- ( Base ` H ) = ( Base ` H )
25		eqid 2196	. . 3 |- ( 0g ` H ) = ( 0g ` H )
26		eqid 2196	. . 3 |- ( +g ` H ) = ( +g ` H )
27	24, 25, 26, 4, 21, 22	igsumvalx 13091	. 2 |- ( ph -> ( H gsumg F ) = ( iota x ( ( dom F = (/) /\ x = ( 0g ` H ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) )
28	17, 23, 27	3eqtr4d 2239	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   (/)c0 3451   dom cdm 4664   iotacio 5218   ` cfv 5259  (class class class)co 5925   ZZ>=cuz 9618   ...cfz 10100   seqcseq 10556   Basecbs 12703   +g cplusg 12780   0gc0g 12958   gsum_g cgsu 12959

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-recs 6372  df-frec 6458  df-neg 8217  df-inn 9008  df-z 9344  df-uz 9619  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-0g 12960  df-igsum 12961

This theorem is referenced by: (None)