Theorem gsumpropd 12978

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 2194	. . . . . . 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 5947	. . . . . . 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 12960	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
8	7	eqeq2d 2205	. . . . 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 10528	. . . . . . . . 9 |- ( ph -> seq m ( ( +g ` G ) , F ) = seq m ( ( +g ` H ) , F ) )
11	10	fveq1d 5557	. . . . . . . 8 |- ( ph -> ( seq m ( ( +g ` G ) , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
12	11	eqeq2d 2205	. . . . . . 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 2495	. . . . 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 1836	. . . 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 5238	. 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 2193	. . 3 |- ( Base ` G ) = ( Base ` G )
19		eqid 2193	. . 3 |- ( 0g ` G ) = ( 0g ` G )
20		eqid 2193	. . 3 |- ( +g ` G ) = ( +g ` G )
21		gsumpropd.f	. . 3 |- ( ph -> F e. V )
22		eqidd 2194	. . 3 |- ( ph -> dom F = dom F )
23	18, 19, 20, 3, 21, 22	igsumvalx 12975	. 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 2193	. . 3 |- ( Base ` H ) = ( Base ` H )
25		eqid 2193	. . 3 |- ( 0g ` H ) = ( 0g ` H )
26		eqid 2193	. . 3 |- ( +g ` H ) = ( +g ` H )
27	24, 25, 26, 4, 21, 22	igsumvalx 12975	. 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 2236	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   (/)c0 3447   dom cdm 4660   iotacio 5214   ` cfv 5255  (class class class)co 5919   ZZ>=cuz 9595   ...cfz 10077   seqcseq 10521   Basecbs 12621   +g cplusg 12698   0gc0g 12870   gsum_g cgsu 12871

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-recs 6360  df-frec 6446  df-neg 8195  df-inn 8985  df-z 9321  df-uz 9596  df-seqfrec 10522  df-ndx 12624  df-slot 12625  df-base 12627  df-0g 12872  df-igsum 12873

This theorem is referenced by: (None)