Theorem gsumpropd 13420

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 2230	. . . . . . 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 6028	. . . . . . 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 13402	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
8	7	eqeq2d 2241	. . . . 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 10671	. . . . . . . . 9 |- ( ph -> seq m ( ( +g ` G ) , F ) = seq m ( ( +g ` H ) , F ) )
11	10	fveq1d 5628	. . . . . . . 8 |- ( ph -> ( seq m ( ( +g ` G ) , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
12	11	eqeq2d 2241	. . . . . . 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 2531	. . . . 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 1871	. . . 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 798	. . 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 5300	. 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 2229	. . 3 |- ( Base ` G ) = ( Base ` G )
19		eqid 2229	. . 3 |- ( 0g ` G ) = ( 0g ` G )
20		eqid 2229	. . 3 |- ( +g ` G ) = ( +g ` G )
21		gsumpropd.f	. . 3 |- ( ph -> F e. V )
22		eqidd 2230	. . 3 |- ( ph -> dom F = dom F )
23	18, 19, 20, 3, 21, 22	igsumvalx 13417	. 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 2229	. . 3 |- ( Base ` H ) = ( Base ` H )
25		eqid 2229	. . 3 |- ( 0g ` H ) = ( 0g ` H )
26		eqid 2229	. . 3 |- ( +g ` H ) = ( +g ` H )
27	24, 25, 26, 4, 21, 22	igsumvalx 13417	. 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 2272	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   (/)c0 3491   dom cdm 4718   iotacio 5275   ` cfv 5317  (class class class)co 6000   ZZ>=cuz 9718   ...cfz 10200   seqcseq 10664   Basecbs 13027   +g cplusg 13105   0gc0g 13284   gsum_g cgsu 13285

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-recs 6449  df-frec 6535  df-neg 8316  df-inn 9107  df-z 9443  df-uz 9719  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-0g 13286  df-igsum 13287

This theorem is referenced by: (None)