Theorem gsumpropd2 13095

Description: A stronger version of gsumpropd 13094, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 13096. (Contributed by Thierry Arnoux, 28-Jun-2017.)

Hypotheses

Ref	Expression
gsumpropd2.f	|- ( ph -> F e. V )
gsumpropd2.g	|- ( ph -> G e. W )
gsumpropd2.h	|- ( ph -> H e. X )
gsumpropd2.b	|- ( ph -> ( Base ` G ) = ( Base ` H ) )
gsumpropd2.c	|- ( ( ph /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) )
gsumpropd2.e	|- ( ( ph /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) = ( s ( +g ` H ) t ) )
gsumpropd2.n	|- ( ph -> Fun F )
gsumpropd2.r	|- ( ph -> ran F C_ ( Base ` G ) )

Assertion

Ref	Expression
gsumpropd2	|- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Distinct variable groups:   F, s, t   G, s, t   H, s, t   ph, s, t

Allowed substitution hints:   V( t, s)   W( t, s)   X( t, s)

Proof of Theorem gsumpropd2

Dummy variables 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		gsumpropd2.b	. . . . . . 7 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
3		gsumpropd2.g	. . . . . . 7 |- ( ph -> G e. W )
4		gsumpropd2.h	. . . . . . 7 |- ( ph -> H e. X )
5		gsumpropd2.e	. . . . . . 7 |- ( ( ph /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) = ( s ( +g ` H ) t ) )
6	1, 2, 3, 4, 5	grpidpropdg 13076	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
7	6	eqeq2d 2208	. . . . 5 |- ( ph -> ( x = ( 0g ` G ) <-> x = ( 0g ` H ) ) )
8	7	anbi2d 464	. . . 4 |- ( ph -> ( ( dom F = (/) /\ x = ( 0g ` G ) ) <-> ( dom F = (/) /\ x = ( 0g ` H ) ) ) )
9		simprl 529	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) -> n e. ( ZZ>= ` m ) )
10		gsumpropd2.r	. . . . . . . . . . . 12 |- ( ph -> ran F C_ ( Base ` G ) )
11	10	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> ran F C_ ( Base ` G ) )
12		gsumpropd2.n	. . . . . . . . . . . . 13 |- ( ph -> Fun F )
13	12	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> Fun F )
14		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> s e. ( m ... n ) )
15		simplrr 536	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> dom F = ( m ... n ) )
16	14, 15	eleqtrrd 2276	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> s e. dom F )
17		fvelrn 5696	. . . . . . . . . . . 12 |- ( ( Fun F /\ s e. dom F ) -> ( F ` s ) e. ran F )
18	13, 16, 17	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> ( F ` s ) e. ran F )
19	11, 18	sseldd 3185	. . . . . . . . . 10 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> ( F ` s ) e. ( Base ` G ) )
20		gsumpropd2.f	. . . . . . . . . . 11 |- ( ph -> F e. V )
21	20	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) -> F e. V )
22		plusgslid 12815	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
23	22	slotex 12730	. . . . . . . . . . . 12 |- ( G e. W -> ( +g ` G ) e. _V )
24	3, 23	syl 14	. . . . . . . . . . 11 |- ( ph -> ( +g ` G ) e. _V )
25	24	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) -> ( +g ` G ) e. _V )
26	22	slotex 12730	. . . . . . . . . . . 12 |- ( H e. X -> ( +g ` H ) e. _V )
27	4, 26	syl 14	. . . . . . . . . . 11 |- ( ph -> ( +g ` H ) e. _V )
28	27	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) -> ( +g ` H ) e. _V )
29		gsumpropd2.c	. . . . . . . . . . 11 |- ( ( ph /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) )
30	29	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) )
31	5	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) = ( s ( +g ` H ) t ) )
32	9, 19, 21, 25, 28, 30, 31	seqfeq4g 10640	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) -> ( seq m ( ( +g ` G ) , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) )
33	32	eqeq2d 2208	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) -> ( x = ( seq m ( ( +g ` G ) , F ) ` n ) <-> x = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
34	33	anassrs 400	. . . . . . 7 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ dom F = ( m ... n ) ) -> ( x = ( seq m ( ( +g ` G ) , F ) ` n ) <-> x = ( seq m ( ( +g ` H ) , F ) ` n ) ) )
35	34	pm5.32da 452	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` m ) ) -> ( ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , F ) ` n ) ) <-> ( dom F = ( m ... n ) /\ x = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) )
36	35	rexbidva 2494	. . . . 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 ) ) ) )
37	36	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 ) ) ) )
38	8, 37	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 ) ) ) ) )
39	38	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 ) ) ) ) )
40		eqid 2196	. . 3 |- ( Base ` G ) = ( Base ` G )
41		eqid 2196	. . 3 |- ( 0g ` G ) = ( 0g ` G )
42		eqid 2196	. . 3 |- ( +g ` G ) = ( +g ` G )
43		eqidd 2197	. . 3 |- ( ph -> dom F = dom F )
44	40, 41, 42, 3, 20, 43	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 ) ) ) ) )
45		eqid 2196	. . 3 |- ( Base ` H ) = ( Base ` H )
46		eqid 2196	. . 3 |- ( 0g ` H ) = ( 0g ` H )
47		eqid 2196	. . 3 |- ( +g ` H ) = ( +g ` H )
48	45, 46, 47, 4, 20, 43	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 ) ) ) ) )
49	39, 44, 48	3eqtr4d 2239	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   _Vcvv 2763   C_ wss 3157   (/)c0 3451   dom cdm 4664   ran crn 4665   iotacio 5218   Fun wfun 5253   ` 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-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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-nel 2463  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-nul 3452  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-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  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-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-igsum 12961

This theorem is referenced by:  gsummgmpropd  13096