Theorem gsumpropd2 12979

Description: A stronger version of gsumpropd 12978, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 12980. (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 2194	. . . . . . 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 12960	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
7	6	eqeq2d 2205	. . . . 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 2273	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> s e. dom F )
17		fvelrn 5690	. . . . . . . . . . . 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 3181	. . . . . . . . . 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 12733	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
23	22	slotex 12648	. . . . . . . . . . . 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 12648	. . . . . . . . . . . 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 10605	. . . . . . . . 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 2205	. . . . . . . 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 2491	. . . . 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 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 ) ) ) )
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 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 ) ) ) ) )
40		eqid 2193	. . 3 |- ( Base ` G ) = ( Base ` G )
41		eqid 2193	. . 3 |- ( 0g ` G ) = ( 0g ` G )
42		eqid 2193	. . 3 |- ( +g ` G ) = ( +g ` G )
43		eqidd 2194	. . 3 |- ( ph -> dom F = dom F )
44	40, 41, 42, 3, 20, 43	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 ) ) ) ) )
45		eqid 2193	. . 3 |- ( Base ` H ) = ( Base ` H )
46		eqid 2193	. . 3 |- ( 0g ` H ) = ( 0g ` H )
47		eqid 2193	. . 3 |- ( +g ` H ) = ( +g ` H )
48	45, 46, 47, 4, 20, 43	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 ) ) ) ) )
49	39, 44, 48	3eqtr4d 2236	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   _Vcvv 2760   C_ wss 3154   (/)c0 3447   dom cdm 4660   ran crn 4661   iotacio 5214   Fun wfun 5249   ` 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-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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-nel 2460  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-nul 3448  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-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  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-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-igsum 12873

This theorem is referenced by:  gsummgmpropd  12980