Theorem gsumpropd2 13475

Description: A stronger version of gsumpropd 13474, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 13476. (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 2232	. . . . . . 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 13456	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
7	6	eqeq2d 2243	. . . . 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 531	. . . . . . . . . 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 538	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> dom F = ( m ... n ) )
16	14, 15	eleqtrrd 2311	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> s e. dom F )
17		fvelrn 5778	. . . . . . . . . . . 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 3228	. . . . . . . . . 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 13194	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
23	22	slotex 13108	. . . . . . . . . . . 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 13108	. . . . . . . . . . . 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 10792	. . . . . . . . 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 2243	. . . . . . . 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 2529	. . . . 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 1873	. . . 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 800	. . 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 5309	. 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 2231	. . 3 |- ( Base ` G ) = ( Base ` G )
41		eqid 2231	. . 3 |- ( 0g ` G ) = ( 0g ` G )
42		eqid 2231	. . 3 |- ( +g ` G ) = ( +g ` G )
43		eqidd 2232	. . 3 |- ( ph -> dom F = dom F )
44	40, 41, 42, 3, 20, 43	igsumvalx 13471	. 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 2231	. . 3 |- ( Base ` H ) = ( Base ` H )
46		eqid 2231	. . 3 |- ( 0g ` H ) = ( 0g ` H )
47		eqid 2231	. . 3 |- ( +g ` H ) = ( +g ` H )
48	45, 46, 47, 4, 20, 43	igsumvalx 13471	. 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 2274	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   E.wex 1540   e. wcel 2202   E.wrex 2511   _Vcvv 2802   C_ wss 3200   (/)c0 3494   dom cdm 4725   ran crn 4726   iotacio 5284   Fun wfun 5320   ` cfv 5326  (class class class)co 6017   ZZ>=cuz 9754   ...cfz 10242   seqcseq 10708   Basecbs 13081   +g cplusg 13159   0gc0g 13338   gsum_g cgsu 13339

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-igsum 13341

This theorem is referenced by:  gsummgmpropd  13476