Theorem gsumpropd2 13656

Description: A stronger version of gsumpropd 13655, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 13657. (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 2235	. . . . . . 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 13637	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
7	6	eqeq2d 2246	. . . . 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 2314	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( n e. ( ZZ>= ` m ) /\ dom F = ( m ... n ) ) ) /\ s e. ( m ... n ) ) -> s e. dom F )
17		fvelrn 5813	. . . . . . . . . . . 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 3243	. . . . . . . . . 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 13409	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
23	22	slotex 13323	. . . . . . . . . . . 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 13323	. . . . . . . . . . . 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 10917	. . . . . . . . 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 2246	. . . . . . . 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 2541	. . . . 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 1874	. . . 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 801	. . 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 5340	. 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 2234	. . 3 |- ( Base ` G ) = ( Base ` G )
41		eqid 2234	. . 3 |- ( 0g ` G ) = ( 0g ` G )
42		eqid 2234	. . 3 |- ( +g ` G ) = ( +g ` G )
43		eqidd 2235	. . 3 |- ( ph -> dom F = dom F )
44	40, 41, 42, 3, 20, 43	igsumvalx 13652	. 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 2234	. . 3 |- ( Base ` H ) = ( Base ` H )
46		eqid 2234	. . 3 |- ( 0g ` H ) = ( 0g ` H )
47		eqid 2234	. . 3 |- ( +g ` H ) = ( +g ` H )
48	45, 46, 47, 4, 20, 43	igsumvalx 13652	. 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 2277	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   _Vcvv 2815   C_ wss 3214   (/)c0 3512   dom cdm 4754   ran crn 4755   iotacio 5315   Fun wfun 5351   ` cfv 5357  (class class class)co 6058   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556

This theorem is referenced by:  gsummgmpropd  13657