Theorem gsumval2 13425

Description: Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
gsumval2.b	|- B = ( Base ` G )
gsumval2.p	|- .+ = ( +g ` G )
gsumval2.g	|- ( ph -> G e. V )
gsumval2.n	|- ( ph -> N e. ( ZZ>= ` M ) )
gsumval2.f	|- ( ph -> F : ( M ... N ) --> B )

Assertion

Ref	Expression
gsumval2	|- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )

Proof of Theorem gsumval2

Dummy variables m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval2.b	. . 3 |- B = ( Base ` G )
2		eqid 2229	. . 3 |- ( 0g ` G ) = ( 0g ` G )
3		gsumval2.p	. . 3 |- .+ = ( +g ` G )
4		gsumval2.g	. . 3 |- ( ph -> G e. V )
5		gsumval2.n	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
6		eluzel2 9723	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7	5, 6	syl 14	. . . 4 |- ( ph -> M e. ZZ )
8		eluzelz 9727	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
9	5, 8	syl 14	. . . 4 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10648	. . 3 |- ( ph -> ( M ... N ) e. Fin )
11		gsumval2.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
12	1, 2, 3, 4, 10, 11	igsumval 13418	. 2 |- ( ph -> ( G gsumg F ) = ( iota x ( ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
13		simprr 531	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq m ( .+ , F ) ` n ) )
14		simprl 529	. . . . . . . . . . . 12 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( M ... N ) = ( m ... n ) )
15		eqcom 2231	. . . . . . . . . . . . . 14 |- ( ( m ... n ) = ( M ... N ) <-> ( M ... N ) = ( m ... n ) )
16		fzopth 10253	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> ( ( m ... n ) = ( M ... N ) <-> ( m = M /\ n = N ) ) )
17	15, 16	bitr3id 194	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` m ) -> ( ( M ... N ) = ( m ... n ) <-> ( m = M /\ n = N ) ) )
18	17	adantr 276	. . . . . . . . . . . 12 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( ( M ... N ) = ( m ... n ) <-> ( m = M /\ n = N ) ) )
19	14, 18	mpbid 147	. . . . . . . . . . 11 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m = M /\ n = N ) )
20	19	simpld 112	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m = M )
21	20	seqeq1d 10670	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> seq m ( .+ , F ) = seq M ( .+ , F ) )
22	19	simprd 114	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> n = N )
23	21, 22	fveq12d 5633	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
24	13, 23	eqtrd 2262	. . . . . . 7 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
25	24	rexlimiva 2643	. . . . . 6 |- ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> x = ( seq M ( .+ , F ) ` N ) )
26	25	exlimiv 1644	. . . . 5 |- ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> x = ( seq M ( .+ , F ) ` N ) )
27	7	elexd 2813	. . . . . . . 8 |- ( ph -> M e. _V )
28	27	adantr 276	. . . . . . 7 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> M e. _V )
29	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> N e. ( ZZ>= ` M ) )
30		oveq2 6008	. . . . . . . . . . 11 |- ( n = N -> ( M ... n ) = ( M ... N ) )
31	30	eqeq2d 2241	. . . . . . . . . 10 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
32		fveq2 5626	. . . . . . . . . . 11 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
33	32	eqeq2d 2241	. . . . . . . . . 10 |- ( n = N -> ( x = ( seq M ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` N ) ) )
34	31, 33	anbi12d 473	. . . . . . . . 9 |- ( n = N -> ( ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
35	34	adantl 277	. . . . . . . 8 |- ( ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) /\ n = N ) -> ( ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
36		eqidd 2230	. . . . . . . . 9 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> ( M ... N ) = ( M ... N ) )
37		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> x = ( seq M ( .+ , F ) ` N ) )
38	36, 37	jca 306	. . . . . . . 8 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) )
39	29, 35, 38	rspcedvd 2913	. . . . . . 7 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) )
40		fveq2 5626	. . . . . . . 8 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
41		oveq1 6007	. . . . . . . . . 10 |- ( m = M -> ( m ... n ) = ( M ... n ) )
42	41	eqeq2d 2241	. . . . . . . . 9 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
43		seqeq1 10667	. . . . . . . . . . 11 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
44	43	fveq1d 5628	. . . . . . . . . 10 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
45	44	eqeq2d 2241	. . . . . . . . 9 |- ( m = M -> ( x = ( seq m ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` n ) ) )
46	42, 45	anbi12d 473	. . . . . . . 8 |- ( m = M -> ( ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) ) )
47	40, 46	rexeqbidv 2745	. . . . . . 7 |- ( m = M -> ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) ) )
48	28, 39, 47	spcedv 2892	. . . . . 6 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) )
49	48	ex 115	. . . . 5 |- ( ph -> ( x = ( seq M ( .+ , F ) ` N ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
50	26, 49	impbid2 143	. . . 4 |- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> x = ( seq M ( .+ , F ) ` N ) ) )
51		eluzfz2 10224	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
52	5, 51	syl 14	. . . . . . 7 |- ( ph -> N e. ( M ... N ) )
53		n0i 3497	. . . . . . 7 |- ( N e. ( M ... N ) -> -. ( M ... N ) = (/) )
54	52, 53	syl 14	. . . . . 6 |- ( ph -> -. ( M ... N ) = (/) )
55	54	intnanrd 937	. . . . 5 |- ( ph -> -. ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) )
56		biorf 749	. . . . 5 |- ( -. ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) -> ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> ( ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
57	55, 56	syl 14	. . . 4 |- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> ( ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
58	50, 57	bitr3d 190	. . 3 |- ( ph -> ( x = ( seq M ( .+ , F ) ` N ) <-> ( ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
59	58	iotabidv 5300	. 2 |- ( ph -> ( iota x x = ( seq M ( .+ , F ) ` N ) ) = ( iota x ( ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
60		eqid 2229	. . 3 |- ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N )
61		seqex 10666	. . . . 5 |- seq M ( .+ , F ) e. _V
62		fvexg 5645	. . . . 5 |- ( ( seq M ( .+ , F ) e. _V /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` N ) e. _V )
63	61, 5, 62	sylancr 414	. . . 4 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. _V )
64		eueq 2974	. . . . 5 |- ( ( seq M ( .+ , F ) ` N ) e. _V <-> E! x x = ( seq M ( .+ , F ) ` N ) )
65	63, 64	sylib 122	. . . 4 |- ( ph -> E! x x = ( seq M ( .+ , F ) ` N ) )
66		eqeq1 2236	. . . . 5 |- ( x = ( seq M ( .+ , F ) ` N ) -> ( x = ( seq M ( .+ , F ) ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) ) )
67	66	iota2 5307	. . . 4 |- ( ( ( seq M ( .+ , F ) ` N ) e. _V /\ E! x x = ( seq M ( .+ , F ) ` N ) ) -> ( ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) <-> ( iota x x = ( seq M ( .+ , F ) ` N ) ) = ( seq M ( .+ , F ) ` N ) ) )
68	63, 65, 67	syl2anc 411	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) <-> ( iota x x = ( seq M ( .+ , F ) ` N ) ) = ( seq M ( .+ , F ) ` N ) ) )
69	60, 68	mpbii 148	. 2 |- ( ph -> ( iota x x = ( seq M ( .+ , F ) ` N ) ) = ( seq M ( .+ , F ) ` N ) )
70	12, 59, 69	3eqtr2d 2268	1 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   E.wrex 2509   _Vcvv 2799   (/)c0 3491   iotacio 5275   -->wf 5313   ` cfv 5317  (class class class)co 6000   Fincfn 6885   ZZcz 9442   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-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  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-nel 2496  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-nul 3492  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-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  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-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-0g 13286  df-igsum 13287

This theorem is referenced by:  gsumsplit1r  13426  gsumprval  13427  gsumwsubmcl  13524  gsumwmhm  13526  mulgnngsum  13659  gsumfzconst  13873