Theorem gsumval2 13040

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 2196	. . 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 9606	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7	5, 6	syl 14	. . . 4 |- ( ph -> M e. ZZ )
8		eluzelz 9610	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
9	5, 8	syl 14	. . . 4 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10523	. . 3 |- ( ph -> ( M ... N ) e. Fin )
11		gsumval2.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
12	1, 2, 3, 4, 10, 11	igsumval 13033	. 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 2198	. . . . . . . . . . . . . 14 |- ( ( m ... n ) = ( M ... N ) <-> ( M ... N ) = ( m ... n ) )
16		fzopth 10136	. . . . . . . . . . . . . 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 10545	. . . . . . . . 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 5565	. . . . . . . 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 2229	. . . . . . 7 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
25	24	rexlimiva 2609	. . . . . 6 |- ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> x = ( seq M ( .+ , F ) ` N ) )
26	25	exlimiv 1612	. . . . 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 2776	. . . . . . . 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 5930	. . . . . . . . . . 11 |- ( n = N -> ( M ... n ) = ( M ... N ) )
31	30	eqeq2d 2208	. . . . . . . . . 10 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
32		fveq2 5558	. . . . . . . . . . 11 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
33	32	eqeq2d 2208	. . . . . . . . . 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 2197	. . . . . . . . 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 2874	. . . . . . 7 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) )
40		fveq2 5558	. . . . . . . 8 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
41		oveq1 5929	. . . . . . . . . 10 |- ( m = M -> ( m ... n ) = ( M ... n ) )
42	41	eqeq2d 2208	. . . . . . . . 9 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
43		seqeq1 10542	. . . . . . . . . . 11 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
44	43	fveq1d 5560	. . . . . . . . . 10 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
45	44	eqeq2d 2208	. . . . . . . . 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 2710	. . . . . . 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 2853	. . . . . 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 10107	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
52	5, 51	syl 14	. . . . . . 7 |- ( ph -> N e. ( M ... N ) )
53		n0i 3456	. . . . . . 7 |- ( N e. ( M ... N ) -> -. ( M ... N ) = (/) )
54	52, 53	syl 14	. . . . . 6 |- ( ph -> -. ( M ... N ) = (/) )
55	54	intnanrd 933	. . . . 5 |- ( ph -> -. ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) )
56		biorf 745	. . . . 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 5241	. 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 2196	. . 3 |- ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N )
61		seqex 10541	. . . . 5 |- seq M ( .+ , F ) e. _V
62		fvexg 5577	. . . . 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 2935	. . . . 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 2203	. . . . 5 |- ( x = ( seq M ( .+ , F ) ` N ) -> ( x = ( seq M ( .+ , F ) ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) ) )
67	66	iota2 5248	. . . 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 2235	1 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1506   E!weu 2045   e. wcel 2167   E.wrex 2476   _Vcvv 2763   (/)c0 3450   iotacio 5217   -->wf 5254   ` cfv 5258  (class class class)co 5922   Fincfn 6799   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   seqcseq 10539   Basecbs 12678   +g cplusg 12755   0gc0g 12927   gsum_g cgsu 12928

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-er 6592  df-en 6800  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-seqfrec 10540  df-ndx 12681  df-slot 12682  df-base 12684  df-0g 12929  df-igsum 12930

This theorem is referenced by:  gsumsplit1r  13041  gsumprval  13042  gsumwsubmcl  13128  gsumwmhm  13130  mulgnngsum  13257  gsumfzconst  13471