Theorem gsumval2 13610

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 2232	. . 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 9858	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7	5, 6	syl 14	. . . 4 |- ( ph -> M e. ZZ )
8		eluzelz 9863	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
9	5, 8	syl 14	. . . 4 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10793	. . 3 |- ( ph -> ( M ... N ) e. Fin )
11		gsumval2.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
12	1, 2, 3, 4, 10, 11	igsumval 13603	. 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 533	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq m ( .+ , F ) ` n ) )
14		simprl 531	. . . . . . . . . . . 12 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( M ... N ) = ( m ... n ) )
15		eqcom 2234	. . . . . . . . . . . . . 14 |- ( ( m ... n ) = ( M ... N ) <-> ( M ... N ) = ( m ... n ) )
16		fzopth 10395	. . . . . . . . . . . . . 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 10815	. . . . . . . . 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 5677	. . . . . . . 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 2265	. . . . . . 7 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
25	24	rexlimiva 2655	. . . . . 6 |- ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> x = ( seq M ( .+ , F ) ` N ) )
26	25	exlimiv 1647	. . . . 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 2827	. . . . . . . 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 6058	. . . . . . . . . . 11 |- ( n = N -> ( M ... n ) = ( M ... N ) )
31	30	eqeq2d 2244	. . . . . . . . . 10 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
32		fveq2 5670	. . . . . . . . . . 11 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
33	32	eqeq2d 2244	. . . . . . . . . 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 2233	. . . . . . . . 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 2927	. . . . . . 7 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) )
40		fveq2 5670	. . . . . . . 8 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
41		oveq1 6057	. . . . . . . . . 10 |- ( m = M -> ( m ... n ) = ( M ... n ) )
42	41	eqeq2d 2244	. . . . . . . . 9 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
43		seqeq1 10812	. . . . . . . . . . 11 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
44	43	fveq1d 5672	. . . . . . . . . 10 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
45	44	eqeq2d 2244	. . . . . . . . 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 2758	. . . . . . 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 2906	. . . . . 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 10366	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
52	5, 51	syl 14	. . . . . . 7 |- ( ph -> N e. ( M ... N ) )
53		n0i 3514	. . . . . . 7 |- ( N e. ( M ... N ) -> -. ( M ... N ) = (/) )
54	52, 53	syl 14	. . . . . 6 |- ( ph -> -. ( M ... N ) = (/) )
55	54	intnanrd 940	. . . . 5 |- ( ph -> -. ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) )
56		biorf 752	. . . . 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 5335	. 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 2232	. . 3 |- ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N )
61		seqex 10811	. . . . 5 |- seq M ( .+ , F ) e. _V
62		fvexg 5689	. . . . 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 2988	. . . . 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 2239	. . . . 5 |- ( x = ( seq M ( .+ , F ) ` N ) -> ( x = ( seq M ( .+ , F ) ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) ) )
67	66	iota2 5342	. . . 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 2271	1 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   E.wex 1541   E!weu 2080   e. wcel 2203   E.wrex 2521   _Vcvv 2813   (/)c0 3508   iotacio 5310   -->wf 5348   ` cfv 5352  (class class class)co 6050   Fincfn 6975   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342   seqcseq 10809   Basecbs 13212   +g cplusg 13290   0gc0g 13469   gsum_g cgsu 13470

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-seqfrec 10810  df-ndx 13215  df-slot 13216  df-base 13218  df-0g 13471  df-igsum 13472

This theorem is referenced by:  gsumsplit1r  13611  gsumprval  13612  gsumwsubmcl  13709  gsumwmhm  13711  mulgnngsum  13844  gsumfzconst  14058  gfsumval  16862  gsumgfsumlem  16865