Theorem gsumval2 13229

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 2205	. . 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 9653	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7	5, 6	syl 14	. . . 4 |- ( ph -> M e. ZZ )
8		eluzelz 9657	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
9	5, 8	syl 14	. . . 4 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10576	. . 3 |- ( ph -> ( M ... N ) e. Fin )
11		gsumval2.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
12	1, 2, 3, 4, 10, 11	igsumval 13222	. 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 2207	. . . . . . . . . . . . . 14 |- ( ( m ... n ) = ( M ... N ) <-> ( M ... N ) = ( m ... n ) )
16		fzopth 10183	. . . . . . . . . . . . . 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 10598	. . . . . . . . 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 5583	. . . . . . . 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 2238	. . . . . . 7 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
25	24	rexlimiva 2618	. . . . . 6 |- ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> x = ( seq M ( .+ , F ) ` N ) )
26	25	exlimiv 1621	. . . . 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 2785	. . . . . . . 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 5952	. . . . . . . . . . 11 |- ( n = N -> ( M ... n ) = ( M ... N ) )
31	30	eqeq2d 2217	. . . . . . . . . 10 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
32		fveq2 5576	. . . . . . . . . . 11 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
33	32	eqeq2d 2217	. . . . . . . . . 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 2206	. . . . . . . . 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 2883	. . . . . . 7 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) )
40		fveq2 5576	. . . . . . . 8 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
41		oveq1 5951	. . . . . . . . . 10 |- ( m = M -> ( m ... n ) = ( M ... n ) )
42	41	eqeq2d 2217	. . . . . . . . 9 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
43		seqeq1 10595	. . . . . . . . . . 11 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
44	43	fveq1d 5578	. . . . . . . . . 10 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
45	44	eqeq2d 2217	. . . . . . . . 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 2719	. . . . . . 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 2862	. . . . . 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 10154	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
52	5, 51	syl 14	. . . . . . 7 |- ( ph -> N e. ( M ... N ) )
53		n0i 3466	. . . . . . 7 |- ( N e. ( M ... N ) -> -. ( M ... N ) = (/) )
54	52, 53	syl 14	. . . . . 6 |- ( ph -> -. ( M ... N ) = (/) )
55	54	intnanrd 934	. . . . 5 |- ( ph -> -. ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) )
56		biorf 746	. . . . 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 5254	. 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 2205	. . 3 |- ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N )
61		seqex 10594	. . . . 5 |- seq M ( .+ , F ) e. _V
62		fvexg 5595	. . . . 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 2944	. . . . 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 2212	. . . . 5 |- ( x = ( seq M ( .+ , F ) ` N ) -> ( x = ( seq M ( .+ , F ) ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) ) )
67	66	iota2 5261	. . . 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 2244	1 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   E.wex 1515   E!weu 2054   e. wcel 2176   E.wrex 2485   _Vcvv 2772   (/)c0 3460   iotacio 5230   -->wf 5267   ` cfv 5271  (class class class)co 5944   Fincfn 6827   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592   Basecbs 12832   +g cplusg 12909   0gc0g 13088   gsum_g cgsu 13089

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-er 6620  df-en 6828  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-seqfrec 10593  df-ndx 12835  df-slot 12836  df-base 12838  df-0g 13090  df-igsum 13091

This theorem is referenced by:  gsumsplit1r  13230  gsumprval  13231  gsumwsubmcl  13328  gsumwmhm  13330  mulgnngsum  13463  gsumfzconst  13677