Theorem gzsumval2 13697

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

Hypotheses

Ref	Expression
gzsumval2.b	|- B = ( Base ` G )
gzsumval2.p	|- .+ = ( +g ` G )
gzsumval2.g	|- ( ph -> G e. V )
gzsumval2.n	|- ( ph -> N e. ( ZZ>= ` M ) )
gzsumval2.f	|- ( ph -> F : ( M ... N ) --> B )

Assertion

Ref	Expression
gzsumval2	|- ( ph -> ( G gzsumgz F ) = ( seq M ( .+ , F ) ` N ) )

Proof of Theorem gzsumval2

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

Step	Hyp	Ref	Expression
1		gzsumval2.b	. . 3 |- B = ( Base ` G )
2		eqid 2238	. . 3 |- ( 0g ` G ) = ( 0g ` G )
3		gzsumval2.p	. . 3 |- .+ = ( +g ` G )
4		gzsumval2.g	. . 3 |- ( ph -> G e. V )
5		gzsumval2.n	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
6		eluzel2 9909	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7	5, 6	syl 14	. . . 4 |- ( ph -> M e. ZZ )
8		eluzelz 9914	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
9	5, 8	syl 14	. . . 4 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10851	. . 3 |- ( ph -> ( M ... N ) e. Fin )
11		gzsumval2.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
12	1, 2, 3, 4, 10, 11	gzsumval 13693	. 2 |- ( ph -> ( G gzsumgz 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 537	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq m ( .+ , F ) ` n ) )
14		simprl 535	. . . . . . . . . . . 12 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( M ... N ) = ( m ... n ) )
15		eqcom 2240	. . . . . . . . . . . . . 14 |- ( ( m ... n ) = ( M ... N ) <-> ( M ... N ) = ( m ... n ) )
16		fzopth 10450	. . . . . . . . . . . . . 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 10873	. . . . . . . . 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 5700	. . . . . . . 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 2271	. . . . . . 7 |- ( ( n e. ( ZZ>= ` m ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
25	24	rexlimiva 2663	. . . . . 6 |- ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> x = ( seq M ( .+ , F ) ` N ) )
26	25	exlimiv 1651	. . . . 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 2835	. . . . . . . 8 |- ( ph -> M e. _V )
28	27	adantr 276	. . . . . . 7 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> M e. _V )
29		oveq2 6087	. . . . . . . . . 10 |- ( n = N -> ( M ... n ) = ( M ... N ) )
30	29	eqeq2d 2250	. . . . . . . . 9 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
31		fveq2 5693	. . . . . . . . . 10 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
32	31	eqeq2d 2250	. . . . . . . . 9 |- ( n = N -> ( x = ( seq M ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` N ) ) )
33	30, 32	anbi12d 477	. . . . . . . 8 |- ( n = N -> ( ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
34	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> N e. ( ZZ>= ` M ) )
35		eqidd 2239	. . . . . . . . 9 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> ( M ... N ) = ( M ... N ) )
36		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> x = ( seq M ( .+ , F ) ` N ) )
37	35, 36	jca 306	. . . . . . . 8 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) )
38	33, 34, 37	rspcedvdw 2936	. . . . . . 7 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) )
39		fveq2 5693	. . . . . . . 8 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
40		oveq1 6086	. . . . . . . . . 10 |- ( m = M -> ( m ... n ) = ( M ... n ) )
41	40	eqeq2d 2250	. . . . . . . . 9 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
42		seqeq1 10870	. . . . . . . . . . 11 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
43	42	fveq1d 5695	. . . . . . . . . 10 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
44	43	eqeq2d 2250	. . . . . . . . 9 |- ( m = M -> ( x = ( seq m ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` n ) ) )
45	41, 44	anbi12d 477	. . . . . . . 8 |- ( m = M -> ( ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) ) )
46	39, 45	rexeqbidv 2766	. . . . . . 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 ) ) ) )
47	28, 38, 46	spcedv 2914	. . . . . 6 |- ( ( ph /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) )
48	47	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 ) ) ) )
49	26, 48	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 ) ) )
50		eluzfz2 10419	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
51		n0i 3527	. . . . . . 7 |- ( N e. ( M ... N ) -> -. ( M ... N ) = (/) )
52	5, 50, 51	3syl 17	. . . . . 6 |- ( ph -> -. ( M ... N ) = (/) )
53	52	intnanrd 944	. . . . 5 |- ( ph -> -. ( ( M ... N ) = (/) /\ x = ( 0g ` G ) ) )
54		biorf 756	. . . . 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 ) ) ) ) )
55	53, 54	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 ) ) ) ) )
56	49, 55	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 ) ) ) ) )
57	56	iotabidv 5358	. 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 ) ) ) ) )
58		eqid 2238	. . 3 |- ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N )
59		seqex 10869	. . . . 5 |- seq M ( .+ , F ) e. _V
60		fvexg 5712	. . . . 5 |- ( ( seq M ( .+ , F ) e. _V /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` N ) e. _V )
61	59, 5, 60	sylancr 418	. . . 4 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. _V )
62		eueq 2997	. . . . 5 |- ( ( seq M ( .+ , F ) ` N ) e. _V <-> E! x x = ( seq M ( .+ , F ) ` N ) )
63	61, 62	sylib 122	. . . 4 |- ( ph -> E! x x = ( seq M ( .+ , F ) ` N ) )
64		eqeq1 2245	. . . . 5 |- ( x = ( seq M ( .+ , F ) ` N ) -> ( x = ( seq M ( .+ , F ) ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) ) )
65	64	iota2 5365	. . . 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 ) ) )
66	61, 63, 65	syl2anc 415	. . 3 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) <-> ( iota x x = ( seq M ( .+ , F ) ` N ) ) = ( seq M ( .+ , F ) ` N ) ) )
67	58, 66	mpbii 148	. 2 |- ( ph -> ( iota x x = ( seq M ( .+ , F ) ` N ) ) = ( seq M ( .+ , F ) ` N ) )
68	12, 57, 67	3eqtr2d 2277	1 |- ( ph -> ( G gzsumgz F ) = ( seq M ( .+ , F ) ` N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   E.wrex 2529   _Vcvv 2821   (/)c0 3520   iotacio 5333   -->wf 5371   ` cfv 5375  (class class class)co 6079   Fincfn 7016   ZZcz 9627   ZZ>=cuz 9904   ...cfz 10394   seqcseq 10867   Basecbs 13335   +g cplusg 13414   0gc0g 13593   gzsum_gz cgzsu 13594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-seqfrec 10868  df-ndx 13338  df-slot 13339  df-base 13341  df-0g 13595  df-gzsum 13596

This theorem is used by:  gzsumsplit1r  13698  gzsumwsubmcl  13784  gzsumwmhm  13786  mulgnngzsum  13913  gzsumconst  14126  gzsumshift  14132  gsumvalfi  14135