Theorem gsumval2a 17279

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 )
gsumval2a.o	|- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }
gsumval2a.f	|- ( ph -> -. ran F C_ O )

Assertion

Ref	Expression
gsumval2a	|- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )

Distinct variable groups:   x, y, B   x, G, y   x, V   x, .+ , y

Allowed substitution hints:   ph( x, y)   F( x, y)   M( x, y)   N( x, y)   O( x, y)   V( y)

Proof of Theorem gsumval2a

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

Step	Hyp	Ref	Expression
1		gsumval2.b	. . . 4 |- B = ( Base ` G )
2		eqid 2622	. . . 4 |- ( 0g ` G ) = ( 0g ` G )
3		gsumval2.p	. . . 4 |- .+ = ( +g ` G )
4		gsumval2a.o	. . . 4 |- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }
5		eqidd 2623	. . . 4 |- ( ph -> ( `' F " ( _V \ O ) ) = ( `' F " ( _V \ O ) ) )
6		gsumval2.g	. . . 4 |- ( ph -> G e. V )
7		ovexd 6680	. . . 4 |- ( ph -> ( M ... N ) e. _V )
8		gsumval2.f	. . . 4 |- ( ph -> F : ( M ... N ) --> B )
9	1, 2, 3, 4, 5, 6, 7, 8	gsumval 17271	. . 3 |- ( ph -> ( G gsumg F ) = if ( ran F C_ O , ( 0g ` G ) , if ( ( M ... N ) e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) ) )
10		gsumval2a.f	. . . . 5 |- ( ph -> -. ran F C_ O )
11	10	iffalsed 4097	. . . 4 |- ( ph -> if ( ran F C_ O , ( 0g ` G ) , if ( ( M ... N ) e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) ) = if ( ( M ... N ) e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) )
12		gsumval2.n	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
13		eluzel2 11692	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
14	12, 13	syl 17	. . . . . 6 |- ( ph -> M e. ZZ )
15		eluzelz 11697	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
16	12, 15	syl 17	. . . . . 6 |- ( ph -> N e. ZZ )
17		fzf 12330	. . . . . . . 8 |- ... : ( ZZ X. ZZ ) --> ~P ZZ
18		ffn 6045	. . . . . . . 8 |- ( ... : ( ZZ X. ZZ ) --> ~P ZZ -> ... Fn ( ZZ X. ZZ ) )
19	17, 18	ax-mp 5	. . . . . . 7 |- ... Fn ( ZZ X. ZZ )
20		fnovrn 6809	. . . . . . 7 |- ( ( ... Fn ( ZZ X. ZZ ) /\ M e. ZZ /\ N e. ZZ ) -> ( M ... N ) e. ran ... )
21	19, 20	mp3an1 1411	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) e. ran ... )
22	14, 16, 21	syl2anc 693	. . . . 5 |- ( ph -> ( M ... N ) e. ran ... )
23	22	iftrued 4094	. . . 4 |- ( ph -> if ( ( M ... N ) e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) = ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) )
24	11, 23	eqtrd 2656	. . 3 |- ( ph -> if ( ran F C_ O , ( 0g ` G ) , if ( ( M ... N ) e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) ) = ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) )
25	9, 24	eqtrd 2656	. 2 |- ( ph -> ( G gsumg F ) = ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) )
26		fvex 6201	. . 3 |- ( seq M ( .+ , F ) ` N ) e. _V
27		fzopth 12378	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( m ... n ) <-> ( M = m /\ N = n ) ) )
28	12, 27	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( M ... N ) = ( m ... n ) <-> ( M = m /\ N = n ) ) )
29		simpl 473	. . . . . . . . . . . . . 14 |- ( ( M = m /\ N = n ) -> M = m )
30	29	seqeq1d 12807	. . . . . . . . . . . . 13 |- ( ( M = m /\ N = n ) -> seq M ( .+ , F ) = seq m ( .+ , F ) )
31		simpr 477	. . . . . . . . . . . . 13 |- ( ( M = m /\ N = n ) -> N = n )
32	30, 31	fveq12d 6197	. . . . . . . . . . . 12 |- ( ( M = m /\ N = n ) -> ( seq M ( .+ , F ) ` N ) = ( seq m ( .+ , F ) ` n ) )
33	32	eqcomd 2628	. . . . . . . . . . 11 |- ( ( M = m /\ N = n ) -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
34		eqeq1 2626	. . . . . . . . . . 11 |- ( z = ( seq m ( .+ , F ) ` n ) -> ( z = ( seq M ( .+ , F ) ` N ) <-> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) ) )
35	33, 34	syl5ibrcom 237	. . . . . . . . . 10 |- ( ( M = m /\ N = n ) -> ( z = ( seq m ( .+ , F ) ` n ) -> z = ( seq M ( .+ , F ) ` N ) ) )
36	28, 35	syl6bi 243	. . . . . . . . 9 |- ( ph -> ( ( M ... N ) = ( m ... n ) -> ( z = ( seq m ( .+ , F ) ` n ) -> z = ( seq M ( .+ , F ) ` N ) ) ) )
37	36	impd 447	. . . . . . . 8 |- ( ph -> ( ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) -> z = ( seq M ( .+ , F ) ` N ) ) )
38	37	rexlimdvw 3034	. . . . . . 7 |- ( ph -> ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) -> z = ( seq M ( .+ , F ) ` N ) ) )
39	38	exlimdv 1861	. . . . . 6 |- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) -> z = ( seq M ( .+ , F ) ` N ) ) )
40	14	adantr 481	. . . . . . . 8 |- ( ( ph /\ z = ( seq M ( .+ , F ) ` N ) ) -> M e. ZZ )
41		oveq2 6658	. . . . . . . . . . . . 13 |- ( n = N -> ( M ... n ) = ( M ... N ) )
42	41	eqcomd 2628	. . . . . . . . . . . 12 |- ( n = N -> ( M ... N ) = ( M ... n ) )
43	42	biantrurd 529	. . . . . . . . . . 11 |- ( n = N -> ( z = ( seq M ( .+ , F ) ` n ) <-> ( ( M ... N ) = ( M ... n ) /\ z = ( seq M ( .+ , F ) ` n ) ) ) )
44		fveq2 6191	. . . . . . . . . . . 12 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
45	44	eqeq2d 2632	. . . . . . . . . . 11 |- ( n = N -> ( z = ( seq M ( .+ , F ) ` n ) <-> z = ( seq M ( .+ , F ) ` N ) ) )
46	43, 45	bitr3d 270	. . . . . . . . . 10 |- ( n = N -> ( ( ( M ... N ) = ( M ... n ) /\ z = ( seq M ( .+ , F ) ` n ) ) <-> z = ( seq M ( .+ , F ) ` N ) ) )
47	46	rspcev 3309	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ z = ( seq M ( .+ , F ) ` N ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ z = ( seq M ( .+ , F ) ` n ) ) )
48	12, 47	sylan 488	. . . . . . . 8 |- ( ( ph /\ z = ( seq M ( .+ , F ) ` N ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ z = ( seq M ( .+ , F ) ` n ) ) )
49		fveq2 6191	. . . . . . . . . 10 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
50		oveq1 6657	. . . . . . . . . . . 12 |- ( m = M -> ( m ... n ) = ( M ... n ) )
51	50	eqeq2d 2632	. . . . . . . . . . 11 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
52		seqeq1 12804	. . . . . . . . . . . . 13 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
53	52	fveq1d 6193	. . . . . . . . . . . 12 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
54	53	eqeq2d 2632	. . . . . . . . . . 11 |- ( m = M -> ( z = ( seq m ( .+ , F ) ` n ) <-> z = ( seq M ( .+ , F ) ` n ) ) )
55	51, 54	anbi12d 747	. . . . . . . . . 10 |- ( m = M -> ( ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... n ) /\ z = ( seq M ( .+ , F ) ` n ) ) ) )
56	49, 55	rexeqbidv 3153	. . . . . . . . 9 |- ( m = M -> ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ z = ( seq M ( .+ , F ) ` n ) ) ) )
57	56	spcegv 3294	. . . . . . . 8 |- ( M e. ZZ -> ( E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ z = ( seq M ( .+ , F ) ` n ) ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) )
58	40, 48, 57	sylc 65	. . . . . . 7 |- ( ( ph /\ z = ( seq M ( .+ , F ) ` N ) ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) )
59	58	ex 450	. . . . . 6 |- ( ph -> ( z = ( seq M ( .+ , F ) ` N ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) )
60	39, 59	impbid 202	. . . . 5 |- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> z = ( seq M ( .+ , F ) ` N ) ) )
61	60	adantr 481	. . . 4 |- ( ( ph /\ ( seq M ( .+ , F ) ` N ) e. _V ) -> ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> z = ( seq M ( .+ , F ) ` N ) ) )
62	61	iota5 5871	. . 3 |- ( ( ph /\ ( seq M ( .+ , F ) ` N ) e. _V ) -> ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) = ( seq M ( .+ , F ) ` N ) )
63	26, 62	mpan2 707	. 2 |- ( ph -> ( iota z E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) = ( seq M ( .+ , F ) ` N ) )
64	25, 63	eqtrd 2656	1 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   ~Pcpw 4158   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   o. ccom 5118   iotacio 5849   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1c1 9937   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-gsum 16103

This theorem is referenced by:  gsumval2  17280