Theorem gsumprval 13453

Description: Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)

Hypotheses

Ref	Expression
gsumprval.b	|- B = ( Base ` G )
gsumprval.p	|- .+ = ( +g ` G )
gsumprval.g	|- ( ph -> G e. V )
gsumprval.m	|- ( ph -> M e. ZZ )
gsumprval.n	|- ( ph -> N = ( M + 1 ) )
gsumprval.f	|- ( ph -> F : { M , N } --> B )

Assertion

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

Proof of Theorem gsumprval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumprval.b	. . 3 |- B = ( Base ` G )
2		gsumprval.p	. . 3 |- .+ = ( +g ` G )
3		gsumprval.g	. . 3 |- ( ph -> G e. V )
4		gsumprval.m	. . . . 5 |- ( ph -> M e. ZZ )
5	4	uzidd 9754	. . . 4 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9795	. . . 4 |- ( M e. ( ZZ>= ` M ) -> ( M + 1 ) e. ( ZZ>= ` M ) )
7	5, 6	syl 14	. . 3 |- ( ph -> ( M + 1 ) e. ( ZZ>= ` M ) )
8		gsumprval.f	. . . 4 |- ( ph -> F : { M , N } --> B )
9		fzpr 10290	. . . . . . 7 |- ( M e. ZZ -> ( M ... ( M + 1 ) ) = { M , ( M + 1 ) } )
10	4, 9	syl 14	. . . . . 6 |- ( ph -> ( M ... ( M + 1 ) ) = { M , ( M + 1 ) } )
11		gsumprval.n	. . . . . . . 8 |- ( ph -> N = ( M + 1 ) )
12	11	eqcomd 2235	. . . . . . 7 |- ( ph -> ( M + 1 ) = N )
13	12	preq2d 3750	. . . . . 6 |- ( ph -> { M , ( M + 1 ) } = { M , N } )
14	10, 13	eqtrd 2262	. . . . 5 |- ( ph -> ( M ... ( M + 1 ) ) = { M , N } )
15	14	feq2d 5464	. . . 4 |- ( ph -> ( F : ( M ... ( M + 1 ) ) --> B <-> F : { M , N } --> B ) )
16	8, 15	mpbird 167	. . 3 |- ( ph -> F : ( M ... ( M + 1 ) ) --> B )
17	1, 2, 3, 7, 16	gsumval2 13451	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( M + 1 ) ) )
18	4	peano2zd 9588	. . . . . . . 8 |- ( ph -> ( M + 1 ) e. ZZ )
19	11, 18	eqeltrd 2306	. . . . . . 7 |- ( ph -> N e. ZZ )
20		prexg 4296	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> { M , N } e. _V )
21	4, 19, 20	syl2anc 411	. . . . . 6 |- ( ph -> { M , N } e. _V )
22	8, 21	fexd 5876	. . . . 5 |- ( ph -> F e. _V )
23		vex 2802	. . . . 5 |- x e. _V
24		fvexg 5651	. . . . 5 |- ( ( F e. _V /\ x e. _V ) -> ( F ` x ) e. _V )
25	22, 23, 24	sylancl 413	. . . 4 |- ( ph -> ( F ` x ) e. _V )
26	25	adantr 276	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. _V )
27		plusgslid 13166	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
28	27	slotex 13080	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
29	3, 28	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
30	2, 29	eqeltrid 2316	. . . . 5 |- ( ph -> .+ e. _V )
31		vex 2802	. . . . . 6 |- y e. _V
32	31	a1i 9	. . . . 5 |- ( ph -> y e. _V )
33		ovexg 6044	. . . . 5 |- ( ( x e. _V /\ .+ e. _V /\ y e. _V ) -> ( x .+ y ) e. _V )
34	23, 30, 32, 33	mp3an2i 1376	. . . 4 |- ( ph -> ( x .+ y ) e. _V )
35	34	adantr 276	. . 3 |- ( ( ph /\ ( x e. _V /\ y e. _V ) ) -> ( x .+ y ) e. _V )
36	5, 26, 35	seq3p1 10704	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( M + 1 ) ) = ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) )
37	4, 26, 35	seq3-1 10701	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
38	12	fveq2d 5636	. . 3 |- ( ph -> ( F ` ( M + 1 ) ) = ( F ` N ) )
39	37, 38	oveq12d 6028	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) = ( ( F ` M ) .+ ( F ` N ) ) )
40	17, 36, 39	3eqtrd 2266	1 |- ( ph -> ( G gsumg F ) = ( ( F ` M ) .+ ( F ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   {cpr 3667   -->wf 5317   ` cfv 5321  (class class class)co 6010   1c1 8016   + caddc 8018   ZZcz 9462   ZZ>=cuz 9738   ...cfz 10221   seqcseq 10686   Basecbs 13053   +g cplusg 13131   gsum_g cgsu 13311

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  df-en 6901  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-seqfrec 10687  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-0g 13312  df-igsum 13313

This theorem is referenced by:  gsumpr12val  13454