Theorem gsumprval 13103

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 9635	. . . 4 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9676	. . . 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 10171	. . . . . . 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 2202	. . . . . . 7 |- ( ph -> ( M + 1 ) = N )
13	12	preq2d 3707	. . . . . 6 |- ( ph -> { M , ( M + 1 ) } = { M , N } )
14	10, 13	eqtrd 2229	. . . . 5 |- ( ph -> ( M ... ( M + 1 ) ) = { M , N } )
15	14	feq2d 5398	. . . 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 13101	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( M + 1 ) ) )
18	4	peano2zd 9470	. . . . . . . 8 |- ( ph -> ( M + 1 ) e. ZZ )
19	11, 18	eqeltrd 2273	. . . . . . 7 |- ( ph -> N e. ZZ )
20		prexg 4245	. . . . . . 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 5795	. . . . 5 |- ( ph -> F e. _V )
23		vex 2766	. . . . 5 |- x e. _V
24		fvexg 5580	. . . . 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 12817	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
28	27	slotex 12732	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
29	3, 28	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
30	2, 29	eqeltrid 2283	. . . . 5 |- ( ph -> .+ e. _V )
31		vex 2766	. . . . . 6 |- y e. _V
32	31	a1i 9	. . . . 5 |- ( ph -> y e. _V )
33		ovexg 5959	. . . . 5 |- ( ( x e. _V /\ .+ e. _V /\ y e. _V ) -> ( x .+ y ) e. _V )
34	23, 30, 32, 33	mp3an2i 1353	. . . 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 10576	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( M + 1 ) ) = ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) )
37	4, 26, 35	seq3-1 10573	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
38	12	fveq2d 5565	. . 3 |- ( ph -> ( F ` ( M + 1 ) ) = ( F ` N ) )
39	37, 38	oveq12d 5943	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` M ) .+ ( F ` ( M + 1 ) ) ) = ( ( F ` M ) .+ ( F ` N ) ) )
40	17, 36, 39	3eqtrd 2233	1 |- ( ph -> ( G gsumg F ) = ( ( F ` M ) .+ ( F ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   {cpr 3624   -->wf 5255   ` cfv 5259  (class class class)co 5925   1c1 7899   + caddc 7901   ZZcz 9345   ZZ>=cuz 9620   ...cfz 10102   seqcseq 10558   Basecbs 12705   +g cplusg 12782   gsum_g cgsu 12961

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-inn 9010  df-2 9068  df-n0 9269  df-z 9346  df-uz 9621  df-fz 10103  df-seqfrec 10559  df-ndx 12708  df-slot 12709  df-base 12711  df-plusg 12795  df-0g 12962  df-igsum 12963

This theorem is referenced by:  gsumpr12val  13104