Theorem gsumsplit1r 13100

Description: Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)

Hypotheses

Ref	Expression
gsumsplit1r.b	|- B = ( Base ` G )
gsumsplit1r.p	|- .+ = ( +g ` G )
gsumsplit1r.g	|- ( ph -> G e. V )
gsumsplit1r.m	|- ( ph -> M e. ZZ )
gsumsplit1r.n	|- ( ph -> N e. ( ZZ>= ` M ) )
gsumsplit1r.f	|- ( ph -> F : ( M ... ( N + 1 ) ) --> B )

Assertion

Ref	Expression
gsumsplit1r	|- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Proof of Theorem gsumsplit1r

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

Step	Hyp	Ref	Expression
1		gsumsplit1r.b	. . 3 |- B = ( Base ` G )
2		gsumsplit1r.p	. . 3 |- .+ = ( +g ` G )
3		gsumsplit1r.g	. . 3 |- ( ph -> G e. V )
4		gsumsplit1r.n	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		peano2uz 9674	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
6	4, 5	syl 14	. . 3 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7		gsumsplit1r.f	. . 3 |- ( ph -> F : ( M ... ( N + 1 ) ) --> B )
8	1, 2, 3, 6, 7	gsumval2 13099	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( N + 1 ) ) )
9		gsumsplit1r.m	. . . . . . 7 |- ( ph -> M e. ZZ )
10		eluzelz 9627	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
11	4, 10	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
12	11	peano2zd 9468	. . . . . . 7 |- ( ph -> ( N + 1 ) e. ZZ )
13	9, 12	fzfigd 10540	. . . . . 6 |- ( ph -> ( M ... ( N + 1 ) ) e. Fin )
14	7, 13	fexd 5795	. . . . 5 |- ( ph -> F e. _V )
15		vex 2766	. . . . 5 |- x e. _V
16		fvexg 5580	. . . . 5 |- ( ( F e. _V /\ x e. _V ) -> ( F ` x ) e. _V )
17	14, 15, 16	sylancl 413	. . . 4 |- ( ph -> ( F ` x ) e. _V )
18	17	adantr 276	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. _V )
19		plusgslid 12815	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
20	19	slotex 12730	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
21	3, 20	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
22	2, 21	eqeltrid 2283	. . . . 5 |- ( ph -> .+ e. _V )
23		vex 2766	. . . . . 6 |- y e. _V
24	23	a1i 9	. . . . 5 |- ( ph -> y e. _V )
25		ovexg 5959	. . . . 5 |- ( ( x e. _V /\ .+ e. _V /\ y e. _V ) -> ( x .+ y ) e. _V )
26	15, 22, 24, 25	mp3an2i 1353	. . . 4 |- ( ph -> ( x .+ y ) e. _V )
27	26	adantr 276	. . 3 |- ( ( ph /\ ( x e. _V /\ y e. _V ) ) -> ( x .+ y ) e. _V )
28	4, 18, 27	seq3p1 10574	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
29		fzssp1 10159	. . . . . . 7 |- ( M ... N ) C_ ( M ... ( N + 1 ) )
30	29	a1i 9	. . . . . 6 |- ( ph -> ( M ... N ) C_ ( M ... ( N + 1 ) ) )
31	7, 30	fssresd 5437	. . . . 5 |- ( ph -> ( F |` ( M ... N ) ) : ( M ... N ) --> B )
32	1, 2, 3, 4, 31	gsumval2 13099	. . . 4 |- ( ph -> ( G gsumg ( F |` ( M ... N ) ) ) = ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) )
33	9	uzidd 9633	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
34		resexg 4987	. . . . . . . . . 10 |- ( F e. _V -> ( F |` ( M ... N ) ) e. _V )
35	14, 34	syl 14	. . . . . . . . 9 |- ( ph -> ( F |` ( M ... N ) ) e. _V )
36		fvexg 5580	. . . . . . . . 9 |- ( ( ( F |` ( M ... N ) ) e. _V /\ x e. _V ) -> ( ( F |` ( M ... N ) ) ` x ) e. _V )
37	35, 15, 36	sylancl 413	. . . . . . . 8 |- ( ph -> ( ( F |` ( M ... N ) ) ` x ) e. _V )
38	37	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( F |` ( M ... N ) ) ` x ) e. _V )
39	9, 38, 27	seq3-1 10571	. . . . . 6 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( ( F |` ( M ... N ) ) ` M ) )
40		eluzfz1 10123	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
41	4, 40	syl 14	. . . . . . 7 |- ( ph -> M e. ( M ... N ) )
42	41	fvresd 5586	. . . . . 6 |- ( ph -> ( ( F |` ( M ... N ) ) ` M ) = ( F ` M ) )
43	39, 42	eqtrd 2229	. . . . 5 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( F ` M ) )
44		fzp1ss 10165	. . . . . . . 8 |- ( M e. ZZ -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
45	9, 44	syl 14	. . . . . . 7 |- ( ph -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
46	45	sselda 3184	. . . . . 6 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> k e. ( M ... N ) )
47	46	fvresd 5586	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> ( ( F |` ( M ... N ) ) ` k ) = ( F ` k ) )
48	33, 43, 38, 18, 27, 4, 47	seq3fveq2 10584	. . . 4 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) = ( seq M ( .+ , F ) ` N ) )
49	32, 48	eqtr2d 2230	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( G gsumg ( F |` ( M ... N ) ) ) )
50	49	oveq1d 5940	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
51	8, 28, 50	3eqtrd 2233	1 |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   |` cres 4666   -->wf 5255   ` cfv 5259  (class class class)co 5925   Fincfn 6808   1c1 7897   + caddc 7899   ZZcz 9343   ZZ>=cuz 9618   ...cfz 10100   seqcseq 10556   Basecbs 12703   +g cplusg 12780   gsum_g cgsu 12959

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 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

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 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-igsum 12961

This theorem is referenced by:  gsumfzconst  13547  gsumfzfsumlemm  14219