Theorem gsumsplit1r 12981

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 9648	. . . 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 12980	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( N + 1 ) ) )
9		gsumsplit1r.m	. . . . . . 7 |- ( ph -> M e. ZZ )
10		eluzelz 9601	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
11	4, 10	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
12	11	peano2zd 9442	. . . . . . 7 |- ( ph -> ( N + 1 ) e. ZZ )
13	9, 12	fzfigd 10502	. . . . . 6 |- ( ph -> ( M ... ( N + 1 ) ) e. Fin )
14	7, 13	fexd 5788	. . . . 5 |- ( ph -> F e. _V )
15		vex 2763	. . . . 5 |- x e. _V
16		fvexg 5573	. . . . 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 12730	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
20	19	slotex 12645	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
21	3, 20	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
22	2, 21	eqeltrid 2280	. . . . 5 |- ( ph -> .+ e. _V )
23		vex 2763	. . . . . 6 |- y e. _V
24	23	a1i 9	. . . . 5 |- ( ph -> y e. _V )
25		ovexg 5952	. . . . 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 10536	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
29		fzssp1 10133	. . . . . . 7 |- ( M ... N ) C_ ( M ... ( N + 1 ) )
30	29	a1i 9	. . . . . 6 |- ( ph -> ( M ... N ) C_ ( M ... ( N + 1 ) ) )
31	7, 30	fssresd 5430	. . . . 5 |- ( ph -> ( F |` ( M ... N ) ) : ( M ... N ) --> B )
32	1, 2, 3, 4, 31	gsumval2 12980	. . . 4 |- ( ph -> ( G gsumg ( F |` ( M ... N ) ) ) = ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) )
33	9	uzidd 9607	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
34		resexg 4982	. . . . . . . . . 10 |- ( F e. _V -> ( F |` ( M ... N ) ) e. _V )
35	14, 34	syl 14	. . . . . . . . 9 |- ( ph -> ( F |` ( M ... N ) ) e. _V )
36		fvexg 5573	. . . . . . . . 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 10533	. . . . . 6 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( ( F |` ( M ... N ) ) ` M ) )
40		eluzfz1 10097	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
41	4, 40	syl 14	. . . . . . 7 |- ( ph -> M e. ( M ... N ) )
42	41	fvresd 5579	. . . . . 6 |- ( ph -> ( ( F |` ( M ... N ) ) ` M ) = ( F ` M ) )
43	39, 42	eqtrd 2226	. . . . 5 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( F ` M ) )
44		fzp1ss 10139	. . . . . . . 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 3179	. . . . . 6 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> k e. ( M ... N ) )
47	46	fvresd 5579	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> ( ( F |` ( M ... N ) ) ` k ) = ( F ` k ) )
48	33, 43, 38, 18, 27, 4, 47	seq3fveq2 10546	. . . 4 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) = ( seq M ( .+ , F ) ` N ) )
49	32, 48	eqtr2d 2227	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( G gsumg ( F |` ( M ... N ) ) ) )
50	49	oveq1d 5933	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
51	8, 28, 50	3eqtrd 2230	1 |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   |` cres 4661   -->wf 5250   ` cfv 5254  (class class class)co 5918   Fincfn 6794   1c1 7873   + caddc 7875   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   Basecbs 12618   +g cplusg 12695   gsum_g cgsu 12868

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870

This theorem is referenced by:  gsumfzconst  13411  gsumfzfsumlemm  14075