Theorem gsumsplit1r 13345

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 9739	. . . 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 13344	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( N + 1 ) ) )
9		gsumsplit1r.m	. . . . . . 7 |- ( ph -> M e. ZZ )
10		eluzelz 9692	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
11	4, 10	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
12	11	peano2zd 9533	. . . . . . 7 |- ( ph -> ( N + 1 ) e. ZZ )
13	9, 12	fzfigd 10613	. . . . . 6 |- ( ph -> ( M ... ( N + 1 ) ) e. Fin )
14	7, 13	fexd 5837	. . . . 5 |- ( ph -> F e. _V )
15		vex 2779	. . . . 5 |- x e. _V
16		fvexg 5618	. . . . 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 13059	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
20	19	slotex 12974	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
21	3, 20	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
22	2, 21	eqeltrid 2294	. . . . 5 |- ( ph -> .+ e. _V )
23		vex 2779	. . . . . 6 |- y e. _V
24	23	a1i 9	. . . . 5 |- ( ph -> y e. _V )
25		ovexg 6001	. . . . 5 |- ( ( x e. _V /\ .+ e. _V /\ y e. _V ) -> ( x .+ y ) e. _V )
26	15, 22, 24, 25	mp3an2i 1355	. . . 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 10647	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
29		fzssp1 10224	. . . . . . 7 |- ( M ... N ) C_ ( M ... ( N + 1 ) )
30	29	a1i 9	. . . . . 6 |- ( ph -> ( M ... N ) C_ ( M ... ( N + 1 ) ) )
31	7, 30	fssresd 5474	. . . . 5 |- ( ph -> ( F |` ( M ... N ) ) : ( M ... N ) --> B )
32	1, 2, 3, 4, 31	gsumval2 13344	. . . 4 |- ( ph -> ( G gsumg ( F |` ( M ... N ) ) ) = ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) )
33	9	uzidd 9698	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
34		resexg 5018	. . . . . . . . . 10 |- ( F e. _V -> ( F |` ( M ... N ) ) e. _V )
35	14, 34	syl 14	. . . . . . . . 9 |- ( ph -> ( F |` ( M ... N ) ) e. _V )
36		fvexg 5618	. . . . . . . . 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 10644	. . . . . 6 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( ( F |` ( M ... N ) ) ` M ) )
40		eluzfz1 10188	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
41	4, 40	syl 14	. . . . . . 7 |- ( ph -> M e. ( M ... N ) )
42	41	fvresd 5624	. . . . . 6 |- ( ph -> ( ( F |` ( M ... N ) ) ` M ) = ( F ` M ) )
43	39, 42	eqtrd 2240	. . . . 5 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( F ` M ) )
44		fzp1ss 10230	. . . . . . . 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 3201	. . . . . 6 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> k e. ( M ... N ) )
47	46	fvresd 5624	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> ( ( F |` ( M ... N ) ) ` k ) = ( F ` k ) )
48	33, 43, 38, 18, 27, 4, 47	seq3fveq2 10657	. . . 4 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) = ( seq M ( .+ , F ) ` N ) )
49	32, 48	eqtr2d 2241	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( G gsumg ( F |` ( M ... N ) ) ) )
50	49	oveq1d 5982	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
51	8, 28, 50	3eqtrd 2244	1 |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   |` cres 4695   -->wf 5286   ` cfv 5290  (class class class)co 5967   Fincfn 6850   1c1 7961   + caddc 7963   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   seqcseq 10629   Basecbs 12947   +g cplusg 13024   gsum_g cgsu 13204

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-seqfrec 10630  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-igsum 13206

This theorem is referenced by:  gsumfzconst  13792  gsumfzfsumlemm  14464