Theorem gsumsplit1r 13471

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 9807	. . . 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 13470	. 2 |- ( ph -> ( G gsumg F ) = ( seq M ( .+ , F ) ` ( N + 1 ) ) )
9		gsumsplit1r.m	. . . . . . 7 |- ( ph -> M e. ZZ )
10		eluzelz 9755	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
11	4, 10	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
12	11	peano2zd 9595	. . . . . . 7 |- ( ph -> ( N + 1 ) e. ZZ )
13	9, 12	fzfigd 10683	. . . . . 6 |- ( ph -> ( M ... ( N + 1 ) ) e. Fin )
14	7, 13	fexd 5879	. . . . 5 |- ( ph -> F e. _V )
15		vex 2803	. . . . 5 |- x e. _V
16		fvexg 5654	. . . . 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 13185	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
20	19	slotex 13099	. . . . . . 7 |- ( G e. V -> ( +g ` G ) e. _V )
21	3, 20	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
22	2, 21	eqeltrid 2316	. . . . 5 |- ( ph -> .+ e. _V )
23		vex 2803	. . . . . 6 |- y e. _V
24	23	a1i 9	. . . . 5 |- ( ph -> y e. _V )
25		ovexg 6047	. . . . 5 |- ( ( x e. _V /\ .+ e. _V /\ y e. _V ) -> ( x .+ y ) e. _V )
26	15, 22, 24, 25	mp3an2i 1376	. . . 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 10717	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
29		fzssp1 10292	. . . . . . 7 |- ( M ... N ) C_ ( M ... ( N + 1 ) )
30	29	a1i 9	. . . . . 6 |- ( ph -> ( M ... N ) C_ ( M ... ( N + 1 ) ) )
31	7, 30	fssresd 5510	. . . . 5 |- ( ph -> ( F |` ( M ... N ) ) : ( M ... N ) --> B )
32	1, 2, 3, 4, 31	gsumval2 13470	. . . 4 |- ( ph -> ( G gsumg ( F |` ( M ... N ) ) ) = ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) )
33	9	uzidd 9761	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
34		resexg 5051	. . . . . . . . . 10 |- ( F e. _V -> ( F |` ( M ... N ) ) e. _V )
35	14, 34	syl 14	. . . . . . . . 9 |- ( ph -> ( F |` ( M ... N ) ) e. _V )
36		fvexg 5654	. . . . . . . . 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 10714	. . . . . 6 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( ( F |` ( M ... N ) ) ` M ) )
40		eluzfz1 10256	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
41	4, 40	syl 14	. . . . . . 7 |- ( ph -> M e. ( M ... N ) )
42	41	fvresd 5660	. . . . . 6 |- ( ph -> ( ( F |` ( M ... N ) ) ` M ) = ( F ` M ) )
43	39, 42	eqtrd 2262	. . . . 5 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` M ) = ( F ` M ) )
44		fzp1ss 10298	. . . . . . . 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 3225	. . . . . 6 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> k e. ( M ... N ) )
47	46	fvresd 5660	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> ( ( F |` ( M ... N ) ) ` k ) = ( F ` k ) )
48	33, 43, 38, 18, 27, 4, 47	seq3fveq2 10727	. . . 4 |- ( ph -> ( seq M ( .+ , ( F |` ( M ... N ) ) ) ` N ) = ( seq M ( .+ , F ) ` N ) )
49	32, 48	eqtr2d 2263	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( G gsumg ( F |` ( M ... N ) ) ) )
50	49	oveq1d 6028	. 2 |- ( ph -> ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
51	8, 28, 50	3eqtrd 2266	1 |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   C_ wss 3198   |` cres 4725   -->wf 5320   ` cfv 5324  (class class class)co 6013   Fincfn 6904   1c1 8023   + caddc 8025   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233   seqcseq 10699   Basecbs 13072   +g cplusg 13150   gsum_g cgsu 13330

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-seqfrec 10700  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-igsum 13332

This theorem is referenced by:  gsumfzconst  13918  gsumfzfsumlemm  14591