Theorem gsumsplit0 13934

Description: Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gsumsplit1r 13482 except that N can equal M - 1. (Contributed by Jim Kingdon, 4-Apr-2026.)

Hypotheses

Ref	Expression
gsumsplit0.b	|- B = ( Base ` G )
gsumsplit0.p	|- .+ = ( +g ` G )
gsumsplit0.g	|- ( ph -> G e. Mnd )
gsumsplit0.m	|- ( ph -> M e. ZZ )
gsumsplit0.n	|- ( ph -> N e. ( ZZ>= ` ( M - 1 ) ) )
gsumsplit0.f	|- ( ph -> F : ( M ... ( N + 1 ) ) --> B )

Assertion

Ref	Expression
gsumsplit0	|- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Proof of Theorem gsumsplit0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> N = ( M - 1 ) )
2	1	oveq1d 6033	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
3		gsumsplit0.m	. . . . . . . 8 |- ( ph -> M e. ZZ )
4	3	zcnd 9603	. . . . . . 7 |- ( ph -> M e. CC )
5		1cnd 8195	. . . . . . 7 |- ( ph -> 1 e. CC )
6	4, 5	npcand 8494	. . . . . 6 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
7	6	adantr 276	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( M - 1 ) + 1 ) = M )
8	2, 7	eqtrd 2264	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = M )
9	8	fveq2d 5643	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` ( N + 1 ) ) = ( F ` M ) )
10	3	zred 9602	. . . . . . . . . . . . 13 |- ( ph -> M e. RR )
11	10	ltm1d 9112	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) < M )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M - 1 ) < M )
13	1, 12	eqbrtrd 4110	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> N < M )
14		peano2zm 9517	. . . . . . . . . . . . . 14 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	3, 14	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( M - 1 ) e. ZZ )
16	15	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M - 1 ) e. ZZ )
17	1, 16	eqeltrd 2308	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> N e. ZZ )
18		fzn 10277	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
19	3, 17, 18	syl2an2r 599	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N < M <-> ( M ... N ) = (/) ) )
20	13, 19	mpbid 147	. . . . . . . . 9 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... N ) = (/) )
21	20	reseq2d 5013	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = ( F |` (/) ) )
22		res0 5017	. . . . . . . 8 |- ( F |` (/) ) = (/)
23	21, 22	eqtrdi 2280	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = (/) )
24	23	oveq2d 6034	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg ( F |` ( M ... N ) ) ) = ( G gsumg (/) ) )
25		gsumsplit0.g	. . . . . . . 8 |- ( ph -> G e. Mnd )
26	25	adantr 276	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> G e. Mnd )
27		eqid 2231	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
28	27	gsum0g 13480	. . . . . . 7 |- ( G e. Mnd -> ( G gsumg (/) ) = ( 0g ` G ) )
29	26, 28	syl 14	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg (/) ) = ( 0g ` G ) )
30	24, 29	eqtrd 2264	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg ( F |` ( M ... N ) ) ) = ( 0g ` G ) )
31	30	oveq1d 6033	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) = ( ( 0g ` G ) .+ ( F ` ( N + 1 ) ) ) )
32		gsumsplit0.f	. . . . . . 7 |- ( ph -> F : ( M ... ( N + 1 ) ) --> B )
33	32	adantr 276	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> F : ( M ... ( N + 1 ) ) --> B )
34	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> M e. ZZ )
35	8, 34	eqeltrd 2308	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) e. ZZ )
36	8	eqcomd 2237	. . . . . . . . 9 |- ( ( ph /\ N = ( M - 1 ) ) -> M = ( N + 1 ) )
37		eqle 8271	. . . . . . . . 9 |- ( ( M e. RR /\ M = ( N + 1 ) ) -> M <_ ( N + 1 ) )
38	10, 36, 37	syl2an2r 599	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> M <_ ( N + 1 ) )
39		eluz2 9761	. . . . . . . 8 |- ( ( N + 1 ) e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ ( N + 1 ) e. ZZ /\ M <_ ( N + 1 ) ) )
40	34, 35, 38, 39	syl3anbrc 1207	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
41		eluzfz2 10267	. . . . . . 7 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) )
42	40, 41	syl 14	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) )
43	33, 42	ffvelcdmd 5783	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` ( N + 1 ) ) e. B )
44		gsumsplit0.b	. . . . . 6 |- B = ( Base ` G )
45		gsumsplit0.p	. . . . . 6 |- .+ = ( +g ` G )
46	44, 45, 27	mndlid 13519	. . . . 5 |- ( ( G e. Mnd /\ ( F ` ( N + 1 ) ) e. B ) -> ( ( 0g ` G ) .+ ( F ` ( N + 1 ) ) ) = ( F ` ( N + 1 ) ) )
47	25, 43, 46	syl2an2r 599	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( 0g ` G ) .+ ( F ` ( N + 1 ) ) ) = ( F ` ( N + 1 ) ) )
48	31, 47	eqtrd 2264	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) = ( F ` ( N + 1 ) ) )
49	8	oveq2d 6034	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( M ... M ) )
50		fzsn 10301	. . . . . . . . . . . . 13 |- ( M e. ZZ -> ( M ... M ) = { M } )
51	3, 50	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( M ... M ) = { M } )
52	51	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... M ) = { M } )
53	49, 52	eqtrd 2264	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = { M } )
54	53	feq2d 5470	. . . . . . . . 9 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F : ( M ... ( N + 1 ) ) --> B <-> F : { M } --> B ) )
55	33, 54	mpbid 147	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> F : { M } --> B )
56		fsn2g 5822	. . . . . . . . . 10 |- ( M e. ZZ -> ( F : { M } --> B <-> ( ( F ` M ) e. B /\ F = { <. M , ( F ` M ) >. } ) ) )
57	3, 56	syl 14	. . . . . . . . 9 |- ( ph -> ( F : { M } --> B <-> ( ( F ` M ) e. B /\ F = { <. M , ( F ` M ) >. } ) ) )
58	57	adantr 276	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F : { M } --> B <-> ( ( F ` M ) e. B /\ F = { <. M , ( F ` M ) >. } ) ) )
59	55, 58	mpbid 147	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( F ` M ) e. B /\ F = { <. M , ( F ` M ) >. } ) )
60	59	simprd 114	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> F = { <. M , ( F ` M ) >. } )
61	59	simpld 112	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` M ) e. B )
62		fmptsn 5843	. . . . . . 7 |- ( ( M e. ZZ /\ ( F ` M ) e. B ) -> { <. M , ( F ` M ) >. } = ( x e. { M } |-> ( F ` M ) ) )
63	3, 61, 62	syl2an2r 599	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> { <. M , ( F ` M ) >. } = ( x e. { M } |-> ( F ` M ) ) )
64	60, 63	eqtrd 2264	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> F = ( x e. { M } |-> ( F ` M ) ) )
65	64	oveq2d 6034	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg F ) = ( G gsumg ( x e. { M } |-> ( F ` M ) ) ) )
66		eqidd 2232	. . . . 5 |- ( ( ( ph /\ N = ( M - 1 ) ) /\ x = M ) -> ( F ` M ) = ( F ` M ) )
67		nfv 1576	. . . . 5 |- F/ x ( ph /\ N = ( M - 1 ) )
68		nfcv 2374	. . . . 5 |- F/_ x ( F ` M )
69	44, 26, 34, 61, 66, 67, 68	gsumfzsnfd 13933	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg ( x e. { M } |-> ( F ` M ) ) ) = ( F ` M ) )
70	65, 69	eqtrd 2264	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg F ) = ( F ` M ) )
71	9, 48, 70	3eqtr4rd 2275	. 2 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
72	25	adantr 276	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> G e. Mnd )
73	3	adantr 276	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> M e. ZZ )
74		simpr 110	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ( ZZ>= ` M ) )
75	32	adantr 276	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> F : ( M ... ( N + 1 ) ) --> B )
76	44, 45, 72, 73, 74, 75	gsumsplit1r 13482	. 2 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
77		gsumsplit0.n	. . . 4 |- ( ph -> N e. ( ZZ>= ` ( M - 1 ) ) )
78		uzp1 9790	. . . 4 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) )
79	77, 78	syl 14	. . 3 |- ( ph -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) )
80	6	fveq2d 5643	. . . . 5 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
81	80	eleq2d 2301	. . . 4 |- ( ph -> ( N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) <-> N e. ( ZZ>= ` M ) ) )
82	81	orbi2d 797	. . 3 |- ( ph -> ( ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) <-> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` M ) ) ) )
83	79, 82	mpbid 147	. 2 |- ( ph -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` M ) ) )
84	71, 76, 83	mpjaodan 805	1 |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   (/)c0 3494   {csn 3669   <.cop 3672   class class class wbr 4088   |-> cmpt 4150   |` cres 4727   -->wf 5322   ` cfv 5326  (class class class)co 6018   RRcr 8031   1c1 8033   + caddc 8035   < clt 8214   <_ cle 8215   - cmin 8350   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243   Basecbs 13083   +g cplusg 13161   0gc0g 13340   gsum_g cgsu 13341   Mndcmnd 13500

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-seqfrec 10710  df-ndx 13086  df-slot 13087  df-base 13089  df-plusg 13174  df-0g 13342  df-igsum 13343  df-mgm 13440  df-sgrp 13486  df-mnd 13501  df-minusg 13588  df-mulg 13708

This theorem is referenced by:  gfsump1  16689