Theorem gsumsplit0 14013

Description: Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gsumsplit1r 13561 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 6043	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
3		gsumsplit0.m	. . . . . . . 8 |- ( ph -> M e. ZZ )
4	3	zcnd 9664	. . . . . . 7 |- ( ph -> M e. CC )
5		1cnd 8255	. . . . . . 7 |- ( ph -> 1 e. CC )
6	4, 5	npcand 8553	. . . . . 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 5652	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` ( N + 1 ) ) = ( F ` M ) )
10	3	zred 9663	. . . . . . . . . . . . 13 |- ( ph -> M e. RR )
11	10	ltm1d 9171	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) < M )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M - 1 ) < M )
13	1, 12	eqbrtrd 4115	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> N < M )
14		peano2zm 9578	. . . . . . . . . . . . . 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 10339	. . . . . . . . . . 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 5019	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = ( F |` (/) ) )
22		res0 5023	. . . . . . . 8 |- ( F |` (/) ) = (/)
23	21, 22	eqtrdi 2280	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = (/) )
24	23	oveq2d 6044	. . . . . 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 13559	. . . . . . 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 6043	. . . 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 8330	. . . . . . . . 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 9822	. . . . . . . 8 |- ( ( N + 1 ) e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ ( N + 1 ) e. ZZ /\ M <_ ( N + 1 ) ) )
40	34, 35, 38, 39	syl3anbrc 1208	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
41		eluzfz2 10329	. . . . . . 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 5791	. . . . 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 13598	. . . . 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 6044	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( M ... M ) )
50		fzsn 10363	. . . . . . . . . . . . 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 5477	. . . . . . . . 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 5830	. . . . . . . . . 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 5851	. . . . . . 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 6044	. . . 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 1577	. . . . 5 |- F/ x ( ph /\ N = ( M - 1 ) )
68		nfcv 2375	. . . . 5 |- F/_ x ( F ` M )
69	44, 26, 34, 61, 66, 67, 68	gsumfzsnfd 14012	. . . 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 13561	. 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 9851	. . . 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 5652	. . . . 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 798	. . 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 806	1 |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   (/)c0 3496   {csn 3673   <.cop 3676   class class class wbr 4093   |-> cmpt 4155   |` cres 4733   -->wf 5329   ` cfv 5333  (class class class)co 6028   RRcr 8091   1c1 8093   + caddc 8095   < clt 8273   <_ cle 8274   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305   Basecbs 13162   +g cplusg 13240   0gc0g 13419   gsum_g cgsu 13420   Mndcmnd 13579

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306  df-seqfrec 10773  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-0g 13421  df-igsum 13422  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-minusg 13667  df-mulg 13787

This theorem is referenced by:  gfsump1  16815