Theorem gzsumsplit0 14148

Description: Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gzsumsplit1r 13715 except that N can equal M - 1. (Contributed by Jim Kingdon, 4-Apr-2026.)

Hypotheses

Ref	Expression
gzsumsplit0.b	|- B = ( Base ` G )
gzsumsplit0.p	|- .+ = ( +g ` G )
gzsumsplit0.g	|- ( ph -> G e. Mnd )
gzsumsplit0.m	|- ( ph -> M e. ZZ )
gzsumsplit0.n	|- ( ph -> N e. ( ZZ>= ` ( M - 1 ) ) )
gzsumsplit0.f	|- ( ph -> F : ( M ... ( N + 1 ) ) --> B )

Assertion

Ref	Expression
gzsumsplit0	|- ( ph -> ( G gzsumgz F ) = ( ( G gzsumgz ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

Proof of Theorem gzsumsplit0

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 6100	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
3		gzsumsplit0.m	. . . . . . . 8 |- ( ph -> M e. ZZ )
4	3	zcnd 9769	. . . . . . 7 |- ( ph -> M e. CC )
5		1cnd 8342	. . . . . . 7 |- ( ph -> 1 e. CC )
6	4, 5	npcand 8641	. . . . . 6 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
7	6	adantr 276	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( M - 1 ) + 1 ) = M )
8	2, 7	eqtrd 2271	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = M )
9	8	fveq2d 5699	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` ( N + 1 ) ) = ( F ` M ) )
10	3	zred 9768	. . . . . . . . . . . . 13 |- ( ph -> M e. RR )
11	10	ltm1d 9262	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) < M )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M - 1 ) < M )
13	1, 12	eqbrtrd 4152	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> N < M )
14		peano2zm 9682	. . . . . . . . . . . . . 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 2315	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> N e. ZZ )
18		fzn 10446	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
19	3, 17, 18	syl2an2r 603	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N < M <-> ( M ... N ) = (/) ) )
20	13, 19	mpbid 147	. . . . . . . . 9 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... N ) = (/) )
21	20	reseq2d 5063	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = ( F |` (/) ) )
22		res0 5067	. . . . . . . 8 |- ( F |` (/) ) = (/)
23	21, 22	eqtrdi 2287	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = (/) )
24	23	oveq2d 6101	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gzsumgz ( F |` ( M ... N ) ) ) = ( G gzsumgz (/) ) )
25		gzsumsplit0.g	. . . . . . . 8 |- ( ph -> G e. Mnd )
26	25	adantr 276	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> G e. Mnd )
27		eqid 2238	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
28	27	gzsum0 13713	. . . . . . 7 |- ( G e. Mnd -> ( G gzsumgz (/) ) = ( 0g ` G ) )
29	26, 28	syl 14	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gzsumgz (/) ) = ( 0g ` G ) )
30	24, 29	eqtrd 2271	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gzsumgz ( F |` ( M ... N ) ) ) = ( 0g ` G ) )
31	30	oveq1d 6100	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( G gzsumgz ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) = ( ( 0g ` G ) .+ ( F ` ( N + 1 ) ) ) )
32		gzsumsplit0.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 2315	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) e. ZZ )
36	8	eqcomd 2244	. . . . . . . . 9 |- ( ( ph /\ N = ( M - 1 ) ) -> M = ( N + 1 ) )
37		eqle 8417	. . . . . . . . 9 |- ( ( M e. RR /\ M = ( N + 1 ) ) -> M <_ ( N + 1 ) )
38	10, 36, 37	syl2an2r 603	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> M <_ ( N + 1 ) )
39		eluz2 9927	. . . . . . . 8 |- ( ( N + 1 ) e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ ( N + 1 ) e. ZZ /\ M <_ ( N + 1 ) ) )
40	34, 35, 38, 39	syl3anbrc 1212	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
41		eluzfz2 10436	. . . . . . 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 5844	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` ( N + 1 ) ) e. B )
44		gzsumsplit0.b	. . . . . 6 |- B = ( Base ` G )
45		gzsumsplit0.p	. . . . . 6 |- .+ = ( +g ` G )
46	44, 45, 27	mndlid 13748	. . . . 5 |- ( ( G e. Mnd /\ ( F ` ( N + 1 ) ) e. B ) -> ( ( 0g ` G ) .+ ( F ` ( N + 1 ) ) ) = ( F ` ( N + 1 ) ) )
47	25, 43, 46	syl2an2r 603	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( 0g ` G ) .+ ( F ` ( N + 1 ) ) ) = ( F ` ( N + 1 ) ) )
48	31, 47	eqtrd 2271	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( G gzsumgz ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) = ( F ` ( N + 1 ) ) )
49	8	oveq2d 6101	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( M ... M ) )
50		fzsn 10472	. . . . . . . . . . . . 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 2271	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = { M } )
54	53	feq2d 5521	. . . . . . . . 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 5883	. . . . . . . . . 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 5904	. . . . . . 7 |- ( ( M e. ZZ /\ ( F ` M ) e. B ) -> { <. M , ( F ` M ) >. } = ( x e. { M } |-> ( F ` M ) ) )
63	3, 61, 62	syl2an2r 603	. . . . . 6 |- ( ( ph /\ N = ( M - 1 ) ) -> { <. M , ( F ` M ) >. } = ( x e. { M } |-> ( F ` M ) ) )
64	60, 63	eqtrd 2271	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> F = ( x e. { M } |-> ( F ` M ) ) )
65	64	oveq2d 6101	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gzsumgz F ) = ( G gzsumgz ( x e. { M } |-> ( F ` M ) ) ) )
66		eqidd 2239	. . . . 5 |- ( ( ( ph /\ N = ( M - 1 ) ) /\ x = M ) -> ( F ` M ) = ( F ` M ) )
67		nfv 1581	. . . . 5 |- F/ x ( ph /\ N = ( M - 1 ) )
68		nfcv 2392	. . . . 5 |- F/_ x ( F ` M )
69	44, 26, 34, 61, 66, 67, 68	gzsumsnfd 14147	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gzsumgz ( x e. { M } |-> ( F ` M ) ) ) = ( F ` M ) )
70	65, 69	eqtrd 2271	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gzsumgz F ) = ( F ` M ) )
71	9, 48, 70	3eqtr4rd 2282	. 2 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gzsumgz F ) = ( ( G gzsumgz ( 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	gzsumsplit1r 13715	. 2 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( G gzsumgz F ) = ( ( G gzsumgz ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
77		gzsumsplit0.n	. . . 4 |- ( ph -> N e. ( ZZ>= ` ( M - 1 ) ) )
78		uzp1 9956	. . . 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 5699	. . . . 5 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
81	80	eleq2d 2308	. . . 4 |- ( ph -> ( N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) <-> N e. ( ZZ>= ` M ) ) )
82	81	orbi2d 802	. . 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 810	1 |- ( ph -> ( G gzsumgz F ) = ( ( G gzsumgz ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   (/)c0 3520   {csn 3709   <.cop 3712   class class class wbr 4130   |-> cmpt 4192   |` cres 4776   -->wf 5373   ` cfv 5377  (class class class)co 6085   RRcr 8178   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   Basecbs 13352   +g cplusg 13431   0gc0g 13610   gzsum_gz cgzsu 13611   Mndcmnd 13729

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-seqfrec 10885  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-gzsum 13613  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-minusg 13809  df-mulg 13923

This theorem is used by:  gsump1  14157