Theorem gzsumsplit0 14125

Description: Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gzsumsplit1r 13692 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 6090	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
3		gzsumsplit0.m	. . . . . . . 8 |- ( ph -> M e. ZZ )
4	3	zcnd 9748	. . . . . . 7 |- ( ph -> M e. CC )
5		1cnd 8332	. . . . . . 7 |- ( ph -> 1 e. CC )
6	4, 5	npcand 8631	. . . . . 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 5694	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` ( N + 1 ) ) = ( F ` M ) )
10	3	zred 9747	. . . . . . . . . . . . 13 |- ( ph -> M e. RR )
11	10	ltm1d 9252	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) < M )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M - 1 ) < M )
13	1, 12	eqbrtrd 4147	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> N < M )
14		peano2zm 9661	. . . . . . . . . . . . . 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 10425	. . . . . . . . . . 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 5058	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = ( F |` (/) ) )
22		res0 5062	. . . . . . . 8 |- ( F |` (/) ) = (/)
23	21, 22	eqtrdi 2287	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = (/) )
24	23	oveq2d 6091	. . . . . 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 13690	. . . . . . 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 6090	. . . 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 8407	. . . . . . . . 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 9906	. . . . . . . 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 10415	. . . . . . 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 5835	. . . . 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 13725	. . . . 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 6091	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( M ... M ) )
50		fzsn 10450	. . . . . . . . . . . . 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 5516	. . . . . . . . 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 5874	. . . . . . . . . 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 5895	. . . . . . 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 6091	. . . 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 14124	. . . 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 13692	. 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 9935	. . . 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 5694	. . . . 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   (/)c0 3520   {csn 3705   <.cop 3708   class class class wbr 4125   |-> cmpt 4187   |` cres 4771   -->wf 5368   ` cfv 5372  (class class class)co 6075   RRcr 8168   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   Basecbs 13330   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588   Mndcmnd 13706

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-minusg 13786  df-mulg 13900

This theorem is referenced by:  gsump1  14134