Theorem gsumsplit0 14099

Description: Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gsumsplit1r 13661 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 6073	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
3		gsumsplit0.m	. . . . . . . 8 |- ( ph -> M e. ZZ )
4	3	zcnd 9719	. . . . . . 7 |- ( ph -> M e. CC )
5		1cnd 8306	. . . . . . 7 |- ( ph -> 1 e. CC )
6	4, 5	npcand 8604	. . . . . 6 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
7	6	adantr 276	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( M - 1 ) + 1 ) = M )
8	2, 7	eqtrd 2267	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) = M )
9	8	fveq2d 5679	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F ` ( N + 1 ) ) = ( F ` M ) )
10	3	zred 9718	. . . . . . . . . . . . 13 |- ( ph -> M e. RR )
11	10	ltm1d 9223	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) < M )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M - 1 ) < M )
13	1, 12	eqbrtrd 4136	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> N < M )
14		peano2zm 9632	. . . . . . . . . . . . . 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 2311	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> N e. ZZ )
18		fzn 10396	. . . . . . . . . . 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 5043	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = ( F |` (/) ) )
22		res0 5047	. . . . . . . 8 |- ( F |` (/) ) = (/)
23	21, 22	eqtrdi 2283	. . . . . . 7 |- ( ( ph /\ N = ( M - 1 ) ) -> ( F |` ( M ... N ) ) = (/) )
24	23	oveq2d 6074	. . . . . 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 2234	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
28	27	gsum0g 13659	. . . . . . 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 2267	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg ( F |` ( M ... N ) ) ) = ( 0g ` G ) )
31	30	oveq1d 6073	. . . 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 2311	. . . . . . . 8 |- ( ( ph /\ N = ( M - 1 ) ) -> ( N + 1 ) e. ZZ )
36	8	eqcomd 2240	. . . . . . . . 9 |- ( ( ph /\ N = ( M - 1 ) ) -> M = ( N + 1 ) )
37		eqle 8381	. . . . . . . . 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 9877	. . . . . . . 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 10386	. . . . . . 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 5818	. . . . 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 13696	. . . . 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 2267	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) = ( F ` ( N + 1 ) ) )
49	8	oveq2d 6074	. . . . . . . . . . 11 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( M ... M ) )
50		fzsn 10421	. . . . . . . . . . . . 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 2267	. . . . . . . . . 10 |- ( ( ph /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = { M } )
54	53	feq2d 5501	. . . . . . . . 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 5857	. . . . . . . . . 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 5878	. . . . . . 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 2267	. . . . 5 |- ( ( ph /\ N = ( M - 1 ) ) -> F = ( x e. { M } |-> ( F ` M ) ) )
65	64	oveq2d 6074	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg F ) = ( G gsumg ( x e. { M } |-> ( F ` M ) ) ) )
66		eqidd 2235	. . . . 5 |- ( ( ( ph /\ N = ( M - 1 ) ) /\ x = M ) -> ( F ` M ) = ( F ` M ) )
67		nfv 1577	. . . . 5 |- F/ x ( ph /\ N = ( M - 1 ) )
68		nfcv 2386	. . . . 5 |- F/_ x ( F ` M )
69	44, 26, 34, 61, 66, 67, 68	gsumfzsnfd 14098	. . . 4 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg ( x e. { M } |-> ( F ` M ) ) ) = ( F ` M ) )
70	65, 69	eqtrd 2267	. . 3 |- ( ( ph /\ N = ( M - 1 ) ) -> ( G gsumg F ) = ( F ` M ) )
71	9, 48, 70	3eqtr4rd 2278	. 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 13661	. 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 9906	. . . 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 5679	. . . . 5 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
81	80	eleq2d 2304	. . . 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 2205   (/)c0 3512   {csn 3694   <.cop 3697   class class class wbr 4114   |-> cmpt 4176   |` cres 4756   -->wf 5353   ` cfv 5357  (class class class)co 6058   RRcr 8142   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554   Mndcmnd 13677

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-minusg 13759  df-mulg 13873

This theorem is referenced by:  gfsump1  14108