Theorem gsumfzsubmcl 13397

Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)

Hypotheses

Ref	Expression
gsumfzsubmcl.g	|- ( ph -> G e. Mnd )
gsumfzsubmcl.m	|- ( ph -> M e. ZZ )
gsumfzsubmcl.n	|- ( ph -> N e. ZZ )
gsumsubmcl.s	|- ( ph -> S e. (SubMnd ` G ) )
gsumfzsubmcl.f	|- ( ph -> F : ( M ... N ) --> S )

Assertion

Ref	Expression
gsumfzsubmcl	|- ( ph -> ( G gsumg F ) e. S )

Proof of Theorem gsumfzsubmcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqid 2193	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2193	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
4		gsumfzsubmcl.g	. . . . . 6 |- ( ph -> G e. Mnd )
5		gsumfzsubmcl.m	. . . . . 6 |- ( ph -> M e. ZZ )
6		gsumfzsubmcl.n	. . . . . 6 |- ( ph -> N e. ZZ )
7		gsumfzsubmcl.f	. . . . . . 7 |- ( ph -> F : ( M ... N ) --> S )
8		gsumsubmcl.s	. . . . . . . 8 |- ( ph -> S e. (SubMnd ` G ) )
9	1	submss 13038	. . . . . . . 8 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
10	8, 9	syl 14	. . . . . . 7 |- ( ph -> S C_ ( Base ` G ) )
11	7, 10	fssd 5408	. . . . . 6 |- ( ph -> F : ( M ... N ) --> ( Base ` G ) )
12	1, 2, 3, 4, 5, 6, 11	gsumfzval 12964	. . . . 5 |- ( ph -> ( G gsumg F ) = if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
13	12	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
14		simpr 110	. . . . 5 |- ( ( ph /\ N < M ) -> N < M )
15	14	iftrued 3564	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( 0g ` G ) )
16	13, 15	eqtrd 2226	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( 0g ` G ) )
17	2	subm0cl 13040	. . . . 5 |- ( S e. (SubMnd ` G ) -> ( 0g ` G ) e. S )
18	8, 17	syl 14	. . . 4 |- ( ph -> ( 0g ` G ) e. S )
19	18	adantr 276	. . 3 |- ( ( ph /\ N < M ) -> ( 0g ` G ) e. S )
20	16, 19	eqeltrd 2270	. 2 |- ( ( ph /\ N < M ) -> ( G gsumg F ) e. S )
21	12	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
22		simpr 110	. . . . 5 |- ( ( ph /\ -. N < M ) -> -. N < M )
23	22	iffalsed 3567	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
24	21, 23	eqtrd 2226	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
25	5	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> M e. ZZ )
26	6	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> N e. ZZ )
27	25	zred 9429	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
28	26	zred 9429	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
29	27, 28, 22	nltled 8130	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
30		eluz2 9588	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
31	25, 26, 29, 30	syl3anbrc 1183	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
32	7	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> S )
33	32	ffvelcdmda 5685	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )
34	8	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> S e. (SubMnd ` G ) )
35		simprl 529	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> x e. S )
36		simprr 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> y e. S )
37	3	submcl 13041	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
38	34, 35, 36, 37	syl3anc 1249	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
39	5, 6	fzfigd 10492	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
40	39	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( M ... N ) e. Fin )
41	32, 40	fexd 5780	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
42		plusgslid 12720	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 12635	. . . . . 6 |- ( G e. Mnd -> ( +g ` G ) e. _V )
44	4, 43	syl 14	. . . . 5 |- ( ph -> ( +g ` G ) e. _V )
45	44	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( +g ` G ) e. _V )
46	31, 33, 38, 41, 45	seqclg 10533	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) e. S )
47	24, 46	eqeltrd 2270	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) e. S )
48		zdclt 9384	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
49	6, 5, 48	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
50		exmiddc 837	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
51	49, 50	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
52	20, 47, 51	mpjaodan 799	1 |- ( ph -> ( G gsumg F ) e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   ifcif 3557   class class class wbr 4029   -->wf 5242   ` cfv 5246  (class class class)co 5910   Fincfn 6785   < clt 8044   <_ cle 8045   ZZcz 9307   ZZ>=cuz 9582   ...cfz 10064   seqcseq 10508   Basecbs 12608   +g cplusg 12685   0gc0g 12857   gsum_g cgsu 12858   Mndcmnd 12987  SubMndcsubmnd 13020

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4322  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-frec 6435  df-1o 6460  df-er 6578  df-en 6786  df-fin 6788  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-inn 8973  df-2 9031  df-n0 9231  df-z 9308  df-uz 9583  df-fz 10065  df-fzo 10199  df-seqfrec 10509  df-ndx 12611  df-slot 12612  df-base 12614  df-plusg 12698  df-0g 12859  df-igsum 12860  df-submnd 13022

This theorem is referenced by:  lgseisenlem3  15136