Theorem gsumfzsubmcl 13744

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 2206	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqid 2206	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2206	. . . . . 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 13378	. . . . . . . 8 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
10	8, 9	syl 14	. . . . . . 7 |- ( ph -> S C_ ( Base ` G ) )
11	7, 10	fssd 5447	. . . . . 6 |- ( ph -> F : ( M ... N ) --> ( Base ` G ) )
12	1, 2, 3, 4, 5, 6, 11	gsumfzval 13293	. . . . 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 3582	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( 0g ` G ) )
16	13, 15	eqtrd 2239	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( 0g ` G ) )
17	2	subm0cl 13380	. . . . 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 2283	. 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 3585	. . . 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 2239	. . 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 9510	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
28	26	zred 9510	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
29	27, 28, 22	nltled 8208	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
30		eluz2 9669	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
31	25, 26, 29, 30	syl3anbrc 1184	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
32	7	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> S )
33	32	ffvelcdmda 5727	. . . 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 13381	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
38	34, 35, 36, 37	syl3anc 1250	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
39	5, 6	fzfigd 10593	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
40	39	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( M ... N ) e. Fin )
41	32, 40	fexd 5826	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
42		plusgslid 13014	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 12929	. . . . . 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 10634	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) e. S )
47	24, 46	eqeltrd 2283	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) e. S )
48		zdclt 9465	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
49	6, 5, 48	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
50		exmiddc 838	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
51	49, 50	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
52	20, 47, 51	mpjaodan 800	1 |- ( ph -> ( G gsumg F ) e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2177   _Vcvv 2773   C_ wss 3170   ifcif 3575   class class class wbr 4050   -->wf 5275   ` cfv 5279  (class class class)co 5956   Fincfn 6839   < clt 8122   <_ cle 8123   ZZcz 9387   ZZ>=cuz 9663   ...cfz 10145   seqcseq 10609   Basecbs 12902   +g cplusg 12979   0gc0g 13158   gsum_g cgsu 13159   Mndcmnd 13318  SubMndcsubmnd 13360

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-addass 8042  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-0id 8048  ax-rnegex 8049  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-1o 6514  df-er 6632  df-en 6840  df-fin 6842  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-inn 9052  df-2 9110  df-n0 9311  df-z 9388  df-uz 9664  df-fz 10146  df-fzo 10280  df-seqfrec 10610  df-ndx 12905  df-slot 12906  df-base 12908  df-plusg 12992  df-0g 13160  df-igsum 13161  df-submnd 13362

This theorem is referenced by:  lgseisenlem3  15619