Theorem gzsumsubmcl 14119

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

Assertion

Ref	Expression
gzsumsubmcl	|- ( ph -> ( G gzsumgz F ) e. S )

Proof of Theorem gzsumsubmcl

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

Step	Hyp	Ref	Expression
1		eqid 2238	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqid 2238	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2238	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
4		gzsumsubmcl.g	. . . . . 6 |- ( ph -> G e. Mnd )
5		gzsumsubmcl.m	. . . . . 6 |- ( ph -> M e. ZZ )
6		gzsumsubmcl.n	. . . . . 6 |- ( ph -> N e. ZZ )
7		gzsumsubmcl.f	. . . . . . 7 |- ( ph -> F : ( M ... N ) --> S )
8		gzsumsubmcl.s	. . . . . . . 8 |- ( ph -> S e. (SubMnd ` G ) )
9	1	submss 13760	. . . . . . . 8 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
10	8, 9	syl 14	. . . . . . 7 |- ( ph -> S C_ ( Base ` G ) )
11	7, 10	fssd 5542	. . . . . 6 |- ( ph -> F : ( M ... N ) --> ( Base ` G ) )
12	1, 2, 3, 4, 5, 6, 11	gzsumfzval 13688	. . . . 5 |- ( ph -> ( G gzsumgz F ) = if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
13	12	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( G gzsumgz F ) = if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
14		simpr 110	. . . . 5 |- ( ( ph /\ N < M ) -> N < M )
15	14	iftrued 3644	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( 0g ` G ) )
16	13, 15	eqtrd 2271	. . 3 |- ( ( ph /\ N < M ) -> ( G gzsumgz F ) = ( 0g ` G ) )
17	2	subm0cl 13762	. . . . 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 2315	. 2 |- ( ( ph /\ N < M ) -> ( G gzsumgz F ) e. S )
21	12	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz F ) = if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
22		simpr 110	. . . . 5 |- ( ( ph /\ -. N < M ) -> -. N < M )
23	22	iffalsed 3647	. . . 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 2271	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz 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 9747	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
28	26	zred 9747	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
29	27, 28, 22	nltled 8437	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
30		eluz2 9906	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
31	25, 26, 29, 30	syl3anbrc 1212	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
32	7	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> S )
33	32	ffvelcdmda 5834	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )
34	8	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> S e. (SubMnd ` G ) )
35		simprl 535	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> x e. S )
36		simprr 537	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> y e. S )
37	3	submcl 13763	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
38	34, 35, 36, 37	syl3anc 1278	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
39	5, 6	fzfigd 10846	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
40	39	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( M ... N ) e. Fin )
41	32, 40	fexd 5938	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
42		plusgslid 13443	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13357	. . . . . 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 10887	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) e. S )
47	24, 46	eqeltrd 2315	. 2 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz F ) e. S )
48		zdclt 9701	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
49	6, 5, 48	syl2anc 415	. . 3 |- ( ph -> DECID N < M )
50		exmiddc 848	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
51	49, 50	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
52	20, 47, 51	mpjaodan 810	1 |- ( ph -> ( G gzsumgz F ) e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ifcif 3635   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   Fincfn 7012   < clt 8350   <_ cle 8351   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   Basecbs 13330   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588   Mndcmnd 13706  SubMndcsubmnd 13742

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-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-fzo 10528  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-submnd 13744

This theorem is referenced by: (None)