Theorem gzsumsubmcl 14232

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 13842	. . . . . . . 8 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
10	8, 9	syl 14	. . . . . . 7 |- ( ph -> S C_ ( Base ` G ) )
11	7, 10	fssd 5550	. . . . . 6 |- ( ph -> F : ( M ... N ) --> ( Base ` G ) )
12	1, 2, 3, 4, 5, 6, 11	gzsumfzval 13770	. . . . 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 3647	. . . 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 13844	. . . . 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 3650	. . . 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 9779	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
28	26	zred 9779	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
29	27, 28, 22	nltled 8454	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
30		eluz2 9943	. . . . 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 5846	. . . 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 13845	. . . . 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 10889	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
40	39	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( M ... N ) e. Fin )
41	32, 40	fexd 5951	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
42		plusgslid 13525	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13437	. . . . . 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 10930	. . 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 9733	. . . 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 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ifcif 3638   class class class wbr 4131   -->wf 5376   ` cfv 5380  (class class class)co 6088   Fincfn 7026   < clt 8366   <_ cle 8367   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428   seqcseq 10905   Basecbs 13410   +g cplusg 13490   0gc0g 13669   gzsum_gz cgzsu 13670   Mndcmnd 13788  SubMndcsubmnd 13824

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-submnd 13826

This theorem is used by: (None)