Theorem gsumfzsubmcl 13930

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 2231	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqid 2231	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2231	. . . . . 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 13564	. . . . . . . 8 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
10	8, 9	syl 14	. . . . . . 7 |- ( ph -> S C_ ( Base ` G ) )
11	7, 10	fssd 5495	. . . . . 6 |- ( ph -> F : ( M ... N ) --> ( Base ` G ) )
12	1, 2, 3, 4, 5, 6, 11	gsumfzval 13479	. . . . 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 3612	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , ( 0g ` G ) , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( 0g ` G ) )
16	13, 15	eqtrd 2264	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( 0g ` G ) )
17	2	subm0cl 13566	. . . . 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 2308	. 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 3615	. . . 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 2264	. . 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 9602	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
28	26	zred 9602	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
29	27, 28, 22	nltled 8300	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
30		eluz2 9761	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
31	25, 26, 29, 30	syl3anbrc 1207	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
32	7	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> S )
33	32	ffvelcdmda 5782	. . . 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 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> x e. S )
36		simprr 533	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> y e. S )
37	3	submcl 13567	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
38	34, 35, 36, 37	syl3anc 1273	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
39	5, 6	fzfigd 10694	. . . . . 6 |- ( ph -> ( M ... N ) e. Fin )
40	39	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( M ... N ) e. Fin )
41	32, 40	fexd 5884	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
42		plusgslid 13200	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13114	. . . . . 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 10735	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) e. S )
47	24, 46	eqeltrd 2308	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) e. S )
48		zdclt 9557	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
49	6, 5, 48	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
50		exmiddc 843	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
51	49, 50	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
52	20, 47, 51	mpjaodan 805	1 |- ( ph -> ( G gsumg F ) e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   ifcif 3605   class class class wbr 4088   -->wf 5322   ` cfv 5326  (class class class)co 6018   Fincfn 6909   < clt 8214   <_ cle 8215   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243   seqcseq 10710   Basecbs 13087   +g cplusg 13165   0gc0g 13344   gsum_g cgsu 13345   Mndcmnd 13504  SubMndcsubmnd 13546

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-0g 13346  df-igsum 13347  df-submnd 13548

This theorem is referenced by:  lgseisenlem3  15807