Theorem gsumfzcl 13203

Description: Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 16-Aug-2025.)

Hypotheses

Ref	Expression
gsumcl.b	|- B = ( Base ` G )
gsumcl.z	|- .0. = ( 0g ` G )
gsumfzcl.g	|- ( ph -> G e. Mnd )
gsumfzcl.m	|- ( ph -> M e. ZZ )
gsumfzcl.n	|- ( ph -> N e. ZZ )
gsumfzcl.f	|- ( ph -> F : ( M ... N ) --> B )

Assertion

Ref	Expression
gsumfzcl	|- ( ph -> ( G gsumg F ) e. B )

Proof of Theorem gsumfzcl

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

Step	Hyp	Ref	Expression
1		gsumcl.b	. . . . . 6 |- B = ( Base ` G )
2		gsumcl.z	. . . . . 6 |- .0. = ( 0g ` G )
3		eqid 2196	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
4		gsumfzcl.g	. . . . . 6 |- ( ph -> G e. Mnd )
5		gsumfzcl.m	. . . . . 6 |- ( ph -> M e. ZZ )
6		gsumfzcl.n	. . . . . 6 |- ( ph -> N e. ZZ )
7		gsumfzcl.f	. . . . . 6 |- ( ph -> F : ( M ... N ) --> B )
8	1, 2, 3, 4, 5, 6, 7	gsumfzval 13095	. . . . 5 |- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
9	8	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
10		simpr 110	. . . . 5 |- ( ( ph /\ N < M ) -> N < M )
11	10	iftrued 3569	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
12	9, 11	eqtrd 2229	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = .0. )
13	1, 2	mndidcl 13134	. . . . 5 |- ( G e. Mnd -> .0. e. B )
14	4, 13	syl 14	. . . 4 |- ( ph -> .0. e. B )
15	14	adantr 276	. . 3 |- ( ( ph /\ N < M ) -> .0. e. B )
16	12, 15	eqeltrd 2273	. 2 |- ( ( ph /\ N < M ) -> ( G gsumg F ) e. B )
17	8	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
18		simpr 110	. . . . 5 |- ( ( ph /\ -. N < M ) -> -. N < M )
19	18	iffalsed 3572	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
20	17, 19	eqtrd 2229	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
21	5	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> M e. ZZ )
22	6	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> N e. ZZ )
23	21	zred 9467	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
24	22	zred 9467	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
25	23, 24, 18	nltled 8166	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
26		eluz2 9626	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
27	21, 22, 25, 26	syl3anbrc 1183	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
28	5, 6	fzfigd 10542	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
29	7, 28	fexd 5795	. . . . . 6 |- ( ph -> F e. _V )
30	29	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ x e. ( ZZ>= ` M ) ) -> F e. _V )
31		vex 2766	. . . . 5 |- x e. _V
32		fvexg 5580	. . . . 5 |- ( ( F e. _V /\ x e. _V ) -> ( F ` x ) e. _V )
33	30, 31, 32	sylancl 413	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. _V )
34	7	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> F : ( M ... N ) --> B )
35		simpr 110	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> x e. ( M ... N ) )
36	34, 35	ffvelcdmd 5701	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. B )
37	4	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> G e. Mnd )
38		simprl 529	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
39		simprr 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
40	1, 3	mndcl 13127	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
41	37, 38, 39, 40	syl3anc 1249	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
42		ssv 3206	. . . . 5 |- B C_ _V
43	42	a1i 9	. . . 4 |- ( ( ph /\ -. N < M ) -> B C_ _V )
44		simprl 529	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> x e. _V )
45		plusgslid 12817	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
46	45	slotex 12732	. . . . . . 7 |- ( G e. Mnd -> ( +g ` G ) e. _V )
47	4, 46	syl 14	. . . . . 6 |- ( ph -> ( +g ` G ) e. _V )
48	47	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> ( +g ` G ) e. _V )
49		simprr 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> y e. _V )
50		ovexg 5959	. . . . 5 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
51	44, 48, 49, 50	syl3anc 1249	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` G ) y ) e. _V )
52	27, 33, 36, 41, 43, 51	seq3clss 10582	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) e. B )
53	20, 52	eqeltrd 2273	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) e. B )
54		zdclt 9422	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
55	6, 5, 54	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
56		exmiddc 837	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
57	55, 56	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
58	16, 53, 57	mpjaodan 799	1 |- ( ph -> ( G gsumg F ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   ifcif 3562   class class class wbr 4034   -->wf 5255   ` cfv 5259  (class class class)co 5925   Fincfn 6808   < clt 8080   <_ cle 8081   ZZcz 9345   ZZ>=cuz 9620   ...cfz 10102   seqcseq 10558   Basecbs 12705   +g cplusg 12782   0gc0g 12960   gsum_g cgsu 12961   Mndcmnd 13120

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-inn 9010  df-2 9068  df-n0 9269  df-z 9346  df-uz 9621  df-fz 10103  df-fzo 10237  df-seqfrec 10559  df-ndx 12708  df-slot 12709  df-base 12711  df-plusg 12795  df-0g 12962  df-igsum 12963  df-mgm 13060  df-sgrp 13106  df-mnd 13121

This theorem is referenced by:  gsumfzmhm2  13552  gsumfzfsumlemm  14221  lgseisenlem3  15399  lgseisenlem4  15400