Theorem gsumfzcl 13575

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 2229	. . . . . 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 13467	. . . . 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 3610	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
12	9, 11	eqtrd 2262	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = .0. )
13	1, 2	mndidcl 13506	. . . . 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 2306	. 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 3613	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
20	17, 19	eqtrd 2262	. . 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 9595	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
24	22	zred 9595	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
25	23, 24, 18	nltled 8293	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
26		eluz2 9754	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
27	21, 22, 25, 26	syl3anbrc 1205	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
28	5, 6	fzfigd 10686	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
29	7, 28	fexd 5879	. . . . . 6 |- ( ph -> F e. _V )
30	29	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ x e. ( ZZ>= ` M ) ) -> F e. _V )
31		vex 2803	. . . . 5 |- x e. _V
32		fvexg 5654	. . . . 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 5779	. . . 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 13499	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
41	37, 38, 39, 40	syl3anc 1271	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
42		ssv 3247	. . . . 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 13188	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
46	45	slotex 13102	. . . . . . 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 6047	. . . . 5 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
51	44, 48, 49, 50	syl3anc 1271	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` G ) y ) e. _V )
52	27, 33, 36, 41, 43, 51	seq3clss 10726	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) e. B )
53	20, 52	eqeltrd 2306	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) e. B )
54		zdclt 9550	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
55	6, 5, 54	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
56		exmiddc 841	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
57	55, 56	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
58	16, 53, 57	mpjaodan 803	1 |- ( ph -> ( G gsumg F ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   _Vcvv 2800   C_ wss 3198   ifcif 3603   class class class wbr 4086   -->wf 5320   ` cfv 5324  (class class class)co 6013   Fincfn 6904   < clt 8207   <_ cle 8208   ZZcz 9472   ZZ>=cuz 9748   ...cfz 10236   seqcseq 10702   Basecbs 13075   +g cplusg 13153   0gc0g 13332   gsum_g cgsu 13333   Mndcmnd 13492

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-fin 6907  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-2 9195  df-n0 9396  df-z 9473  df-uz 9749  df-fz 10237  df-fzo 10371  df-seqfrec 10703  df-ndx 13078  df-slot 13079  df-base 13081  df-plusg 13166  df-0g 13334  df-igsum 13335  df-mgm 13432  df-sgrp 13478  df-mnd 13493

This theorem is referenced by:  gsumfzmhm2  13924  gsumfzfsumlemm  14594  lgseisenlem3  15794  lgseisenlem4  15795