Theorem gsumfzcl 13796

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 2234	. . . . . 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 13688	. . . . 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 3633	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
12	9, 11	eqtrd 2267	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = .0. )
13	1, 2	mndidcl 13727	. . . . 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 2311	. 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 3636	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
20	17, 19	eqtrd 2267	. . 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 9718	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
24	22	zred 9718	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
25	23, 24, 18	nltled 8410	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
26		eluz2 9877	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
27	21, 22, 25, 26	syl3anbrc 1208	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
28	5, 6	fzfigd 10817	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
29	7, 28	fexd 5921	. . . . . 6 |- ( ph -> F e. _V )
30	29	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ x e. ( ZZ>= ` M ) ) -> F e. _V )
31		vex 2818	. . . . 5 |- x e. _V
32		fvexg 5694	. . . . 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 5818	. . . 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 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
39		simprr 533	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
40	1, 3	mndcl 13720	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
41	37, 38, 39, 40	syl3anc 1274	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
42		ssv 3264	. . . . 5 |- B C_ _V
43	42	a1i 9	. . . 4 |- ( ( ph /\ -. N < M ) -> B C_ _V )
44		simprl 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> x e. _V )
45		plusgslid 13409	. . . . . . . 8 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
46	45	slotex 13323	. . . . . . 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 533	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> y e. _V )
50		ovexg 6092	. . . . 5 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
51	44, 48, 49, 50	syl3anc 1274	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` G ) y ) e. _V )
52	27, 33, 36, 41, 43, 51	seq3clss 10857	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) e. B )
53	20, 52	eqeltrd 2311	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) e. B )
54		zdclt 9672	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
55	6, 5, 54	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
56		exmiddc 844	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
57	55, 56	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
58	16, 53, 57	mpjaodan 806	1 |- ( ph -> ( G gsumg F ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   ifcif 3624   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   Fincfn 6988   < clt 8324   <_ cle 8325   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554   Mndcmnd 13713

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556  df-mgm 13653  df-sgrp 13699  df-mnd 13714

This theorem is referenced by:  gsumfzmhm2  14145  gsumfzfsumlemm  14847  lgseisenlem3  16057  lgseisenlem4  16058