Theorem gsumwsubmcl 13068

Description: Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)

Assertion

Ref	Expression
gsumwsubmcl	|- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( G gsumg W ) e. S )

Proof of Theorem gsumwsubmcl

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

Step	Hyp	Ref	Expression
1		oveq2 5926	. . . . 5 |- ( W = (/) -> ( G gsumg W ) = ( G gsumg (/) ) )
2	1	adantl 277	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) = ( G gsumg (/) ) )
3		submrcl 13043	. . . . . 6 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
4		eqid 2193	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
5	4	gsum0g 12979	. . . . . 6 |- ( G e. Mnd -> ( G gsumg (/) ) = ( 0g ` G ) )
6	3, 5	syl 14	. . . . 5 |- ( S e. (SubMnd ` G ) -> ( G gsumg (/) ) = ( 0g ` G ) )
7	6	ad2antrr 488	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg (/) ) = ( 0g ` G ) )
8	2, 7	eqtrd 2226	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) = ( 0g ` G ) )
9	4	subm0cl 13050	. . . 4 |- ( S e. (SubMnd ` G ) -> ( 0g ` G ) e. S )
10	9	ad2antrr 488	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( 0g ` G ) e. S )
11	8, 10	eqeltrd 2270	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) e. S )
12		eqid 2193	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2193	. . . 4 |- ( +g ` G ) = ( +g ` G )
14	3	ad2antrr 488	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> G e. Mnd )
15		lennncl 10934	. . . . . . 7 |- ( ( W e. Word S /\ W =/= (/) ) -> (♯ ` W ) e. NN )
16	15	adantll 476	. . . . . 6 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> (♯ ` W ) e. NN )
17		nnm1nn0 9281	. . . . . 6 |- ( (♯ ` W ) e. NN -> ( (♯ ` W ) - 1 ) e. NN0 )
18	16, 17	syl 14	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. NN0 )
19		nn0uz 9627	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
20	18, 19	eleqtrdi 2286	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
21		wrdf 10920	. . . . . . 7 |- ( W e. Word S -> W : ( 0..^ (♯ ` W ) ) --> S )
22	21	ad2antlr 489	. . . . . 6 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0..^ (♯ ` W ) ) --> S )
23	16	nnzd 9438	. . . . . . . 8 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
24		fzoval 10214	. . . . . . . 8 |- ( (♯ ` W ) e. ZZ -> ( 0..^ (♯ ` W ) ) = ( 0 ... ( (♯ ` W ) - 1 ) ) )
25	23, 24	syl 14	. . . . . . 7 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( 0..^ (♯ ` W ) ) = ( 0 ... ( (♯ ` W ) - 1 ) ) )
26	25	feq2d 5391	. . . . . 6 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( W : ( 0..^ (♯ ` W ) ) --> S <-> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> S ) )
27	22, 26	mpbid 147	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> S )
28	12	submss 13048	. . . . . 6 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
29	28	ad2antrr 488	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> S C_ ( Base ` G ) )
30	27, 29	fssd 5416	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` G ) )
31	12, 13, 14, 20, 30	gsumval2 12980	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) = ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) )
32		fvexg 5573	. . . . 5 |- ( ( W e. Word S /\ x e. ( ZZ>= ` 0 ) ) -> ( W ` x ) e. _V )
33	32	ad4ant24 516	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ x e. ( ZZ>= ` 0 ) ) -> ( W ` x ) e. _V )
34	27	ffvelcdmda 5693	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. S )
35	13	submcl 13051	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
36	35	3expb 1206	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
37	36	ad4ant14 514	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
38		ssv 3201	. . . . 5 |- S C_ _V
39	38	a1i 9	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> S C_ _V )
40		simprl 529	. . . . 5 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. _V /\ y e. _V ) ) -> x e. _V )
41	14	adantr 276	. . . . . 6 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. _V /\ y e. _V ) ) -> G e. Mnd )
42		plusgslid 12730	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 12645	. . . . . 6 |- ( G e. Mnd -> ( +g ` G ) e. _V )
44	41, 43	syl 14	. . . . 5 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. _V /\ y e. _V ) ) -> ( +g ` G ) e. _V )
45		simprr 531	. . . . 5 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. _V /\ y e. _V ) ) -> y e. _V )
46		ovexg 5952	. . . . 5 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
47	40, 44, 45, 46	syl3anc 1249	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` G ) y ) e. _V )
48	20, 33, 34, 37, 39, 47	seq3clss 10542	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) e. S )
49	31, 48	eqeltrd 2270	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) e. S )
50		wrdfin 10933	. . . . 5 |- ( W e. Word S -> W e. Fin )
51		fin0or 6942	. . . . 5 |- ( W e. Fin -> ( W = (/) \/ E. j j e. W ) )
52	50, 51	syl 14	. . . 4 |- ( W e. Word S -> ( W = (/) \/ E. j j e. W ) )
53		n0r 3460	. . . . 5 |- ( E. j j e. W -> W =/= (/) )
54	53	orim2i 762	. . . 4 |- ( ( W = (/) \/ E. j j e. W ) -> ( W = (/) \/ W =/= (/) ) )
55	52, 54	syl 14	. . 3 |- ( W e. Word S -> ( W = (/) \/ W =/= (/) ) )
56	55	adantl 277	. 2 |- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( W = (/) \/ W =/= (/) ) )
57	11, 49, 56	mpjaodan 799	1 |- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( G gsumg W ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   =/= wne 2364   _Vcvv 2760   C_ wss 3153   (/)c0 3446   -->wf 5250   ` cfv 5254  (class class class)co 5918   Fincfn 6794   0cc0 7872   1c1 7873   - cmin 8190   NNcn 8982   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074  ..^cfzo 10208   seqcseq 10518  ♯chash 10846  Word cword 10914   Basecbs 12618   +g cplusg 12695   0gc0g 12867   gsum_g cgsu 12868   Mndcmnd 12997  SubMndcsubmnd 13030

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-ihash 10847  df-word 10915  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870  df-submnd 13032

This theorem is referenced by:  gsumwcl  13069