Theorem gsumwsubmcl 13640

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 6036	. . . . 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 13615	. . . . . 6 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
4		eqid 2231	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
5	4	gsum0g 13540	. . . . . 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 2264	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) = ( 0g ` G ) )
9	4	subm0cl 13622	. . . 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 2308	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) e. S )
12		eqid 2231	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2231	. . . 4 |- ( +g ` G ) = ( +g ` G )
14	3	ad2antrr 488	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> G e. Mnd )
15		lennncl 11180	. . . . . . 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 9486	. . . . . 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 9834	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
20	18, 19	eleqtrdi 2324	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
21		wrdf 11166	. . . . . . 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 9644	. . . . . . . 8 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
24		fzoval 10426	. . . . . . . 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 5477	. . . . . 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 13620	. . . . . 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 5502	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` G ) )
31	12, 13, 14, 20, 30	gsumval2 13541	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) = ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) )
32		fvexg 5667	. . . . 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 5790	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. S )
35	13	submcl 13623	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
36	35	3expb 1231	. . . . 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 3250	. . . . 5 |- S C_ _V
39	38	a1i 9	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> S C_ _V )
40		simprl 531	. . . . 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 13256	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13170	. . . . . 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 533	. . . . 5 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. _V /\ y e. _V ) ) -> y e. _V )
46		ovexg 6062	. . . . 5 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
47	40, 44, 45, 46	syl3anc 1274	. . . 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 10777	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) e. S )
49	31, 48	eqeltrd 2308	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) e. S )
50		wrdfin 11179	. . . . 5 |- ( W e. Word S -> W e. Fin )
51		fin0or 7118	. . . . 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 3510	. . . . 5 |- ( E. j j e. W -> W =/= (/) )
54	53	orim2i 769	. . . 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 806	1 |- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( G gsumg W ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2202   =/= wne 2403   _Vcvv 2803   C_ wss 3201   (/)c0 3496   -->wf 5329   ` cfv 5333  (class class class)co 6028   Fincfn 6952   0cc0 8075   1c1 8076   - cmin 8393   NNcn 9186   NN0cn0 9445   ZZcz 9522   ZZ>=cuz 9798   ...cfz 10286  ..^cfzo 10420   seqcseq 10753  ♯chash 11081  Word cword 11160   Basecbs 13143   +g cplusg 13221   0gc0g 13400   gsum_g cgsu 13401   Mndcmnd 13560  SubMndcsubmnd 13602

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-n0 9446  df-z 9523  df-uz 9799  df-fz 10287  df-fzo 10421  df-seqfrec 10754  df-ihash 11082  df-word 11161  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-0g 13402  df-igsum 13403  df-submnd 13604

This theorem is referenced by:  gsumwcl  13641