Theorem gsumwsubmcl 13550

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 6018	. . . . 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 13525	. . . . . 6 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
4		eqid 2229	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
5	4	gsum0g 13450	. . . . . 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 2262	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) = ( 0g ` G ) )
9	4	subm0cl 13532	. . . 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 2306	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) e. S )
12		eqid 2229	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2229	. . . 4 |- ( +g ` G ) = ( +g ` G )
14	3	ad2antrr 488	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> G e. Mnd )
15		lennncl 11109	. . . . . . 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 9426	. . . . . 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 9774	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
20	18, 19	eleqtrdi 2322	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
21		wrdf 11095	. . . . . . 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 9584	. . . . . . . 8 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
24		fzoval 10361	. . . . . . . 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 5464	. . . . . 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 13530	. . . . . 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 5489	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` G ) )
31	12, 13, 14, 20, 30	gsumval2 13451	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) = ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) )
32		fvexg 5651	. . . . 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 5775	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. S )
35	13	submcl 13533	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
36	35	3expb 1228	. . . . 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 3246	. . . . 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 13166	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13080	. . . . . 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 6044	. . . . 5 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
47	40, 44, 45, 46	syl3anc 1271	. . . 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 10710	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) e. S )
49	31, 48	eqeltrd 2306	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) e. S )
50		wrdfin 11108	. . . . 5 |- ( W e. Word S -> W e. Fin )
51		fin0or 7061	. . . . 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 3505	. . . . 5 |- ( E. j j e. W -> W =/= (/) )
54	53	orim2i 766	. . . 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 803	1 |- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( G gsumg W ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   =/= wne 2400   _Vcvv 2799   C_ wss 3197   (/)c0 3491   -->wf 5317   ` cfv 5321  (class class class)co 6010   Fincfn 6900   0cc0 8015   1c1 8016   - cmin 8333   NNcn 9126   NN0cn0 9385   ZZcz 9462   ZZ>=cuz 9738   ...cfz 10221  ..^cfzo 10355   seqcseq 10686  ♯chash 11014  Word cword 11089   Basecbs 13053   +g cplusg 13131   0gc0g 13310   gsum_g cgsu 13311   Mndcmnd 13470  SubMndcsubmnd 13512

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

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-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-ihash 11015  df-word 11090  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-0g 13312  df-igsum 13313  df-submnd 13514

This theorem is referenced by:  gsumwcl  13551