Theorem gsumwsubmcl 13515

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 6002	. . . . 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 13490	. . . . . 6 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
4		eqid 2229	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
5	4	gsum0g 13415	. . . . . 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 13497	. . . 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 11078	. . . . . . 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 9398	. . . . . 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 9745	. . . . 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 11064	. . . . . . 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 9556	. . . . . . . 8 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
24		fzoval 10332	. . . . . . . 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 5457	. . . . . 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 13495	. . . . . 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 5482	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` G ) )
31	12, 13, 14, 20, 30	gsumval2 13416	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) = ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) )
32		fvexg 5642	. . . . 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 5763	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. S )
35	13	submcl 13498	. . . . . 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 13131	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13045	. . . . . 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 6028	. . . . 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 10680	. . 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 11077	. . . . 5 |- ( W e. Word S -> W e. Fin )
51		fin0or 7036	. . . . 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 5310   ` cfv 5314  (class class class)co 5994   Fincfn 6877   0cc0 7987   1c1 7988   - cmin 8305   NNcn 9098   NN0cn0 9357   ZZcz 9434   ZZ>=cuz 9710   ...cfz 10192  ..^cfzo 10326   seqcseq 10656  ♯chash 10984  Word cword 11058   Basecbs 13018   +g cplusg 13096   0gc0g 13275   gsum_g cgsu 13276   Mndcmnd 13435  SubMndcsubmnd 13477

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-2 9157  df-n0 9358  df-z 9435  df-uz 9711  df-fz 10193  df-fzo 10327  df-seqfrec 10657  df-ihash 10985  df-word 11059  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-0g 13277  df-igsum 13278  df-submnd 13479

This theorem is referenced by:  gsumwcl  13516