Theorem gsumwsubmcl 13793

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 6066	. . . . 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 13768	. . . . . 6 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
4		eqid 2234	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
5	4	gsum0g 13693	. . . . . 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 2267	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) = ( 0g ` G ) )
9	4	subm0cl 13775	. . . 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 2311	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W = (/) ) -> ( G gsumg W ) e. S )
12		eqid 2234	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2234	. . . 4 |- ( +g ` G ) = ( +g ` G )
14	3	ad2antrr 488	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> G e. Mnd )
15		lennncl 11269	. . . . . . 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 9554	. . . . . 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 9907	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
20	18, 19	eleqtrdi 2327	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
21		wrdf 11255	. . . . . . 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 9717	. . . . . . . 8 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
24		fzoval 10504	. . . . . . . 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 5501	. . . . . 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 13773	. . . . . 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 5527	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` G ) )
31	12, 13, 14, 20, 30	gsumval2 13694	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) = ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) )
32		fvexg 5694	. . . . 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 5817	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. S )
35	13	submcl 13776	. . . . . 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 3264	. . . . 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 13409	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13323	. . . . . 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 6092	. . . . 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 10857	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( seq 0 ( ( +g ` G ) , W ) ` ( (♯ ` W ) - 1 ) ) e. S )
49	31, 48	eqeltrd 2311	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) e. S )
50		wrdfin 11268	. . . . 5 |- ( W e. Word S -> W e. Fin )
51		fin0or 7156	. . . . 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 3526	. . . . 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 2205   =/= wne 2414   _Vcvv 2815   C_ wss 3214   (/)c0 3512   -->wf 5353   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   1c1 8144   - cmin 8460   NNcn 9254   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361  ..^cfzo 10498   seqcseq 10833  ♯chash 11163  Word cword 11249   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554   Mndcmnd 13713  SubMndcsubmnd 13755

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-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-dom 6990  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-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556  df-submnd 13757

This theorem is referenced by:  gsumwcl  13794