Theorem gsumwsubmcl 17375

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . 4 |- ( W = (/) -> ( G gsumg W ) = ( G gsumg (/) ) )
2		eqid 2622	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
3	2	gsum0 17278	. . . 4 |- ( G gsumg (/) ) = ( 0g ` G )
4	1, 3	syl6eq 2672	. . 3 |- ( W = (/) -> ( G gsumg W ) = ( 0g ` G ) )
5	4	eleq1d 2686	. 2 |- ( W = (/) -> ( ( G gsumg W ) e. S <-> ( 0g ` G ) e. S ) )
6		eqid 2622	. . . 4 |- ( Base ` G ) = ( Base ` G )
7		eqid 2622	. . . 4 |- ( +g ` G ) = ( +g ` G )
8		submrcl 17346	. . . . 5 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
9	8	ad2antrr 762	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> G e. Mnd )
10		lennncl 13325	. . . . . . 7 |- ( ( W e. Word S /\ W =/= (/) ) -> ( # ` W ) e. NN )
11	10	adantll 750	. . . . . 6 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( # ` W ) e. NN )
12		nnm1nn0 11334	. . . . . 6 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. NN0 )
13	11, 12	syl 17	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. NN0 )
14		nn0uz 11722	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
15	13, 14	syl6eleq 2711	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
16		wrdf 13310	. . . . . . 7 |- ( W e. Word S -> W : ( 0..^ ( # ` W ) ) --> S )
17	16	ad2antlr 763	. . . . . 6 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0..^ ( # ` W ) ) --> S )
18	11	nnzd 11481	. . . . . . . 8 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( # ` W ) e. ZZ )
19		fzoval 12471	. . . . . . . 8 |- ( ( # ` W ) e. ZZ -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
20	18, 19	syl 17	. . . . . . 7 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
21	20	feq2d 6031	. . . . . 6 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( W : ( 0..^ ( # ` W ) ) --> S <-> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> S ) )
22	17, 21	mpbid 222	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> S )
23	6	submss 17350	. . . . . 6 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
24	23	ad2antrr 762	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> S C_ ( Base ` G ) )
25	22, 24	fssd 6057	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> ( Base ` G ) )
26	6, 7, 9, 15, 25	gsumval2 17280	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) = ( seq 0 ( ( +g ` G ) , W ) ` ( ( # ` W ) - 1 ) ) )
27	22	ffvelrnda 6359	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> ( W ` x ) e. S )
28		simpll 790	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> S e. (SubMnd ` G ) )
29	7	submcl 17353	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) e. S )
30	29	3expb 1266	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
31	28, 30	sylan 488	. . . 4 |- ( ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) e. S )
32	15, 27, 31	seqcl 12821	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( seq 0 ( ( +g ` G ) , W ) ` ( ( # ` W ) - 1 ) ) e. S )
33	26, 32	eqeltrd 2701	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ W e. Word S ) /\ W =/= (/) ) -> ( G gsumg W ) e. S )
34	2	subm0cl 17352	. . 3 |- ( S e. (SubMnd ` G ) -> ( 0g ` G ) e. S )
35	34	adantr 481	. 2 |- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( 0g ` G ) e. S )
36	5, 33, 35	pm2.61ne 2879	1 |- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( G gsumg W ) e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   seqcseq 12801   #chash 13117  Word cword 13291   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  SubMndcsubmnd 17334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336

This theorem is referenced by:  gsumwcl  17377  gsumwspan  17383  frmdss2  17400  psgnunilem5  17914