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Theorem gsumwsubmcl 14815
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.
StepHypRef Expression
1 oveq2 6118 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
2 eqid 2442 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
32gsum0 14811 . . . 4  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
41, 3syl6eq 2490 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
54eleq1d 2508 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  e.  S  <->  ( 0g `  G )  e.  S ) )
6 eqid 2442 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2442 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
8 submrcl 14778 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  G  e.  Mnd )
98ad2antrr 708 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  G  e.  Mnd )
10 lennncl 11767 . . . . . . 7  |-  ( ( W  e. Word  S  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
1110adantll 696 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
12 nnm1nn0 10292 . . . . . 6  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  NN0 )
1311, 12syl 16 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (
( # `  W )  -  1 )  e. 
NN0 )
14 nn0uz 10551 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
1513, 14syl6eleq 2532 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (
( # `  W )  -  1 )  e.  ( ZZ>= `  0 )
)
16 wrdf 11764 . . . . . . 7  |-  ( W  e. Word  S  ->  W : ( 0..^ (
# `  W )
) --> S )
1716ad2antlr 709 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0..^ (
# `  W )
) --> S )
1811nnzd 10405 . . . . . . . 8  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ZZ )
19 fzoval 11172 . . . . . . . 8  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
2018, 19syl 16 . . . . . . 7  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (
0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
2120feq2d 5610 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( W : ( 0..^ (
# `  W )
) --> S  <->  W :
( 0 ... (
( # `  W )  -  1 ) ) --> S ) )
2217, 21mpbid 203 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> S )
236submss 14781 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
2423ad2antrr 708 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  S  C_  ( Base `  G
) )
25 fss 5628 . . . . 5  |-  ( ( W : ( 0 ... ( ( # `  W )  -  1 ) ) --> S  /\  S  C_  ( Base `  G
) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> ( Base `  G
) )
2622, 24, 25syl2anc 644 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> ( Base `  G
) )
276, 7, 9, 15, 26gsumval2 14814 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( G  gsumg  W )  =  (  seq  0 ( ( +g  `  G ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
2822ffvelrnda 5899 . . . 4  |-  ( ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )  -> 
( W `  x
)  e.  S )
29 simpll 732 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  S  e.  (SubMnd `  G )
)
307submcl 14784 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( +g  `  G
) y )  e.  S )
31303expb 1155 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( +g  `  G ) y )  e.  S
)
3229, 31sylan 459 . . . 4  |-  ( ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x ( +g  `  G
) y )  e.  S )
3315, 28, 32seqcl 11374 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (  seq  0 ( ( +g  `  G ) ,  W
) `  ( ( # `
 W )  - 
1 ) )  e.  S )
3427, 33eqeltrd 2516 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( G  gsumg  W )  e.  S
)
352subm0cl 14783 . . 3  |-  ( S  e.  (SubMnd `  G
)  ->  ( 0g `  G )  e.  S
)
3635adantr 453 . 2  |-  ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  ->  ( 0g `  G )  e.  S )
375, 34, 36pm2.61ne 2685 1  |-  ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  ->  ( G  gsumg  W )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605    C_ wss 3306   (/)c0 3613   -->wf 5479   ` cfv 5483  (class class class)co 6110   0cc0 9021   1c1 9022    - cmin 9322   NNcn 10031   NN0cn0 10252   ZZcz 10313   ZZ>=cuz 10519   ...cfz 11074  ..^cfzo 11166    seq cseq 11354   #chash 11649  Word cword 11748   Basecbs 13500   +g cplusg 13560   0gc0g 13754    gsumg cgsu 13755   Mndcmnd 14715  SubMndcsubmnd 14768
This theorem is referenced by:  gsumwcl  14817  gsumwspan  14822  frmdss2  14839  psgnunilem5  27432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-fzo 11167  df-seq 11355  df-hash 11650  df-word 11754  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-0g 13758  df-gsum 13759  df-mnd 14721  df-submnd 14770
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