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Theorem gsumwsubmcl 16205
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 6278 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
2 eqid 2454 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
32gsum0 16104 . . . 4  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
41, 3syl6eq 2511 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
54eleq1d 2523 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  e.  S  <->  ( 0g `  G )  e.  S ) )
6 eqid 2454 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2454 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
8 submrcl 16176 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  G  e.  Mnd )
98ad2antrr 723 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  G  e.  Mnd )
10 lennncl 12550 . . . . . . 7  |-  ( ( W  e. Word  S  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
1110adantll 711 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
12 nnm1nn0 10833 . . . . . 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 11116 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
1513, 14syl6eleq 2552 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (
( # `  W )  -  1 )  e.  ( ZZ>= `  0 )
)
16 wrdf 12538 . . . . . . 7  |-  ( W  e. Word  S  ->  W : ( 0..^ (
# `  W )
) --> S )
1716ad2antlr 724 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0..^ (
# `  W )
) --> S )
1811nnzd 10964 . . . . . . . 8  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ZZ )
19 fzoval 11805 . . . . . . . 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 5700 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( W : ( 0..^ (
# `  W )
) --> S  <->  W :
( 0 ... (
( # `  W )  -  1 ) ) --> S ) )
2217, 21mpbid 210 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> S )
236submss 16180 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
2423ad2antrr 723 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  S  C_  ( Base `  G
) )
2522, 24fssd 5722 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> ( Base `  G
) )
266, 7, 9, 15, 25gsumval2 16106 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( G  gsumg  W )  =  (  seq 0 ( ( +g  `  G ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
2722ffvelrnda 6007 . . . 4  |-  ( ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )  -> 
( W `  x
)  e.  S )
28 simpll 751 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  S  e.  (SubMnd `  G )
)
297submcl 16183 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( +g  `  G
) y )  e.  S )
30293expb 1195 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( +g  `  G ) y )  e.  S
)
3128, 30sylan 469 . . . 4  |-  ( ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x ( +g  `  G
) y )  e.  S )
3215, 27, 31seqcl 12109 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (  seq 0 ( ( +g  `  G ) ,  W
) `  ( ( # `
 W )  - 
1 ) )  e.  S )
3326, 32eqeltrd 2542 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( G  gsumg  W )  e.  S
)
342subm0cl 16182 . . 3  |-  ( S  e.  (SubMnd `  G
)  ->  ( 0g `  G )  e.  S
)
3534adantr 463 . 2  |-  ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  ->  ( 0g `  G )  e.  S )
365, 33, 35pm2.61ne 2769 1  |-  ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  ->  ( G  gsumg  W )  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783   -->wf 5566   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    - cmin 9796   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675  ..^cfzo 11799    seqcseq 12089   #chash 12387  Word cword 12518   Basecbs 14716   +g cplusg 14784   0gc0g 14929    gsumg cgsu 14930   Mndcmnd 16118  SubMndcsubmnd 16164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-word 12526  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166
This theorem is referenced by:  gsumwcl  16207  gsumwspan  16213  frmdss2  16230  psgnunilem5  16718
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