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Theorem gsumwsubmcl 14477
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 5882 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
2 eqid 2296 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
32gsum0 14473 . . . 4  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
41, 3syl6eq 2344 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
54eleq1d 2362 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  e.  S  <->  ( 0g `  G )  e.  S ) )
6 eqid 2296 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2296 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
8 submrcl 14440 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  G  e.  Mnd )
98ad2antrr 706 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  G  e.  Mnd )
10 lennncl 11438 . . . . . . 7  |-  ( ( W  e. Word  S  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
1110adantll 694 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
12 nnm1nn0 10021 . . . . . 6  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  NN0 )
1311, 12syl 15 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (
( # `  W )  -  1 )  e. 
NN0 )
14 nn0uz 10278 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
1513, 14syl6eleq 2386 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (
( # `  W )  -  1 )  e.  ( ZZ>= `  0 )
)
16 wrdf 11435 . . . . . . 7  |-  ( W  e. Word  S  ->  W : ( 0..^ (
# `  W )
) --> S )
1716ad2antlr 707 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0..^ (
# `  W )
) --> S )
1811nnzd 10132 . . . . . . . 8  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ZZ )
19 fzoval 10892 . . . . . . . 8  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
2018, 19syl 15 . . . . . . 7  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (
0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
2120feq2d 5396 . . . . . 6  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( W : ( 0..^ (
# `  W )
) --> S  <->  W :
( 0 ... (
( # `  W )  -  1 ) ) --> S ) )
2217, 21mpbid 201 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> S )
236submss 14443 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
2423ad2antrr 706 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  S  C_  ( Base `  G
) )
25 fss 5413 . . . . 5  |-  ( ( W : ( 0 ... ( ( # `  W )  -  1 ) ) --> S  /\  S  C_  ( Base `  G
) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> ( Base `  G
) )
2622, 24, 25syl2anc 642 . . . 4  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> ( Base `  G
) )
276, 7, 9, 15, 26gsumval2 14476 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( G  gsumg  W )  =  (  seq  0 ( ( +g  `  G ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
28 ffvelrn 5679 . . . . 5  |-  ( ( W : ( 0 ... ( ( # `  W )  -  1 ) ) --> S  /\  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )  -> 
( W `  x
)  e.  S )
2922, 28sylan 457 . . . 4  |-  ( ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )  -> 
( W `  x
)  e.  S )
30 simpll 730 . . . . 5  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  S  e.  (SubMnd `  G )
)
317submcl 14446 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( +g  `  G
) y )  e.  S )
32313expb 1152 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( +g  `  G ) y )  e.  S
)
3330, 32sylan 457 . . . 4  |-  ( ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x ( +g  `  G
) y )  e.  S )
3415, 29, 33seqcl 11082 . . 3  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  (  seq  0 ( ( +g  `  G ) ,  W
) `  ( ( # `
 W )  - 
1 ) )  e.  S )
3527, 34eqeltrd 2370 . 2  |-  ( ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  /\  W  =/=  (/) )  ->  ( G  gsumg  W )  e.  S
)
362subm0cl 14445 . . 3  |-  ( S  e.  (SubMnd `  G
)  ->  ( 0g `  G )  e.  S
)
3736adantr 451 . 2  |-  ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  ->  ( 0g `  G )  e.  S )
385, 35, 37pm2.61ne 2534 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   -->wf 5267   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798  ..^cfzo 10886    seq cseq 11062   #chash 11353  Word cword 11419   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377  SubMndcsubmnd 14430
This theorem is referenced by:  gsumwcl  14479  gsumwspan  14484  frmdss2  14501  psgnunilem5  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-word 11425  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432
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