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Theorem frmdmnd 15845
Description: A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypothesis
Ref Expression
frmdmnd.m  |-  M  =  (freeMnd `  I )
Assertion
Ref Expression
frmdmnd  |-  ( I  e.  V  ->  M  e.  Mnd )

Proof of Theorem frmdmnd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2463 . 2  |-  ( I  e.  V  ->  ( Base `  M )  =  ( Base `  M
) )
2 eqidd 2463 . 2  |-  ( I  e.  V  ->  ( +g  `  M )  =  ( +g  `  M
) )
3 frmdmnd.m . . . . . 6  |-  M  =  (freeMnd `  I )
4 eqid 2462 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
5 eqid 2462 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
63, 4, 5frmdadd 15841 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  =  ( x concat  y ) )
73, 4frmdelbas 15839 . . . . . 6  |-  ( x  e.  ( Base `  M
)  ->  x  e. Word  I )
83, 4frmdelbas 15839 . . . . . 6  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
9 ccatcl 12547 . . . . . 6  |-  ( ( x  e. Word  I  /\  y  e. Word  I )  ->  ( x concat  y )  e. Word  I )
107, 8, 9syl2an 477 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x concat  y )  e. Word  I
)
116, 10eqeltrd 2550 . . . 4  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
12113adant1 1009 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
133, 4frmdbas 15838 . . . 4  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
14133ad2ant1 1012 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  ( Base `  M )  = Word 
I )
1512, 14eleqtrrd 2553 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e.  ( Base `  M
) )
16 simpr1 997 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  ->  x  e.  ( Base `  M ) )
1716, 7syl 16 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  ->  x  e. Word  I )
18 simpr2 998 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
y  e.  ( Base `  M ) )
1918, 8syl 16 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
y  e. Word  I )
20 simpr3 999 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
z  e.  ( Base `  M ) )
213, 4frmdelbas 15839 . . . . . 6  |-  ( z  e.  ( Base `  M
)  ->  z  e. Word  I )
2220, 21syl 16 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
z  e. Word  I )
23 ccatass 12559 . . . . 5  |-  ( ( x  e. Word  I  /\  y  e. Word  I  /\  z  e. Word  I )  ->  (
( x concat  y ) concat  z )  =  ( x concat 
( y concat  z )
) )
2417, 19, 22, 23syl3anc 1223 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x concat  y
) concat  z )  =  ( x concat  ( y concat  z
) ) )
2516, 18, 10syl2anc 661 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x concat  y )  e. Word  I )
2613adantr 465 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( Base `  M )  = Word  I )
2725, 26eleqtrrd 2553 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x concat  y )  e.  ( Base `  M
) )
283, 4, 5frmdadd 15841 . . . . 5  |-  ( ( ( x concat  y )  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
)  ->  ( (
x concat  y ) ( +g  `  M ) z )  =  ( ( x concat 
y ) concat  z )
)
2927, 20, 28syl2anc 661 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x concat  y
) ( +g  `  M
) z )  =  ( ( x concat  y
) concat  z ) )
30 ccatcl 12547 . . . . . . 7  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( y concat  z )  e. Word  I )
3119, 22, 30syl2anc 661 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y concat  z )  e. Word  I )
3231, 26eleqtrrd 2553 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y concat  z )  e.  ( Base `  M
) )
333, 4, 5frmdadd 15841 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  (
y concat  z )  e.  (
Base `  M )
)  ->  ( x
( +g  `  M ) ( y concat  z ) )  =  ( x concat 
( y concat  z )
) )
3416, 32, 33syl2anc 661 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ( +g  `  M ) ( y concat 
z ) )  =  ( x concat  ( y concat 
z ) ) )
3524, 29, 343eqtr4d 2513 . . 3  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x concat  y
) ( +g  `  M
) z )  =  ( x ( +g  `  M ) ( y concat 
z ) ) )
3616, 18, 6syl2anc 661 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ( +g  `  M ) y )  =  ( x concat  y
) )
3736oveq1d 6292 . . 3  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x ( +g  `  M ) y ) ( +g  `  M ) z )  =  ( ( x concat 
y ) ( +g  `  M ) z ) )
383, 4, 5frmdadd 15841 . . . . 5  |-  ( ( y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
3918, 20, 38syl2anc 661 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y ( +g  `  M ) z )  =  ( y concat  z
) )
4039oveq2d 6293 . . 3  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ( +g  `  M ) ( y ( +g  `  M
) z ) )  =  ( x ( +g  `  M ) ( y concat  z ) ) )
4135, 37, 403eqtr4d 2513 . 2  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x ( +g  `  M ) y ) ( +g  `  M ) z )  =  ( x ( +g  `  M ) ( y ( +g  `  M ) z ) ) )
42 wrd0 12520 . . 3  |-  (/)  e. Word  I
4342, 13syl5eleqr 2557 . 2  |-  ( I  e.  V  ->  (/)  e.  (
Base `  M )
)
443, 4, 5frmdadd 15841 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  ( (/) ( +g  `  M
) x )  =  ( (/) concat  x ) )
4543, 44sylan 471 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  ( (/) concat  x ) )
467adantl 466 . . . 4  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  ->  x  e. Word  I )
47 ccatlid 12557 . . . 4  |-  ( x  e. Word  I  ->  ( (/) concat  x )  =  x )
4846, 47syl 16 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) concat  x )  =  x )
4945, 48eqtrd 2503 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  x )
503, 4, 5frmdadd 15841 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  (/)  e.  (
Base `  M )
)  ->  ( x
( +g  `  M )
(/) )  =  ( x concat  (/) ) )
5150ancoms 453 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) (/) )  =  ( x concat  (/) ) )
5243, 51sylan 471 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  ( x concat  (/) ) )
53 ccatrid 12558 . . . 4  |-  ( x  e. Word  I  ->  (
x concat  (/) )  =  x )
5446, 53syl 16 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x concat  (/) )  =  x )
5552, 54eqtrd 2503 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  x )
561, 2, 15, 41, 43, 49, 55ismndd 15752 1  |-  ( I  e.  V  ->  M  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   (/)c0 3780   ` cfv 5581  (class class class)co 6277  Word cword 12489   concat cconcat 12491   Basecbs 14481   +g cplusg 14546   Mndcmnd 15717  freeMndcfrmd 15833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-concat 12499  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-mnd 15723  df-frmd 15835
This theorem is referenced by:  frmdsssubm  15847  frmdgsum  15848  frmdup1  15850  frgp0  16569  frgpadd  16572  frgpmhm  16574
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