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Theorem frmdmnd 14805
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 2438 . 2  |-  ( I  e.  V  ->  ( Base `  M )  =  ( Base `  M
) )
2 eqidd 2438 . 2  |-  ( I  e.  V  ->  ( +g  `  M )  =  ( +g  `  M
) )
3 frmdmnd.m . . . . . 6  |-  M  =  (freeMnd `  I )
4 eqid 2437 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
5 eqid 2437 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
63, 4, 5frmdadd 14801 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  =  ( x concat  y ) )
73, 4frmdelbas 14799 . . . . . 6  |-  ( x  e.  ( Base `  M
)  ->  x  e. Word  I )
83, 4frmdelbas 14799 . . . . . 6  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
9 ccatcl 11744 . . . . . 6  |-  ( ( x  e. Word  I  /\  y  e. Word  I )  ->  ( x concat  y )  e. Word  I )
107, 8, 9syl2an 465 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x concat  y )  e. Word  I
)
116, 10eqeltrd 2511 . . . 4  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
12113adant1 976 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
133, 4frmdbas 14798 . . . 4  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
14133ad2ant1 979 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  ( Base `  M )  = Word 
I )
1512, 14eleqtrrd 2514 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e.  ( Base `  M
) )
16 simpr1 964 . . . . . 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 965 . . . . . 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 966 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
z  e.  ( Base `  M ) )
213, 4frmdelbas 14799 . . . . . 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 11751 . . . . 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 1185 . . . 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 644 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x concat  y )  e. Word  I )
2613adantr 453 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( Base `  M )  = Word  I )
2725, 26eleqtrrd 2514 . . . . 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 14801 . . . . 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 644 . . . 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 11744 . . . . . . 7  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( y concat  z )  e. Word  I )
3119, 22, 30syl2anc 644 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y concat  z )  e. Word  I )
3231, 26eleqtrrd 2514 . . . . 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 14801 . . . . 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 644 . . . 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 2479 . . 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 644 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ( +g  `  M ) y )  =  ( x concat  y
) )
3736oveq1d 6097 . . 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 14801 . . . . 5  |-  ( ( y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
3918, 20, 38syl2anc 644 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y ( +g  `  M ) z )  =  ( y concat  z
) )
4039oveq2d 6098 . . 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 2479 . 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 11733 . . 3  |-  (/)  e. Word  I
4342, 13syl5eleqr 2524 . 2  |-  ( I  e.  V  ->  (/)  e.  (
Base `  M )
)
443, 4, 5frmdadd 14801 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  ( (/) ( +g  `  M
) x )  =  ( (/) concat  x ) )
4543, 44sylan 459 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  ( (/) concat  x ) )
467adantl 454 . . . 4  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  ->  x  e. Word  I )
47 ccatlid 11749 . . . 4  |-  ( x  e. Word  I  ->  ( (/) concat  x )  =  x )
4846, 47syl 16 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) concat  x )  =  x )
4945, 48eqtrd 2469 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  x )
503, 4, 5frmdadd 14801 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  (/)  e.  (
Base `  M )
)  ->  ( x
( +g  `  M )
(/) )  =  ( x concat  (/) ) )
5150ancoms 441 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) (/) )  =  ( x concat  (/) ) )
5243, 51sylan 459 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  ( x concat  (/) ) )
53 ccatrid 11750 . . . 4  |-  ( x  e. Word  I  ->  (
x concat  (/) )  =  x )
5446, 53syl 16 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x concat  (/) )  =  x )
5552, 54eqtrd 2469 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  x )
561, 2, 15, 41, 43, 49, 55ismndd 14720 1  |-  ( I  e.  V  ->  M  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   (/)c0 3629   ` cfv 5455  (class class class)co 6082  Word cword 11718   concat cconcat 11719   Basecbs 13470   +g cplusg 13530   Mndcmnd 14685  freeMndcfrmd 14793
This theorem is referenced by:  frmdsssubm  14807  frmdgsum  14808  frmdup1  14810  frgp0  15393  frgpadd  15396  frgpmhm  15398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-fzo 11137  df-hash 11620  df-word 11724  df-concat 11725  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-plusg 13543  df-mnd 14691  df-frmd 14795
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