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Theorem frmdmnd 14497
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 2297 . 2  |-  ( I  e.  V  ->  ( Base `  M )  =  ( Base `  M
) )
2 eqidd 2297 . 2  |-  ( I  e.  V  ->  ( +g  `  M )  =  ( +g  `  M
) )
3 frmdmnd.m . . . . . 6  |-  M  =  (freeMnd `  I )
4 eqid 2296 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
5 eqid 2296 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
63, 4, 5frmdadd 14493 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  =  ( x concat  y ) )
73, 4frmdelbas 14491 . . . . . 6  |-  ( x  e.  ( Base `  M
)  ->  x  e. Word  I )
83, 4frmdelbas 14491 . . . . . 6  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
9 ccatcl 11445 . . . . . 6  |-  ( ( x  e. Word  I  /\  y  e. Word  I )  ->  ( x concat  y )  e. Word  I )
107, 8, 9syl2an 463 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x concat  y )  e. Word  I
)
116, 10eqeltrd 2370 . . . 4  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
12113adant1 973 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
133, 4frmdbas 14490 . . . 4  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
14133ad2ant1 976 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  ( Base `  M )  = Word 
I )
1512, 14eleqtrrd 2373 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e.  ( Base `  M
) )
16 simpr1 961 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  ->  x  e.  ( Base `  M ) )
1716, 7syl 15 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  ->  x  e. Word  I )
18 simpr2 962 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
y  e.  ( Base `  M ) )
1918, 8syl 15 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
y  e. Word  I )
20 simpr3 963 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
z  e.  ( Base `  M ) )
213, 4frmdelbas 14491 . . . . . 6  |-  ( z  e.  ( Base `  M
)  ->  z  e. Word  I )
2220, 21syl 15 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
z  e. Word  I )
23 ccatass 11452 . . . . 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 1182 . . . 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 642 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x concat  y )  e. Word  I )
2613adantr 451 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( Base `  M )  = Word  I )
2725, 26eleqtrrd 2373 . . . . 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 14493 . . . . 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 642 . . . 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 11445 . . . . . . 7  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( y concat  z )  e. Word  I )
3119, 22, 30syl2anc 642 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y concat  z )  e. Word  I )
3231, 26eleqtrrd 2373 . . . . 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 14493 . . . . 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 642 . . . 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 2338 . . 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 642 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ( +g  `  M ) y )  =  ( x concat  y
) )
3736oveq1d 5889 . . 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 14493 . . . . 5  |-  ( ( y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
3918, 20, 38syl2anc 642 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y ( +g  `  M ) z )  =  ( y concat  z
) )
4039oveq2d 5890 . . 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 2338 . 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 11434 . . 3  |-  (/)  e. Word  I
4342, 13syl5eleqr 2383 . 2  |-  ( I  e.  V  ->  (/)  e.  (
Base `  M )
)
443, 4, 5frmdadd 14493 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  ( (/) ( +g  `  M
) x )  =  ( (/) concat  x ) )
4543, 44sylan 457 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  ( (/) concat  x ) )
467adantl 452 . . . 4  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  ->  x  e. Word  I )
47 ccatlid 11450 . . . 4  |-  ( x  e. Word  I  ->  ( (/) concat  x )  =  x )
4846, 47syl 15 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) concat  x )  =  x )
4945, 48eqtrd 2328 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  x )
503, 4, 5frmdadd 14493 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  (/)  e.  (
Base `  M )
)  ->  ( x
( +g  `  M )
(/) )  =  ( x concat  (/) ) )
5150ancoms 439 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) (/) )  =  ( x concat  (/) ) )
5243, 51sylan 457 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  ( x concat  (/) ) )
53 ccatrid 11451 . . . 4  |-  ( x  e. Word  I  ->  (
x concat  (/) )  =  x )
5446, 53syl 15 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x concat  (/) )  =  x )
5552, 54eqtrd 2328 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  x )
561, 2, 15, 41, 43, 49, 55ismndd 14412 1  |-  ( I  e.  V  ->  M  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   (/)c0 3468   ` cfv 5271  (class class class)co 5874  Word cword 11419   concat cconcat 11420   Basecbs 13164   +g cplusg 13224   Mndcmnd 14377  freeMndcfrmd 14485
This theorem is referenced by:  frmdsssubm  14499  frmdgsum  14500  frmdup1  14502  frgp0  15085  frgpadd  15088  frgpmhm  15090
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-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-map 6790  df-pm 6791  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-hash 11354  df-word 11425  df-concat 11426  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mnd 14383  df-frmd 14487
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