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Theorem frmdmnd 14481
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 2284 . 2  |-  ( I  e.  V  ->  ( Base `  M )  =  ( Base `  M
) )
2 eqidd 2284 . 2  |-  ( I  e.  V  ->  ( +g  `  M )  =  ( +g  `  M
) )
3 frmdmnd.m . . . . . 6  |-  M  =  (freeMnd `  I )
4 eqid 2283 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
5 eqid 2283 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
63, 4, 5frmdadd 14477 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  =  ( x concat  y ) )
73, 4frmdelbas 14475 . . . . . 6  |-  ( x  e.  ( Base `  M
)  ->  x  e. Word  I )
83, 4frmdelbas 14475 . . . . . 6  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
9 ccatcl 11429 . . . . . 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 2357 . . . 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 14474 . . . 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 2360 . 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 14475 . . . . . 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 11436 . . . . 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 2360 . . . . 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 14477 . . . . 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 11429 . . . . . . 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 2360 . . . . 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 14477 . . . . 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 2325 . . 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 5873 . . 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 14477 . . . . 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 5874 . . 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 2325 . 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 11418 . . 3  |-  (/)  e. Word  I
4342, 13syl5eleqr 2370 . 2  |-  ( I  e.  V  ->  (/)  e.  (
Base `  M )
)
443, 4, 5frmdadd 14477 . . . 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 11434 . . . 4  |-  ( x  e. Word  I  ->  ( (/) concat  x )  =  x )
4846, 47syl 15 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) concat  x )  =  x )
4945, 48eqtrd 2315 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  x )
503, 4, 5frmdadd 14477 . . . . 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 11435 . . . 4  |-  ( x  e. Word  I  ->  (
x concat  (/) )  =  x )
5446, 53syl 15 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x concat  (/) )  =  x )
5552, 54eqtrd 2315 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  x )
561, 2, 15, 41, 43, 49, 55ismndd 14396 1  |-  ( I  e.  V  ->  M  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   (/)c0 3455   ` cfv 5255  (class class class)co 5858  Word cword 11403   concat cconcat 11404   Basecbs 13148   +g cplusg 13208   Mndcmnd 14361  freeMndcfrmd 14469
This theorem is referenced by:  frmdsssubm  14483  frmdgsum  14484  frmdup1  14486  frgp0  15069  frgpadd  15072  frgpmhm  15074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mnd 14367  df-frmd 14471
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