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Theorem frmdmnd 16241
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 2401 . 2  |-  ( I  e.  V  ->  ( Base `  M )  =  ( Base `  M
) )
2 eqidd 2401 . 2  |-  ( I  e.  V  ->  ( +g  `  M )  =  ( +g  `  M
) )
3 frmdmnd.m . . . . . 6  |-  M  =  (freeMnd `  I )
4 eqid 2400 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
5 eqid 2400 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
63, 4, 5frmdadd 16237 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  =  ( x ++  y ) )
73, 4frmdelbas 16235 . . . . . 6  |-  ( x  e.  ( Base `  M
)  ->  x  e. Word  I )
83, 4frmdelbas 16235 . . . . . 6  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
9 ccatcl 12552 . . . . . 6  |-  ( ( x  e. Word  I  /\  y  e. Word  I )  ->  ( x ++  y )  e. Word  I )
107, 8, 9syl2an 475 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ++  y )  e. Word 
I )
116, 10eqeltrd 2488 . . . 4  |-  ( ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
12113adant1 1013 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e. Word 
I )
133, 4frmdbas 16234 . . . 4  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
14133ad2ant1 1016 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  ( Base `  M )  = Word 
I )
1512, 14eleqtrrd 2491 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) y )  e.  ( Base `  M
) )
16 simpr1 1001 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  ->  x  e.  ( Base `  M ) )
1716, 7syl 17 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  ->  x  e. Word  I )
18 simpr2 1002 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
y  e.  ( Base `  M ) )
1918, 8syl 17 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
y  e. Word  I )
20 simpr3 1003 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
z  e.  ( Base `  M ) )
213, 4frmdelbas 16235 . . . . . 6  |-  ( z  e.  ( Base `  M
)  ->  z  e. Word  I )
2220, 21syl 17 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
z  e. Word  I )
23 ccatass 12564 . . . . 5  |-  ( ( x  e. Word  I  /\  y  e. Word  I  /\  z  e. Word  I )  ->  (
( x ++  y ) ++  z )  =  ( x ++  ( y ++  z ) ) )
2417, 19, 22, 23syl3anc 1228 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x ++  y
) ++  z )  =  ( x ++  ( y ++  z ) ) )
2516, 18, 10syl2anc 659 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ++  y )  e. Word  I )
2613adantr 463 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( Base `  M )  = Word  I )
2725, 26eleqtrrd 2491 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ++  y )  e.  ( Base `  M
) )
283, 4, 5frmdadd 16237 . . . . 5  |-  ( ( ( x ++  y )  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
)  ->  ( (
x ++  y ) ( +g  `  M ) z )  =  ( ( x ++  y ) ++  z ) )
2927, 20, 28syl2anc 659 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x ++  y
) ( +g  `  M
) z )  =  ( ( x ++  y
) ++  z ) )
30 ccatcl 12552 . . . . . . 7  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( y ++  z )  e. Word  I )
3119, 22, 30syl2anc 659 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y ++  z )  e. Word  I )
3231, 26eleqtrrd 2491 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y ++  z )  e.  ( Base `  M
) )
333, 4, 5frmdadd 16237 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  (
y ++  z )  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) ( y ++  z ) )  =  ( x ++  ( y ++  z ) ) )
3416, 32, 33syl2anc 659 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ( +g  `  M ) ( y ++  z ) )  =  ( x ++  ( y ++  z ) ) )
3524, 29, 343eqtr4d 2451 . . 3  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x ++  y
) ( +g  `  M
) z )  =  ( x ( +g  `  M ) ( y ++  z ) ) )
3616, 18, 6syl2anc 659 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( x ( +g  `  M ) y )  =  ( x ++  y
) )
3736oveq1d 6247 . . 3  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( ( x ( +g  `  M ) y ) ( +g  `  M ) z )  =  ( ( x ++  y ) ( +g  `  M ) z ) )
383, 4, 5frmdadd 16237 . . . . 5  |-  ( ( y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) )  ->  (
y ( +g  `  M
) z )  =  ( y ++  z ) )
3918, 20, 38syl2anc 659 . . . 4  |-  ( ( I  e.  V  /\  ( x  e.  ( Base `  M )  /\  y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) ) )  -> 
( y ( +g  `  M ) z )  =  ( y ++  z ) )
4039oveq2d 6248 . . 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 ++  z ) ) )
4135, 37, 403eqtr4d 2451 . 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 12523 . . 3  |-  (/)  e. Word  I
4342, 13syl5eleqr 2495 . 2  |-  ( I  e.  V  ->  (/)  e.  (
Base `  M )
)
443, 4, 5frmdadd 16237 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  ( (/) ( +g  `  M
) x )  =  ( (/) ++  x )
)
4543, 44sylan 469 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  ( (/) ++  x )
)
467adantl 464 . . . 4  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  ->  x  e. Word  I )
47 ccatlid 12562 . . . 4  |-  ( x  e. Word  I  ->  ( (/) ++ 
x )  =  x )
4846, 47syl 17 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ++  x )  =  x )
4945, 48eqtrd 2441 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( (/) ( +g  `  M
) x )  =  x )
503, 4, 5frmdadd 16237 . . . . 5  |-  ( ( x  e.  ( Base `  M )  /\  (/)  e.  (
Base `  M )
)  ->  ( x
( +g  `  M )
(/) )  =  ( x ++  (/) ) )
5150ancoms 451 . . . 4  |-  ( (
(/)  e.  ( Base `  M )  /\  x  e.  ( Base `  M
) )  ->  (
x ( +g  `  M
) (/) )  =  ( x ++  (/) ) )
5243, 51sylan 469 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  ( x ++  (/) ) )
53 ccatrid 12563 . . . 4  |-  ( x  e. Word  I  ->  (
x ++  (/) )  =  x )
5446, 53syl 17 . . 3  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ++  (/) )  =  x )
5552, 54eqtrd 2441 . 2  |-  ( ( I  e.  V  /\  x  e.  ( Base `  M ) )  -> 
( x ( +g  `  M ) (/) )  =  x )
561, 2, 15, 41, 43, 49, 55ismndd 16157 1  |-  ( I  e.  V  ->  M  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   (/)c0 3735   ` cfv 5523  (class class class)co 6232  Word cword 12488   ++ cconcat 12490   Basecbs 14731   +g cplusg 14799   Mndcmnd 16133  freeMndcfrmd 16229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-hash 12358  df-word 12496  df-concat 12498  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-plusg 14812  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-frmd 16231
This theorem is referenced by:  frmdsssubm  16243  frmdgsum  16244  frmdup1  16246  frgp0  16992  frgpadd  16995  frgpmhm  16997  mrsubff  29600  mrsubccat  29606  elmrsubrn  29608
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