MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frmdmnd Structured version   Visualization version   GIF version

Theorem frmdmnd 18012
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 (𝐼𝑉𝑀 ∈ Mnd)

Proof of Theorem frmdmnd
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2819 . 2 (𝐼𝑉 → (Base‘𝑀) = (Base‘𝑀))
2 eqidd 2819 . 2 (𝐼𝑉 → (+g𝑀) = (+g𝑀))
3 frmdmnd.m . . . . . 6 𝑀 = (freeMnd‘𝐼)
4 eqid 2818 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
5 eqid 2818 . . . . . 6 (+g𝑀) = (+g𝑀)
63, 4, 5frmdadd 18008 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
73, 4frmdelbas 18006 . . . . . 6 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
83, 4frmdelbas 18006 . . . . . 6 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9 ccatcl 13914 . . . . . 6 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
107, 8, 9syl2an 595 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
116, 10eqeltrd 2910 . . . 4 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
12113adant1 1122 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
133, 4frmdbas 18005 . . . 4 (𝐼𝑉 → (Base‘𝑀) = Word 𝐼)
14133ad2ant1 1125 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
1512, 14eleqtrrd 2913 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
16 simpr1 1186 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
1716, 7syl 17 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ Word 𝐼)
18 simpr2 1187 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
1918, 8syl 17 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
20 simpr3 1188 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
213, 4frmdelbas 18006 . . . . . 6 (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
2220, 21syl 17 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
23 ccatass 13930 . . . . 5 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2417, 19, 22, 23syl3anc 1363 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2516, 18, 10syl2anc 584 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
2613adantr 481 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
2725, 26eleqtrrd 2913 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
283, 4, 5frmdadd 18008 . . . . 5 (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
2927, 20, 28syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30 ccatcl 13914 . . . . . . 7 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3119, 22, 30syl2anc 584 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3231, 26eleqtrrd 2913 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
333, 4, 5frmdadd 18008 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3416, 32, 33syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3524, 29, 343eqtr4d 2863 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
3616, 18, 6syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
3736oveq1d 7160 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦)(+g𝑀)𝑧))
383, 4, 5frmdadd 18008 . . . . 5 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
3918, 20, 38syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4039oveq2d 7161 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
4135, 37, 403eqtr4d 2863 . 2 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
42 wrd0 13877 . . 3 ∅ ∈ Word 𝐼
4342, 13eleqtrrid 2917 . 2 (𝐼𝑉 → ∅ ∈ (Base‘𝑀))
443, 4, 5frmdadd 18008 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
4543, 44sylan 580 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
467adantl 482 . . . 4 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47 ccatlid 13928 . . . 4 (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
4846, 47syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
4945, 48eqtrd 2853 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = 𝑥)
503, 4, 5frmdadd 18008 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5150ancoms 459 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5243, 51sylan 580 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
53 ccatrid 13929 . . . 4 (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
5446, 53syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
5552, 54eqtrd 2853 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = 𝑥)
561, 2, 15, 41, 43, 49, 55ismndd 17921 1 (𝐼𝑉𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  c0 4288  cfv 6348  (class class class)co 7145  Word cword 13849   ++ cconcat 13910  Basecbs 16471  +gcplusg 16553  Mndcmnd 17899  freeMndcfrmd 18000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-concat 13911  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-plusg 16566  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-frmd 18002
This theorem is referenced by:  frmdsssubm  18014  frmdgsum  18015  frmdup1  18017  frgp0  18815  frgpadd  18818  frgpmhm  18820  mrsubff  32656  mrsubccat  32662  elmrsubrn  32664
  Copyright terms: Public domain W3C validator