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Theorem frmdmnd 18000
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 2821 . 2 (𝐼𝑉 → (Base‘𝑀) = (Base‘𝑀))
2 eqidd 2821 . 2 (𝐼𝑉 → (+g𝑀) = (+g𝑀))
3 frmdmnd.m . . . . . 6 𝑀 = (freeMnd‘𝐼)
4 eqid 2820 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
5 eqid 2820 . . . . . 6 (+g𝑀) = (+g𝑀)
63, 4, 5frmdadd 17996 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
73, 4frmdelbas 17994 . . . . . 6 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
83, 4frmdelbas 17994 . . . . . 6 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9 ccatcl 13904 . . . . . 6 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
107, 8, 9syl2an 597 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
116, 10eqeltrd 2911 . . . 4 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
12113adant1 1126 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
133, 4frmdbas 17993 . . . 4 (𝐼𝑉 → (Base‘𝑀) = Word 𝐼)
14133ad2ant1 1129 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
1512, 14eleqtrrd 2914 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
16 simpr1 1190 . . . . . 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 1191 . . . . . 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 1192 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
213, 4frmdelbas 17994 . . . . . 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 13920 . . . . 5 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2417, 19, 22, 23syl3anc 1367 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2516, 18, 10syl2anc 586 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
2613adantr 483 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
2725, 26eleqtrrd 2914 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
283, 4, 5frmdadd 17996 . . . . 5 (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
2927, 20, 28syl2anc 586 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30 ccatcl 13904 . . . . . . 7 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3119, 22, 30syl2anc 586 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3231, 26eleqtrrd 2914 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
333, 4, 5frmdadd 17996 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3416, 32, 33syl2anc 586 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3524, 29, 343eqtr4d 2865 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
3616, 18, 6syl2anc 586 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
3736oveq1d 7146 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦)(+g𝑀)𝑧))
383, 4, 5frmdadd 17996 . . . . 5 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
3918, 20, 38syl2anc 586 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4039oveq2d 7147 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
4135, 37, 403eqtr4d 2865 . 2 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
42 wrd0 13869 . . 3 ∅ ∈ Word 𝐼
4342, 13eleqtrrid 2918 . 2 (𝐼𝑉 → ∅ ∈ (Base‘𝑀))
443, 4, 5frmdadd 17996 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
4543, 44sylan 582 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
467adantl 484 . . . 4 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47 ccatlid 13918 . . . 4 (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
4846, 47syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
4945, 48eqtrd 2855 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = 𝑥)
503, 4, 5frmdadd 17996 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5150ancoms 461 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5243, 51sylan 582 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
53 ccatrid 13919 . . . 4 (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
5446, 53syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
5552, 54eqtrd 2855 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = 𝑥)
561, 2, 15, 41, 43, 49, 55ismndd 17909 1 (𝐼𝑉𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1537  wcel 2114  c0 4267  cfv 6329  (class class class)co 7131  Word cword 13844   ++ cconcat 13900  Basecbs 16459  +gcplusg 16541  Mndcmnd 17887  freeMndcfrmd 17988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437  ax-cnex 10569  ax-resscn 10570  ax-1cn 10571  ax-icn 10572  ax-addcl 10573  ax-addrcl 10574  ax-mulcl 10575  ax-mulrcl 10576  ax-mulcom 10577  ax-addass 10578  ax-mulass 10579  ax-distr 10580  ax-i2m1 10581  ax-1ne0 10582  ax-1rid 10583  ax-rnegex 10584  ax-rrecex 10585  ax-cnre 10586  ax-pre-lttri 10587  ax-pre-lttrn 10588  ax-pre-ltadd 10589  ax-pre-mulgt0 10590
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-int 4851  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6122  df-ord 6168  df-on 6169  df-lim 6170  df-suc 6171  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7089  df-ov 7134  df-oprab 7135  df-mpo 7136  df-om 7557  df-1st 7665  df-2nd 7666  df-wrecs 7923  df-recs 7984  df-rdg 8022  df-1o 8078  df-oadd 8082  df-er 8265  df-map 8384  df-en 8486  df-dom 8487  df-sdom 8488  df-fin 8489  df-card 9344  df-pnf 10653  df-mnf 10654  df-xr 10655  df-ltxr 10656  df-le 10657  df-sub 10848  df-neg 10849  df-nn 11615  df-2 11677  df-n0 11875  df-z 11959  df-uz 12221  df-fz 12875  df-fzo 13016  df-hash 13674  df-word 13845  df-concat 13901  df-struct 16461  df-ndx 16462  df-slot 16463  df-base 16465  df-plusg 16554  df-mgm 17828  df-sgrp 17877  df-mnd 17888  df-frmd 17990
This theorem is referenced by:  frmdsssubm  18002  frmdgsum  18003  frmdup1  18005  frgp0  18862  frgpadd  18865  frgpmhm  18867  mrsubff  32764  mrsubccat  32770  elmrsubrn  32772
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