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Theorem frmdmnd 18819
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 2726 . 2 (𝐼𝑉 → (Base‘𝑀) = (Base‘𝑀))
2 eqidd 2726 . 2 (𝐼𝑉 → (+g𝑀) = (+g𝑀))
3 frmdmnd.m . . . . . 6 𝑀 = (freeMnd‘𝐼)
4 eqid 2725 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
5 eqid 2725 . . . . . 6 (+g𝑀) = (+g𝑀)
63, 4, 5frmdadd 18815 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
73, 4frmdelbas 18813 . . . . . 6 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
83, 4frmdelbas 18813 . . . . . 6 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9 ccatcl 14560 . . . . . 6 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
107, 8, 9syl2an 594 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
116, 10eqeltrd 2825 . . . 4 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
12113adant1 1127 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
133, 4frmdbas 18812 . . . 4 (𝐼𝑉 → (Base‘𝑀) = Word 𝐼)
14133ad2ant1 1130 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
1512, 14eleqtrrd 2828 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
16 simpr1 1191 . . . . . 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 1192 . . . . . 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 1193 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
213, 4frmdelbas 18813 . . . . . 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 14574 . . . . 5 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2417, 19, 22, 23syl3anc 1368 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2516, 18, 10syl2anc 582 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
2613adantr 479 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
2725, 26eleqtrrd 2828 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
283, 4, 5frmdadd 18815 . . . . 5 (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
2927, 20, 28syl2anc 582 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30 ccatcl 14560 . . . . . . 7 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3119, 22, 30syl2anc 582 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3231, 26eleqtrrd 2828 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
333, 4, 5frmdadd 18815 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3416, 32, 33syl2anc 582 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3524, 29, 343eqtr4d 2775 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
3616, 18, 6syl2anc 582 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
3736oveq1d 7434 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦)(+g𝑀)𝑧))
383, 4, 5frmdadd 18815 . . . . 5 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
3918, 20, 38syl2anc 582 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4039oveq2d 7435 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
4135, 37, 403eqtr4d 2775 . 2 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
42 wrd0 14525 . . 3 ∅ ∈ Word 𝐼
4342, 13eleqtrrid 2832 . 2 (𝐼𝑉 → ∅ ∈ (Base‘𝑀))
443, 4, 5frmdadd 18815 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
4543, 44sylan 578 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
467adantl 480 . . . 4 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47 ccatlid 14572 . . . 4 (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
4846, 47syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
4945, 48eqtrd 2765 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = 𝑥)
503, 4, 5frmdadd 18815 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5150ancoms 457 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5243, 51sylan 578 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
53 ccatrid 14573 . . . 4 (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
5446, 53syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
5552, 54eqtrd 2765 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = 𝑥)
561, 2, 15, 41, 43, 49, 55ismndd 18719 1 (𝐼𝑉𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  c0 4322  cfv 6549  (class class class)co 7419  Word cword 14500   ++ cconcat 14556  Basecbs 17183  +gcplusg 17236  Mndcmnd 18697  freeMndcfrmd 18807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-fzo 13663  df-hash 14326  df-word 14501  df-concat 14557  df-struct 17119  df-slot 17154  df-ndx 17166  df-base 17184  df-plusg 17249  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-frmd 18809
This theorem is referenced by:  frmdsssubm  18821  frmdgsum  18822  frmdup1  18824  frgp0  19727  frgpadd  19730  frgpmhm  19732  mrsubff  35253  mrsubccat  35259  elmrsubrn  35261
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