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Theorem frmdmnd 17597
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 2761 . 2 (𝐼𝑉 → (Base‘𝑀) = (Base‘𝑀))
2 eqidd 2761 . 2 (𝐼𝑉 → (+g𝑀) = (+g𝑀))
3 frmdmnd.m . . . . . 6 𝑀 = (freeMnd‘𝐼)
4 eqid 2760 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
5 eqid 2760 . . . . . 6 (+g𝑀) = (+g𝑀)
63, 4, 5frmdadd 17593 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
73, 4frmdelbas 17591 . . . . . 6 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
83, 4frmdelbas 17591 . . . . . 6 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9 ccatcl 13546 . . . . . 6 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
107, 8, 9syl2an 495 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
116, 10eqeltrd 2839 . . . 4 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
12113adant1 1125 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
133, 4frmdbas 17590 . . . 4 (𝐼𝑉 → (Base‘𝑀) = Word 𝐼)
14133ad2ant1 1128 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
1512, 14eleqtrrd 2842 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
16 simpr1 1234 . . . . . 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 1236 . . . . . 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 1238 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
213, 4frmdelbas 17591 . . . . . 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 13560 . . . . 5 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2417, 19, 22, 23syl3anc 1477 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2516, 18, 10syl2anc 696 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
2613adantr 472 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
2725, 26eleqtrrd 2842 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
283, 4, 5frmdadd 17593 . . . . 5 (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
2927, 20, 28syl2anc 696 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30 ccatcl 13546 . . . . . . 7 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3119, 22, 30syl2anc 696 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3231, 26eleqtrrd 2842 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
333, 4, 5frmdadd 17593 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3416, 32, 33syl2anc 696 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3524, 29, 343eqtr4d 2804 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
3616, 18, 6syl2anc 696 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
3736oveq1d 6828 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦)(+g𝑀)𝑧))
383, 4, 5frmdadd 17593 . . . . 5 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
3918, 20, 38syl2anc 696 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4039oveq2d 6829 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
4135, 37, 403eqtr4d 2804 . 2 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
42 wrd0 13516 . . 3 ∅ ∈ Word 𝐼
4342, 13syl5eleqr 2846 . 2 (𝐼𝑉 → ∅ ∈ (Base‘𝑀))
443, 4, 5frmdadd 17593 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
4543, 44sylan 489 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
467adantl 473 . . . 4 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47 ccatlid 13558 . . . 4 (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
4846, 47syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
4945, 48eqtrd 2794 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = 𝑥)
503, 4, 5frmdadd 17593 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5150ancoms 468 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5243, 51sylan 489 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
53 ccatrid 13559 . . . 4 (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
5446, 53syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
5552, 54eqtrd 2794 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = 𝑥)
561, 2, 15, 41, 43, 49, 55ismndd 17514 1 (𝐼𝑉𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  c0 4058  cfv 6049  (class class class)co 6813  Word cword 13477   ++ cconcat 13479  Basecbs 16059  +gcplusg 16143  Mndcmnd 17495  freeMndcfrmd 17585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-concat 13487  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-plusg 16156  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-frmd 17587
This theorem is referenced by:  frmdsssubm  17599  frmdgsum  17600  frmdup1  17602  frgp0  18373  frgpadd  18376  frgpmhm  18378  mrsubff  31716  mrsubccat  31722  elmrsubrn  31724
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