Theorem frmdmnd 18894

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 ∈ 𝑉 → 𝑀 ∈ Mnd)

Proof of Theorem frmdmnd

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2741	. 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = (Base‘𝑀))
2		eqidd 2741	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘𝑀) = (+g‘𝑀))
3		frmdmnd.m	. . . . . 6 ⊢ 𝑀 = (freeMnd‘𝐼)
4		eqid 2740	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
5		eqid 2740	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
6	3, 4, 5	frmdadd 18890	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
7	3, 4	frmdelbas 18888	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
8	3, 4	frmdelbas 18888	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9		ccatcl 14622	. . . . . 6 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
10	7, 8, 9	syl2an 595	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
11	6, 10	eqeltrd 2844	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ Word 𝐼)
12	11	3adant1 1130	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ Word 𝐼)
13	3, 4	frmdbas 18887	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = Word 𝐼)
14	13	3ad2ant1 1133	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
15	12, 14	eleqtrrd 2847	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
16		simpr1 1194	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
17	16, 7	syl 17	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ Word 𝐼)
18		simpr2 1195	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
19	18, 8	syl 17	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
20		simpr3 1196	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
21	3, 4	frmdelbas 18888	. . . . . 6 ⊢ (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
23		ccatass 14636	. . . . 5 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
24	17, 19, 22, 23	syl3anc 1371	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
25	16, 18, 10	syl2anc 583	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
26	13	adantr 480	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
27	25, 26	eleqtrrd 2847	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
28	3, 4, 5	frmdadd 18890	. . . . 5 ⊢ (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
29	27, 20, 28	syl2anc 583	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30		ccatcl 14622	. . . . . . 7 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
31	19, 22, 30	syl2anc 583	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
32	31, 26	eleqtrrd 2847	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
33	3, 4, 5	frmdadd 18890	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
34	16, 32, 33	syl2anc 583	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
35	24, 29, 34	3eqtr4d 2790	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦 ++ 𝑧)))
36	16, 18, 6	syl2anc 583	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
37	36	oveq1d 7463	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧))
38	3, 4, 5	frmdadd 18890	. . . . 5 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
39	18, 20, 38	syl2anc 583	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
40	39	oveq2d 7464	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)) = (𝑥(+g‘𝑀)(𝑦 ++ 𝑧)))
41	35, 37, 40	3eqtr4d 2790	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
42		wrd0 14587	. . 3 ⊢ ∅ ∈ Word 𝐼
43	42, 13	eleqtrrid 2851	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘𝑀))
44	3, 4, 5	frmdadd 18890	. . . 4 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥))
45	43, 44	sylan 579	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥))
46	7	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47		ccatlid 14634	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
48	46, 47	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
49	45, 48	eqtrd 2780	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥)
50	3, 4, 5	frmdadd 18890	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅))
51	50	ancoms 458	. . . 4 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅))
52	43, 51	sylan 579	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅))
53		ccatrid 14635	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
54	46, 53	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
55	52, 54	eqtrd 2780	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥)
56	1, 2, 15, 41, 43, 49, 55	ismndd 18794	1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∅c0 4352  ‘cfv 6573  (class class class)co 7448  Word cword 14562   ++ cconcat 14618  Basecbs 17258  +_gcplusg 17311  Mndcmnd 18772  freeMndcfrmd 18882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-frmd 18884

This theorem is referenced by:  frmdsssubm  18896  frmdgsum  18897  frmdup1  18899  frgp0  19802  frgpadd  19805  frgpmhm  19807  mrsubff  35480  mrsubccat  35486  elmrsubrn  35488