Theorem frmdmnd 18918

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 2770	. 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = (Base‘𝑀))
2		eqidd 2770	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘𝑀) = (+g‘𝑀))
3		frmdmnd.m	. . . . . 6 ⊢ 𝑀 = (freeMnd‘𝐼)
4		eqid 2769	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
5		eqid 2769	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
6	3, 4, 5	frmdadd 18914	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
7	3, 4	frmdelbas 18912	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
8	3, 4	frmdelbas 18912	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9		ccatcl 14611	. . . . . 6 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
10	7, 8, 9	syl2an 607	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
11	6, 10	eqeltrd 2869	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ Word 𝐼)
12	11	3adant1 1146	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ Word 𝐼)
13	3, 4	frmdbas 18911	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = Word 𝐼)
14	13	3ad2ant1 1149	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
15	12, 14	eleqtrrd 2872	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
16		simpr1 1211	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
17	16, 7	syl 18	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ Word 𝐼)
18		simpr2 1212	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
19	18, 8	syl 18	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
20		simpr3 1213	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
21	3, 4	frmdelbas 18912	. . . . . 6 ⊢ (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
22	20, 21	syl 18	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
23		ccatass 14626	. . . . 5 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
24	17, 19, 22, 23	syl3anc 1396	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
25	16, 18, 10	syl2anc 595	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
26	13	adantr 485	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
27	25, 26	eleqtrrd 2872	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
28	3, 4, 5	frmdadd 18914	. . . . 5 ⊢ (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
29	27, 20, 28	syl2anc 595	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30		ccatcl 14611	. . . . . . 7 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
31	19, 22, 30	syl2anc 595	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
32	31, 26	eleqtrrd 2872	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
33	3, 4, 5	frmdadd 18914	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
34	16, 32, 33	syl2anc 595	. . . 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 2814	. . 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 595	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
37	36	oveq1d 7426	. . 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 18914	. . . . 5 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
39	18, 20, 38	syl2anc 595	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
40	39	oveq2d 7427	. . 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 2814	. 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 14576	. . 3 ⊢ ∅ ∈ Word 𝐼
43	42, 13	eleqtrrid 2876	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘𝑀))
44	3, 4, 5	frmdadd 18914	. . . 4 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥))
45	43, 44	sylan 591	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥))
46	7	adantl 486	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47		ccatlid 14624	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
48	46, 47	syl 18	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
49	45, 48	eqtrd 2804	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥)
50	3, 4, 5	frmdadd 18914	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅))
51	50	ancoms 463	. . . 4 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅))
52	43, 51	sylan 591	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅))
53		ccatrid 14625	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
54	46, 53	syl 18	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
55	52, 54	eqtrd 2804	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥)
56	1, 2, 15, 41, 43, 49, 55	ismndd 18814	1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∅c0 4294  ‘cfv 6537  (class class class)co 7411  Word cword 14550   ++ cconcat 14607  Basecbs 17269  +_gcplusg 17310  Mndcmnd 18792  freeMndcfrmd 18906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-hash 14367  df-word 14551  df-concat 14608  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-plusg 17323  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-frmd 18908

This theorem is referenced by:  frmdsssubm  18920  frmdgsum  18921  frmdup1  18923  frgp0  19830  frgpadd  19833  frgpmhm  19835  mrsubff  35903  mrsubccat  35909  elmrsubrn  35911