Theorem frmdmnd 18913

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 2764	. 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = (Base‘𝑀))
2		eqidd 2764	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘𝑀) = (+g‘𝑀))
3		frmdmnd.m	. . . . . 6 ⊢ 𝑀 = (freeMnd‘𝐼)
4		eqid 2763	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
5		eqid 2763	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
6	3, 4, 5	frmdadd 18909	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
7	3, 4	frmdelbas 18907	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
8	3, 4	frmdelbas 18907	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9		ccatcl 14607	. . . . . 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 2863	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ Word 𝐼)
12	11	3adant1 1148	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ Word 𝐼)
13	3, 4	frmdbas 18906	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = Word 𝐼)
14	13	3ad2ant1 1151	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
15	12, 14	eleqtrrd 2866	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
16		simpr1 1213	. . . . . 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 1214	. . . . . 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 1215	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
21	3, 4	frmdelbas 18907	. . . . . 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 14622	. . . . 5 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
24	17, 19, 22, 23	syl3anc 1398	. . . 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 2866	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
28	3, 4, 5	frmdadd 18909	. . . . 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 14607	. . . . . . 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 2866	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
33	3, 4, 5	frmdadd 18909	. . . . 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 2808	. . 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 7425	. . 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 18909	. . . . 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 7426	. . 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 2808	. 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 14572	. . 3 ⊢ ∅ ∈ Word 𝐼
43	42, 13	eleqtrrid 2870	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘𝑀))
44	3, 4, 5	frmdadd 18909	. . . 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 14620	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
48	46, 47	syl 18	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
49	45, 48	eqtrd 2798	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥)
50	3, 4, 5	frmdadd 18909	. . . . 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 14621	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
54	46, 53	syl 18	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
55	52, 54	eqtrd 2798	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥)
56	1, 2, 15, 41, 43, 49, 55	ismndd 18809	1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∅c0 4286  ‘cfv 6536  (class class class)co 7410  Word cword 14546   ++ cconcat 14603  Basecbs 17264  +_gcplusg 17305  Mndcmnd 18787  freeMndcfrmd 18901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-hash 14363  df-word 14547  df-concat 14604  df-struct 17202  df-slot 17237  df-ndx 17249  df-base 17265  df-plusg 17318  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-frmd 18903

This theorem is referenced by:  frmdsssubm  18915  frmdgsum  18916  frmdup1  18918  frgp0  19825  frgpadd  19828  frgpmhm  19830  mrsubff  36004  mrsubccat  36010  elmrsubrn  36012