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Theorem frmdmnd 18818

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 2729	. 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = (Base‘𝑀))
2		eqidd 2729	. 2 ⊢ (𝐼 ∈ 𝑉 → (+_g‘𝑀) = (+_g‘𝑀))
3		frmdmnd.m	. . . . . 6 ⊢ 𝑀 = (freeMnd‘𝐼)
4		eqid 2728	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
5		eqid 2728	. . . . . 6 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
6	3, 4, 5	frmdadd 18814	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
7	3, 4	frmdelbas 18812	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
8	3, 4	frmdelbas 18812	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9		ccatcl 14564	. . . . . 6 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
10	7, 8, 9	syl2an 594	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
11	6, 10	eqeltrd 2829	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)𝑦) ∈ Word 𝐼)
12	11	3adant1 1127	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)𝑦) ∈ Word 𝐼)
13	3, 4	frmdbas 18811	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = Word 𝐼)
14	13	3ad2ant1 1130	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
15	12, 14	eleqtrrd 2832	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)𝑦) ∈ (Base‘𝑀))
16		simpr1 1191	. . . . . 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 1192	. . . . . 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 1193	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
21	3, 4	frmdelbas 18812	. . . . . 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 14578	. . . . 5 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
24	17, 19, 22, 23	syl3anc 1368	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
25	16, 18, 10	syl2anc 582	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
26	13	adantr 479	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
27	25, 26	eleqtrrd 2832	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
28	3, 4, 5	frmdadd 18814	. . . . 5 ⊢ (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+_g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
29	27, 20, 28	syl2anc 582	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+_g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30		ccatcl 14564	. . . . . . 7 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
31	19, 22, 30	syl2anc 582	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
32	31, 26	eleqtrrd 2832	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
33	3, 4, 5	frmdadd 18814	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
34	16, 32, 33	syl2anc 582	. . . 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 2778	. . 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 582	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+_g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
37	36	oveq1d 7441	. . 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 18814	. . . . 5 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+_g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
39	18, 20, 38	syl2anc 582	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+_g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
40	39	oveq2d 7442	. . 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 2778	. 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 14529	. . 3 ⊢ ∅ ∈ Word 𝐼
43	42, 13	eleqtrrid 2836	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘𝑀))
44	3, 4, 5	frmdadd 18814	. . . 4 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+_g‘𝑀)𝑥) = (∅ ++ 𝑥))
45	43, 44	sylan 578	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+_g‘𝑀)𝑥) = (∅ ++ 𝑥))
46	7	adantl 480	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47		ccatlid 14576	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
48	46, 47	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
49	45, 48	eqtrd 2768	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+_g‘𝑀)𝑥) = 𝑥)
50	3, 4, 5	frmdadd 18814	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)∅) = (𝑥 ++ ∅))
51	50	ancoms 457	. . . 4 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)∅) = (𝑥 ++ ∅))
52	43, 51	sylan 578	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)∅) = (𝑥 ++ ∅))
53		ccatrid 14577	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
54	46, 53	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
55	52, 54	eqtrd 2768	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+_g‘𝑀)∅) = 𝑥)
56	1, 2, 15, 41, 43, 49, 55	ismndd 18723	1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∅c0 4326 ‘cfv 6553 (class class class)co 7426 Word cword 14504 ++ cconcat 14560 Basecbs 17187 +_gcplusg 17240 Mndcmnd 18701 freeMndcfrmd 18806

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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-frmd 18808

This theorem is referenced by: frmdsssubm 18820 frmdgsum 18821 frmdup1 18823 frgp0 19722 frgpadd 19725 frgpmhm 19727 mrsubff 35155 mrsubccat 35161 elmrsubrn 35163

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