Theorem frmdmnd 18768

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 2730	. 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = (Base‘𝑀))
2		eqidd 2730	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘𝑀) = (+g‘𝑀))
3		frmdmnd.m	. . . . . 6 ⊢ 𝑀 = (freeMnd‘𝐼)
4		eqid 2729	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
5		eqid 2729	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
6	3, 4, 5	frmdadd 18764	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
7	3, 4	frmdelbas 18762	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
8	3, 4	frmdelbas 18762	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9		ccatcl 14515	. . . . . 6 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
10	7, 8, 9	syl2an 596	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
11	6, 10	eqeltrd 2828	. . . 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 18761	. . . 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 2831	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
16		simpr1 1195	. . . . . 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 1196	. . . . . 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 1197	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
21	3, 4	frmdelbas 18762	. . . . . 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 14529	. . . . 5 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
24	17, 19, 22, 23	syl3anc 1373	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
25	16, 18, 10	syl2anc 584	. . . . . 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 2831	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
28	3, 4, 5	frmdadd 18764	. . . . 5 ⊢ (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
29	27, 20, 28	syl2anc 584	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g‘𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30		ccatcl 14515	. . . . . . 7 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
31	19, 22, 30	syl2anc 584	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
32	31, 26	eleqtrrd 2831	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
33	3, 4, 5	frmdadd 18764	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
34	16, 32, 33	syl2anc 584	. . . 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 2774	. . 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 584	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ++ 𝑦))
37	36	oveq1d 7384	. . 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 18764	. . . . 5 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
39	18, 20, 38	syl2anc 584	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
40	39	oveq2d 7385	. . 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 2774	. 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 14480	. . 3 ⊢ ∅ ∈ Word 𝐼
43	42, 13	eleqtrrid 2835	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘𝑀))
44	3, 4, 5	frmdadd 18764	. . . 4 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥))
45	43, 44	sylan 580	. . 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 14527	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
48	46, 47	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
49	45, 48	eqtrd 2764	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥)
50	3, 4, 5	frmdadd 18764	. . . . 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 580	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅))
53		ccatrid 14528	. . . 4 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
54	46, 53	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
55	52, 54	eqtrd 2764	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥)
56	1, 2, 15, 41, 43, 49, 55	ismndd 18665	1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∅c0 4292  ‘cfv 6499  (class class class)co 7369  Word cword 14454   ++ cconcat 14511  Basecbs 17155  +_gcplusg 17196  Mndcmnd 18643  freeMndcfrmd 18756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-hash 14272  df-word 14455  df-concat 14512  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-plusg 17209  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-frmd 18758

This theorem is referenced by:  frmdsssubm  18770  frmdgsum  18771  frmdup1  18773  frgp0  19674  frgpadd  19677  frgpmhm  19679  mrsubff  35492  mrsubccat  35498  elmrsubrn  35500