Theorem frmdmnd 17396

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 e. V -> M e. 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 2623	. 2 |- ( I e. V -> ( Base ` M ) = ( Base ` M ) )
2		eqidd 2623	. 2 |- ( I e. V -> ( +g ` M ) = ( +g ` M ) )
3		frmdmnd.m	. . . . . 6 |- M = (freeMnd ` I )
4		eqid 2622	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
5		eqid 2622	. . . . . 6 |- ( +g ` M ) = ( +g ` M )
6	3, 4, 5	frmdadd 17392	. . . . 5 |- ( ( x e. ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( x ( +g ` M ) y ) = ( x ++ y ) )
7	3, 4	frmdelbas 17390	. . . . . 6 |- ( x e. ( Base ` M ) -> x e. Word I )
8	3, 4	frmdelbas 17390	. . . . . 6 |- ( y e. ( Base ` M ) -> y e. Word I )
9		ccatcl 13359	. . . . . 6 |- ( ( x e. Word I /\ y e. Word I ) -> ( x ++ y ) e. Word I )
10	7, 8, 9	syl2an 494	. . . . 5 |- ( ( x e. ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( x ++ y ) e. Word I )
11	6, 10	eqeltrd 2701	. . . 4 |- ( ( x e. ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( x ( +g ` M ) y ) e. Word I )
12	11	3adant1 1079	. . 3 |- ( ( I e. V /\ x e. ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( x ( +g ` M ) y ) e. Word I )
13	3, 4	frmdbas 17389	. . . 4 |- ( I e. V -> ( Base ` M ) = Word I )
14	13	3ad2ant1 1082	. . 3 |- ( ( I e. V /\ x e. ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( Base ` M ) = Word I )
15	12, 14	eleqtrrd 2704	. 2 |- ( ( I e. V /\ x e. ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( x ( +g ` M ) y ) e. ( Base ` M ) )
16		simpr1 1067	. . . . . 6 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> x e. ( Base ` M ) )
17	16, 7	syl 17	. . . . 5 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> x e. Word I )
18		simpr2 1068	. . . . . 6 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> y e. ( Base ` M ) )
19	18, 8	syl 17	. . . . 5 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> y e. Word I )
20		simpr3 1069	. . . . . 6 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> z e. ( Base ` M ) )
21	3, 4	frmdelbas 17390	. . . . . 6 |- ( z e. ( Base ` M ) -> z e. Word I )
22	20, 21	syl 17	. . . . 5 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> z e. Word I )
23		ccatass 13371	. . . . 5 |- ( ( x e. Word I /\ y e. Word I /\ z e. Word I ) -> ( ( x ++ y ) ++ z ) = ( x ++ ( y ++ z ) ) )
24	17, 19, 22, 23	syl3anc 1326	. . . 4 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( ( x ++ y ) ++ z ) = ( x ++ ( y ++ z ) ) )
25	16, 18, 10	syl2anc 693	. . . . . 6 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( x ++ y ) e. Word I )
26	13	adantr 481	. . . . . 6 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( Base ` M ) = Word I )
27	25, 26	eleqtrrd 2704	. . . . 5 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( x ++ y ) e. ( Base ` M ) )
28	3, 4, 5	frmdadd 17392	. . . . 5 |- ( ( ( x ++ y ) e. ( Base ` M ) /\ z e. ( Base ` M ) ) -> ( ( x ++ y ) ( +g ` M ) z ) = ( ( x ++ y ) ++ z ) )
29	27, 20, 28	syl2anc 693	. . . 4 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( ( x ++ y ) ( +g ` M ) z ) = ( ( x ++ y ) ++ z ) )
30		ccatcl 13359	. . . . . . 7 |- ( ( y e. Word I /\ z e. Word I ) -> ( y ++ z ) e. Word I )
31	19, 22, 30	syl2anc 693	. . . . . 6 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( y ++ z ) e. Word I )
32	31, 26	eleqtrrd 2704	. . . . 5 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( y ++ z ) e. ( Base ` M ) )
33	3, 4, 5	frmdadd 17392	. . . . 5 |- ( ( x e. ( Base ` M ) /\ ( y ++ z ) e. ( Base ` M ) ) -> ( x ( +g ` M ) ( y ++ z ) ) = ( x ++ ( y ++ z ) ) )
34	16, 32, 33	syl2anc 693	. . . 4 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( x ( +g ` M ) ( y ++ z ) ) = ( x ++ ( y ++ z ) ) )
35	24, 29, 34	3eqtr4d 2666	. . 3 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( ( x ++ y ) ( +g ` M ) z ) = ( x ( +g ` M ) ( y ++ z ) ) )
36	16, 18, 6	syl2anc 693	. . . 4 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( x ( +g ` M ) y ) = ( x ++ y ) )
37	36	oveq1d 6665	. . 3 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( ( x ( +g ` M ) y ) ( +g ` M ) z ) = ( ( x ++ y ) ( +g ` M ) z ) )
38	3, 4, 5	frmdadd 17392	. . . . 5 |- ( ( y e. ( Base ` M ) /\ z e. ( Base ` M ) ) -> ( y ( +g ` M ) z ) = ( y ++ z ) )
39	18, 20, 38	syl2anc 693	. . . 4 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( y ( +g ` M ) z ) = ( y ++ z ) )
40	39	oveq2d 6666	. . 3 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( x ( +g ` M ) ( y ( +g ` M ) z ) ) = ( x ( +g ` M ) ( y ++ z ) ) )
41	35, 37, 40	3eqtr4d 2666	. 2 |- ( ( I e. V /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) /\ z e. ( Base ` M ) ) ) -> ( ( x ( +g ` M ) y ) ( +g ` M ) z ) = ( x ( +g ` M ) ( y ( +g ` M ) z ) ) )
42		wrd0 13330	. . 3 |- (/) e. Word I
43	42, 13	syl5eleqr 2708	. 2 |- ( I e. V -> (/) e. ( Base ` M ) )
44	3, 4, 5	frmdadd 17392	. . . 4 |- ( ( (/) e. ( Base ` M ) /\ x e. ( Base ` M ) ) -> ( (/) ( +g ` M ) x ) = ( (/) ++ x ) )
45	43, 44	sylan 488	. . 3 |- ( ( I e. V /\ x e. ( Base ` M ) ) -> ( (/) ( +g ` M ) x ) = ( (/) ++ x ) )
46	7	adantl 482	. . . 4 |- ( ( I e. V /\ x e. ( Base ` M ) ) -> x e. Word I )
47		ccatlid 13369	. . . 4 |- ( x e. Word I -> ( (/) ++ x ) = x )
48	46, 47	syl 17	. . 3 |- ( ( I e. V /\ x e. ( Base ` M ) ) -> ( (/) ++ x ) = x )
49	45, 48	eqtrd 2656	. 2 |- ( ( I e. V /\ x e. ( Base ` M ) ) -> ( (/) ( +g ` M ) x ) = x )
50	3, 4, 5	frmdadd 17392	. . . . 5 |- ( ( x e. ( Base ` M ) /\ (/) e. ( Base ` M ) ) -> ( x ( +g ` M ) (/) ) = ( x ++ (/) ) )
51	50	ancoms 469	. . . 4 |- ( ( (/) e. ( Base ` M ) /\ x e. ( Base ` M ) ) -> ( x ( +g ` M ) (/) ) = ( x ++ (/) ) )
52	43, 51	sylan 488	. . 3 |- ( ( I e. V /\ x e. ( Base ` M ) ) -> ( x ( +g ` M ) (/) ) = ( x ++ (/) ) )
53		ccatrid 13370	. . . 4 |- ( x e. Word I -> ( x ++ (/) ) = x )
54	46, 53	syl 17	. . 3 |- ( ( I e. V /\ x e. ( Base ` M ) ) -> ( x ++ (/) ) = x )
55	52, 54	eqtrd 2656	. 2 |- ( ( I e. V /\ x e. ( Base ` M ) ) -> ( x ( +g ` M ) (/) ) = x )
56	1, 2, 15, 41, 43, 49, 55	ismndd 17313	1 |- ( I e. V -> M e. Mnd )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   (/)c0 3915   ` cfv 5888  (class class class)co 6650  Word cword 13291   ++ cconcat 13293   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294  freeMndcfrmd 17384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-frmd 17386

This theorem is referenced by:  frmdsssubm  17398  frmdgsum  17399  frmdup1  17401  frgp0  18173  frgpadd  18176  frgpmhm  18178  mrsubff  31409  mrsubccat  31415  elmrsubrn  31417