Theorem frmdgsum 17399

Description: Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)

Hypotheses

Ref	Expression
frmdmnd.m	|- M = (freeMnd ` I )
frmdgsum.u	|- U = (varFMnd ` I )

Assertion

Ref	Expression
frmdgsum	|- ( ( I e. V /\ W e. Word I ) -> ( M gsumg ( U o. W ) ) = W )

Proof of Theorem frmdgsum

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coeq2 5280	. . . . . . 7 |- ( x = (/) -> ( U o. x ) = ( U o. (/) ) )
2		co02 5649	. . . . . . 7 |- ( U o. (/) ) = (/)
3	1, 2	syl6eq 2672	. . . . . 6 |- ( x = (/) -> ( U o. x ) = (/) )
4	3	oveq2d 6666	. . . . 5 |- ( x = (/) -> ( M gsumg ( U o. x ) ) = ( M gsumg (/) ) )
5		id 22	. . . . 5 |- ( x = (/) -> x = (/) )
6	4, 5	eqeq12d 2637	. . . 4 |- ( x = (/) -> ( ( M gsumg ( U o. x ) ) = x <-> ( M gsumg (/) ) = (/) ) )
7	6	imbi2d 330	. . 3 |- ( x = (/) -> ( ( I e. V -> ( M gsumg ( U o. x ) ) = x ) <-> ( I e. V -> ( M gsumg (/) ) = (/) ) ) )
8		coeq2 5280	. . . . . 6 |- ( x = y -> ( U o. x ) = ( U o. y ) )
9	8	oveq2d 6666	. . . . 5 |- ( x = y -> ( M gsumg ( U o. x ) ) = ( M gsumg ( U o. y ) ) )
10		id 22	. . . . 5 |- ( x = y -> x = y )
11	9, 10	eqeq12d 2637	. . . 4 |- ( x = y -> ( ( M gsumg ( U o. x ) ) = x <-> ( M gsumg ( U o. y ) ) = y ) )
12	11	imbi2d 330	. . 3 |- ( x = y -> ( ( I e. V -> ( M gsumg ( U o. x ) ) = x ) <-> ( I e. V -> ( M gsumg ( U o. y ) ) = y ) ) )
13		coeq2 5280	. . . . . 6 |- ( x = ( y ++ <" z "> ) -> ( U o. x ) = ( U o. ( y ++ <" z "> ) ) )
14	13	oveq2d 6666	. . . . 5 |- ( x = ( y ++ <" z "> ) -> ( M gsumg ( U o. x ) ) = ( M gsumg ( U o. ( y ++ <" z "> ) ) ) )
15		id 22	. . . . 5 |- ( x = ( y ++ <" z "> ) -> x = ( y ++ <" z "> ) )
16	14, 15	eqeq12d 2637	. . . 4 |- ( x = ( y ++ <" z "> ) -> ( ( M gsumg ( U o. x ) ) = x <-> ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( y ++ <" z "> ) ) )
17	16	imbi2d 330	. . 3 |- ( x = ( y ++ <" z "> ) -> ( ( I e. V -> ( M gsumg ( U o. x ) ) = x ) <-> ( I e. V -> ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( y ++ <" z "> ) ) ) )
18		coeq2 5280	. . . . . 6 |- ( x = W -> ( U o. x ) = ( U o. W ) )
19	18	oveq2d 6666	. . . . 5 |- ( x = W -> ( M gsumg ( U o. x ) ) = ( M gsumg ( U o. W ) ) )
20		id 22	. . . . 5 |- ( x = W -> x = W )
21	19, 20	eqeq12d 2637	. . . 4 |- ( x = W -> ( ( M gsumg ( U o. x ) ) = x <-> ( M gsumg ( U o. W ) ) = W ) )
22	21	imbi2d 330	. . 3 |- ( x = W -> ( ( I e. V -> ( M gsumg ( U o. x ) ) = x ) <-> ( I e. V -> ( M gsumg ( U o. W ) ) = W ) ) )
23		frmdmnd.m	. . . . . 6 |- M = (freeMnd ` I )
24	23	frmd0 17397	. . . . 5 |- (/) = ( 0g ` M )
25	24	gsum0 17278	. . . 4 |- ( M gsumg (/) ) = (/)
26	25	a1i 11	. . 3 |- ( I e. V -> ( M gsumg (/) ) = (/) )
27		oveq1 6657	. . . . . 6 |- ( ( M gsumg ( U o. y ) ) = y -> ( ( M gsumg ( U o. y ) ) ++ <" z "> ) = ( y ++ <" z "> ) )
28		simprl 794	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> y e. Word I )
29		simprr 796	. . . . . . . . . . . 12 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> z e. I )
30	29	s1cld 13383	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> <" z "> e. Word I )
31		frmdgsum.u	. . . . . . . . . . . . 13 |- U = (varFMnd ` I )
32	31	vrmdf 17395	. . . . . . . . . . . 12 |- ( I e. V -> U : I -->Word I )
33	32	adantr 481	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> U : I -->Word I )
34		ccatco 13581	. . . . . . . . . . 11 |- ( ( y e. Word I /\ <" z "> e. Word I /\ U : I -->Word I ) -> ( U o. ( y ++ <" z "> ) ) = ( ( U o. y ) ++ ( U o. <" z "> ) ) )
35	28, 30, 33, 34	syl3anc 1326	. . . . . . . . . 10 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( U o. ( y ++ <" z "> ) ) = ( ( U o. y ) ++ ( U o. <" z "> ) ) )
36		s1co 13579	. . . . . . . . . . . . 13 |- ( ( z e. I /\ U : I -->Word I ) -> ( U o. <" z "> ) = <" ( U ` z ) "> )
37	29, 33, 36	syl2anc 693	. . . . . . . . . . . 12 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( U o. <" z "> ) = <" ( U ` z ) "> )
38	31	vrmdval 17394	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ z e. I ) -> ( U ` z ) = <" z "> )
39	38	adantrl 752	. . . . . . . . . . . . 13 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( U ` z ) = <" z "> )
40	39	s1eqd 13381	. . . . . . . . . . . 12 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> <" ( U ` z ) "> = <" <" z "> "> )
41	37, 40	eqtrd 2656	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( U o. <" z "> ) = <" <" z "> "> )
42	41	oveq2d 6666	. . . . . . . . . 10 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( ( U o. y ) ++ ( U o. <" z "> ) ) = ( ( U o. y ) ++ <" <" z "> "> ) )
43	35, 42	eqtrd 2656	. . . . . . . . 9 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( U o. ( y ++ <" z "> ) ) = ( ( U o. y ) ++ <" <" z "> "> ) )
44	43	oveq2d 6666	. . . . . . . 8 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( M gsumg ( ( U o. y ) ++ <" <" z "> "> ) ) )
45	23	frmdmnd 17396	. . . . . . . . . . 11 |- ( I e. V -> M e. Mnd )
46	45	adantr 481	. . . . . . . . . 10 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> M e. Mnd )
47		wrdco 13577	. . . . . . . . . . . 12 |- ( ( y e. Word I /\ U : I -->Word I ) -> ( U o. y ) e. Word Word I )
48	28, 33, 47	syl2anc 693	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( U o. y ) e. Word Word I )
49		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` M ) = ( Base ` M )
50	23, 49	frmdbas 17389	. . . . . . . . . . . . 13 |- ( I e. V -> ( Base ` M ) = Word I )
51	50	adantr 481	. . . . . . . . . . . 12 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( Base ` M ) = Word I )
52		wrdeq 13327	. . . . . . . . . . . 12 |- ( ( Base ` M ) = Word I -> Word ( Base ` M ) = Word Word I )
53	51, 52	syl 17	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> Word ( Base ` M ) = Word Word I )
54	48, 53	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( U o. y ) e. Word ( Base ` M ) )
55	30, 51	eleqtrrd 2704	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> <" z "> e. ( Base ` M ) )
56	55	s1cld 13383	. . . . . . . . . 10 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> <" <" z "> "> e. Word ( Base ` M ) )
57		eqid 2622	. . . . . . . . . . 11 |- ( +g ` M ) = ( +g ` M )
58	49, 57	gsumccat 17378	. . . . . . . . . 10 |- ( ( M e. Mnd /\ ( U o. y ) e. Word ( Base ` M ) /\ <" <" z "> "> e. Word ( Base ` M ) ) -> ( M gsumg ( ( U o. y ) ++ <" <" z "> "> ) ) = ( ( M gsumg ( U o. y ) ) ( +g ` M ) ( M gsumg <" <" z "> "> ) ) )
59	46, 54, 56, 58	syl3anc 1326	. . . . . . . . 9 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( M gsumg ( ( U o. y ) ++ <" <" z "> "> ) ) = ( ( M gsumg ( U o. y ) ) ( +g ` M ) ( M gsumg <" <" z "> "> ) ) )
60	49	gsumws1 17376	. . . . . . . . . . . 12 |- ( <" z "> e. ( Base ` M ) -> ( M gsumg <" <" z "> "> ) = <" z "> )
61	55, 60	syl 17	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( M gsumg <" <" z "> "> ) = <" z "> )
62	61	oveq2d 6666	. . . . . . . . . 10 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( ( M gsumg ( U o. y ) ) ( +g ` M ) ( M gsumg <" <" z "> "> ) ) = ( ( M gsumg ( U o. y ) ) ( +g ` M ) <" z "> ) )
63	49	gsumwcl 17377	. . . . . . . . . . . 12 |- ( ( M e. Mnd /\ ( U o. y ) e. Word ( Base ` M ) ) -> ( M gsumg ( U o. y ) ) e. ( Base ` M ) )
64	46, 54, 63	syl2anc 693	. . . . . . . . . . 11 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( M gsumg ( U o. y ) ) e. ( Base ` M ) )
65	23, 49, 57	frmdadd 17392	. . . . . . . . . . 11 |- ( ( ( M gsumg ( U o. y ) ) e. ( Base ` M ) /\ <" z "> e. ( Base ` M ) ) -> ( ( M gsumg ( U o. y ) ) ( +g ` M ) <" z "> ) = ( ( M gsumg ( U o. y ) ) ++ <" z "> ) )
66	64, 55, 65	syl2anc 693	. . . . . . . . . 10 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( ( M gsumg ( U o. y ) ) ( +g ` M ) <" z "> ) = ( ( M gsumg ( U o. y ) ) ++ <" z "> ) )
67	62, 66	eqtrd 2656	. . . . . . . . 9 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( ( M gsumg ( U o. y ) ) ( +g ` M ) ( M gsumg <" <" z "> "> ) ) = ( ( M gsumg ( U o. y ) ) ++ <" z "> ) )
68	59, 67	eqtrd 2656	. . . . . . . 8 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( M gsumg ( ( U o. y ) ++ <" <" z "> "> ) ) = ( ( M gsumg ( U o. y ) ) ++ <" z "> ) )
69	44, 68	eqtrd 2656	. . . . . . 7 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( ( M gsumg ( U o. y ) ) ++ <" z "> ) )
70	69	eqeq1d 2624	. . . . . 6 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( y ++ <" z "> ) <-> ( ( M gsumg ( U o. y ) ) ++ <" z "> ) = ( y ++ <" z "> ) ) )
71	27, 70	syl5ibr 236	. . . . 5 |- ( ( I e. V /\ ( y e. Word I /\ z e. I ) ) -> ( ( M gsumg ( U o. y ) ) = y -> ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( y ++ <" z "> ) ) )
72	71	expcom 451	. . . 4 |- ( ( y e. Word I /\ z e. I ) -> ( I e. V -> ( ( M gsumg ( U o. y ) ) = y -> ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( y ++ <" z "> ) ) ) )
73	72	a2d 29	. . 3 |- ( ( y e. Word I /\ z e. I ) -> ( ( I e. V -> ( M gsumg ( U o. y ) ) = y ) -> ( I e. V -> ( M gsumg ( U o. ( y ++ <" z "> ) ) ) = ( y ++ <" z "> ) ) ) )
74	7, 12, 17, 22, 26, 73	wrdind 13476	. 2 |- ( W e. Word I -> ( I e. V -> ( M gsumg ( U o. W ) ) = W ) )
75	74	impcom 446	1 |- ( ( I e. V /\ W e. Word I ) -> ( M gsumg ( U o. W ) ) = W )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650  Word cword 13291   ++ cconcat 13293   <"cs1 13294   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101   Mndcmnd 17294  freeMndcfrmd 17384  var_FMndcvrmd 17385

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-rmo 2920  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-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-frmd 17386  df-vrmd 17387

This theorem is referenced by:  frmdss2  17400  frmdup3lem  17403  frgpup3lem  18190