Theorem frgpmhm 18178

Description: The "natural map" from words of the free monoid to their cosets in the free group is a surjective monoid homomorphism. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
frgpmhm.m	|- M = (freeMnd ` ( I X. 2o ) )
frgpmhm.w	|- W = ( Base ` M )
frgpmhm.g	|- G = (freeGrp ` I )
frgpmhm.r	|- .~ = ( ~FG ` I )
frgpmhm.f	|- F = ( x e. W |-> [ x ] .~ )

Assertion

Ref	Expression
frgpmhm	|- ( I e. V -> F e. ( M MndHom G ) )

Distinct variable groups:   x, G   x, I   x, V   x, W   x, .~

Allowed substitution hints:   F( x)   M( x)

Proof of Theorem frgpmhm

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2on 7568	. . . . 5 |- 2o e. On
2		xpexg 6960	. . . . 5 |- ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
3	1, 2	mpan2 707	. . . 4 |- ( I e. V -> ( I X. 2o ) e. _V )
4		frgpmhm.m	. . . . 5 |- M = (freeMnd ` ( I X. 2o ) )
5	4	frmdmnd 17396	. . . 4 |- ( ( I X. 2o ) e. _V -> M e. Mnd )
6	3, 5	syl 17	. . 3 |- ( I e. V -> M e. Mnd )
7		frgpmhm.g	. . . . 5 |- G = (freeGrp ` I )
8	7	frgpgrp 18175	. . . 4 |- ( I e. V -> G e. Grp )
9		grpmnd 17429	. . . 4 |- ( G e. Grp -> G e. Mnd )
10	8, 9	syl 17	. . 3 |- ( I e. V -> G e. Mnd )
11	6, 10	jca 554	. 2 |- ( I e. V -> ( M e. Mnd /\ G e. Mnd ) )
12		frgpmhm.w	. . . . . . . . . 10 |- W = ( Base ` M )
13	4, 12	frmdbas 17389	. . . . . . . . 9 |- ( ( I X. 2o ) e. _V -> W = Word ( I X. 2o ) )
14		wrdexg 13315	. . . . . . . . . 10 |- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V )
15		fvi 6255	. . . . . . . . . 10 |- (Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
16	14, 15	syl 17	. . . . . . . . 9 |- ( ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
17	13, 16	eqtr4d 2659	. . . . . . . 8 |- ( ( I X. 2o ) e. _V -> W = ( _I ` Word ( I X. 2o ) ) )
18	3, 17	syl 17	. . . . . . 7 |- ( I e. V -> W = ( _I ` Word ( I X. 2o ) ) )
19	18	eleq2d 2687	. . . . . 6 |- ( I e. V -> ( x e. W <-> x e. ( _I ` Word ( I X. 2o ) ) ) )
20	19	biimpa 501	. . . . 5 |- ( ( I e. V /\ x e. W ) -> x e. ( _I ` Word ( I X. 2o ) ) )
21		frgpmhm.r	. . . . . 6 |- .~ = ( ~FG ` I )
22		eqid 2622	. . . . . 6 |- ( _I ` Word ( I X. 2o ) ) = ( _I ` Word ( I X. 2o ) )
23		eqid 2622	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
24	7, 21, 22, 23	frgpeccl 18174	. . . . 5 |- ( x e. ( _I ` Word ( I X. 2o ) ) -> [ x ] .~ e. ( Base ` G ) )
25	20, 24	syl 17	. . . 4 |- ( ( I e. V /\ x e. W ) -> [ x ] .~ e. ( Base ` G ) )
26		frgpmhm.f	. . . 4 |- F = ( x e. W |-> [ x ] .~ )
27	25, 26	fmptd 6385	. . 3 |- ( I e. V -> F : W --> ( Base ` G ) )
28	22, 21	efger 18131	. . . . . . . 8 |- .~ Er ( _I ` Word ( I X. 2o ) )
29		ereq2 7750	. . . . . . . . 9 |- ( W = ( _I ` Word ( I X. 2o ) ) -> ( .~ Er W <-> .~ Er ( _I ` Word ( I X. 2o ) ) ) )
30	18, 29	syl 17	. . . . . . . 8 |- ( I e. V -> ( .~ Er W <-> .~ Er ( _I ` Word ( I X. 2o ) ) ) )
31	28, 30	mpbiri 248	. . . . . . 7 |- ( I e. V -> .~ Er W )
32	31	adantr 481	. . . . . 6 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> .~ Er W )
33		fvex 6201	. . . . . . . 8 |- ( Base ` M ) e. _V
34	12, 33	eqeltri 2697	. . . . . . 7 |- W e. _V
35	34	a1i 11	. . . . . 6 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> W e. _V )
36	32, 35, 26	divsfval 16207	. . . . 5 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( F ` ( a ++ b ) ) = [ ( a ++ b ) ] .~ )
37		eqid 2622	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
38	4, 12, 37	frmdadd 17392	. . . . . . 7 |- ( ( a e. W /\ b e. W ) -> ( a ( +g ` M ) b ) = ( a ++ b ) )
39	38	adantl 482	. . . . . 6 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( a ( +g ` M ) b ) = ( a ++ b ) )
40	39	fveq2d 6195	. . . . 5 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( F ` ( a ( +g ` M ) b ) ) = ( F ` ( a ++ b ) ) )
41	32, 35, 26	divsfval 16207	. . . . . . 7 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( F ` a ) = [ a ] .~ )
42	32, 35, 26	divsfval 16207	. . . . . . 7 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( F ` b ) = [ b ] .~ )
43	41, 42	oveq12d 6668	. . . . . 6 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( ( F ` a ) ( +g ` G ) ( F ` b ) ) = ( [ a ] .~ ( +g ` G ) [ b ] .~ ) )
44	18	eleq2d 2687	. . . . . . . . 9 |- ( I e. V -> ( a e. W <-> a e. ( _I ` Word ( I X. 2o ) ) ) )
45	18	eleq2d 2687	. . . . . . . . 9 |- ( I e. V -> ( b e. W <-> b e. ( _I ` Word ( I X. 2o ) ) ) )
46	44, 45	anbi12d 747	. . . . . . . 8 |- ( I e. V -> ( ( a e. W /\ b e. W ) <-> ( a e. ( _I ` Word ( I X. 2o ) ) /\ b e. ( _I ` Word ( I X. 2o ) ) ) ) )
47	46	biimpa 501	. . . . . . 7 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( a e. ( _I ` Word ( I X. 2o ) ) /\ b e. ( _I ` Word ( I X. 2o ) ) ) )
48		eqid 2622	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
49	22, 7, 21, 48	frgpadd 18176	. . . . . . 7 |- ( ( a e. ( _I ` Word ( I X. 2o ) ) /\ b e. ( _I ` Word ( I X. 2o ) ) ) -> ( [ a ] .~ ( +g ` G ) [ b ] .~ ) = [ ( a ++ b ) ] .~ )
50	47, 49	syl 17	. . . . . 6 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( [ a ] .~ ( +g ` G ) [ b ] .~ ) = [ ( a ++ b ) ] .~ )
51	43, 50	eqtrd 2656	. . . . 5 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( ( F ` a ) ( +g ` G ) ( F ` b ) ) = [ ( a ++ b ) ] .~ )
52	36, 40, 51	3eqtr4d 2666	. . . 4 |- ( ( I e. V /\ ( a e. W /\ b e. W ) ) -> ( F ` ( a ( +g ` M ) b ) ) = ( ( F ` a ) ( +g ` G ) ( F ` b ) ) )
53	52	ralrimivva 2971	. . 3 |- ( I e. V -> A. a e. W A. b e. W ( F ` ( a ( +g ` M ) b ) ) = ( ( F ` a ) ( +g ` G ) ( F ` b ) ) )
54	34	a1i 11	. . . . 5 |- ( I e. V -> W e. _V )
55	31, 54, 26	divsfval 16207	. . . 4 |- ( I e. V -> ( F ` (/) ) = [ (/) ] .~ )
56	7, 21	frgp0 18173	. . . . 5 |- ( I e. V -> ( G e. Grp /\ [ (/) ] .~ = ( 0g ` G ) ) )
57	56	simprd 479	. . . 4 |- ( I e. V -> [ (/) ] .~ = ( 0g ` G ) )
58	55, 57	eqtrd 2656	. . 3 |- ( I e. V -> ( F ` (/) ) = ( 0g ` G ) )
59	27, 53, 58	3jca 1242	. 2 |- ( I e. V -> ( F : W --> ( Base ` G ) /\ A. a e. W A. b e. W ( F ` ( a ( +g ` M ) b ) ) = ( ( F ` a ) ( +g ` G ) ( F ` b ) ) /\ ( F ` (/) ) = ( 0g ` G ) ) )
60	4	frmd0 17397	. . 3 |- (/) = ( 0g ` M )
61		eqid 2622	. . 3 |- ( 0g ` G ) = ( 0g ` G )
62	12, 23, 37, 48, 60, 61	ismhm 17337	. 2 |- ( F e. ( M MndHom G ) <-> ( ( M e. Mnd /\ G e. Mnd ) /\ ( F : W --> ( Base ` G ) /\ A. a e. W A. b e. W ( F ` ( a ( +g ` M ) b ) ) = ( ( F ` a ) ( +g ` G ) ( F ` b ) ) /\ ( F ` (/) ) = ( 0g ` G ) ) ) )
63	11, 59, 62	sylanbrc 698	1 |- ( I e. V -> F e. ( M MndHom G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   _I cid 5023   X. cxp 5112   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   2oc2o 7554   Er wer 7739   [cec 7740  Word cword 13291   ++ cconcat 13293   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294   MndHom cmhm 17333  freeMndcfrmd 17384   Grpcgrp 17422   ~_FG cefg 18119  freeGrpcfrgp 18120

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-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  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-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-frmd 17386  df-grp 17425  df-efg 18122  df-frgp 18123

This theorem is referenced by:  frgpup3lem  18190