Theorem frmdss2 17400

Description: A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of J is Word J". (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
frmdss2	|- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> ( ( U " J ) C_ A <-> Word J C_ A ) )

Proof of Theorem frmdss2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . . 7 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> I e. V )
2		simpl2 1065	. . . . . . . . 9 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> J C_ I )
3		sswrd 13313	. . . . . . . . 9 |- ( J C_ I -> Word J C_ Word I )
4	2, 3	syl 17	. . . . . . . 8 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> Word J C_ Word I )
5		simprr 796	. . . . . . . 8 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> x e. Word J )
6	4, 5	sseldd 3604	. . . . . . 7 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> x e. Word I )
7		frmdmnd.m	. . . . . . . 8 |- M = (freeMnd ` I )
8		frmdgsum.u	. . . . . . . 8 |- U = (varFMnd ` I )
9	7, 8	frmdgsum 17399	. . . . . . 7 |- ( ( I e. V /\ x e. Word I ) -> ( M gsumg ( U o. x ) ) = x )
10	1, 6, 9	syl2anc 693	. . . . . 6 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( M gsumg ( U o. x ) ) = x )
11		simpl3 1066	. . . . . . 7 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> A e. (SubMnd ` M ) )
12		wrdf 13310	. . . . . . . . . . 11 |- ( x e. Word J -> x : ( 0..^ ( # ` x ) ) --> J )
13	12	ad2antll 765	. . . . . . . . . 10 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> x : ( 0..^ ( # ` x ) ) --> J )
14		frn 6053	. . . . . . . . . 10 |- ( x : ( 0..^ ( # ` x ) ) --> J -> ran x C_ J )
15	13, 14	syl 17	. . . . . . . . 9 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ran x C_ J )
16		cores 5638	. . . . . . . . 9 |- ( ran x C_ J -> ( ( U |` J ) o. x ) = ( U o. x ) )
17	15, 16	syl 17	. . . . . . . 8 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( ( U |` J ) o. x ) = ( U o. x ) )
18	8	vrmdf 17395	. . . . . . . . . . . . . 14 |- ( I e. V -> U : I -->Word I )
19	18	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> U : I -->Word I )
20		ffn 6045	. . . . . . . . . . . . 13 |- ( U : I -->Word I -> U Fn I )
21	19, 20	syl 17	. . . . . . . . . . . 12 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> U Fn I )
22	21	adantr 481	. . . . . . . . . . 11 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> U Fn I )
23		fnssres 6004	. . . . . . . . . . 11 |- ( ( U Fn I /\ J C_ I ) -> ( U |` J ) Fn J )
24	22, 2, 23	syl2anc 693	. . . . . . . . . 10 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( U |` J ) Fn J )
25		df-ima 5127	. . . . . . . . . . 11 |- ( U " J ) = ran ( U |` J )
26		simprl 794	. . . . . . . . . . 11 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( U " J ) C_ A )
27	25, 26	syl5eqssr 3650	. . . . . . . . . 10 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ran ( U |` J ) C_ A )
28		df-f 5892	. . . . . . . . . 10 |- ( ( U |` J ) : J --> A <-> ( ( U |` J ) Fn J /\ ran ( U |` J ) C_ A ) )
29	24, 27, 28	sylanbrc 698	. . . . . . . . 9 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( U |` J ) : J --> A )
30		wrdco 13577	. . . . . . . . 9 |- ( ( x e. Word J /\ ( U |` J ) : J --> A ) -> ( ( U |` J ) o. x ) e. Word A )
31	5, 29, 30	syl2anc 693	. . . . . . . 8 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( ( U |` J ) o. x ) e. Word A )
32	17, 31	eqeltrrd 2702	. . . . . . 7 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( U o. x ) e. Word A )
33		gsumwsubmcl 17375	. . . . . . 7 |- ( ( A e. (SubMnd ` M ) /\ ( U o. x ) e. Word A ) -> ( M gsumg ( U o. x ) ) e. A )
34	11, 32, 33	syl2anc 693	. . . . . 6 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> ( M gsumg ( U o. x ) ) e. A )
35	10, 34	eqeltrrd 2702	. . . . 5 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( ( U " J ) C_ A /\ x e. Word J ) ) -> x e. A )
36	35	expr 643	. . . 4 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( U " J ) C_ A ) -> ( x e. Word J -> x e. A ) )
37	36	ssrdv 3609	. . 3 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ ( U " J ) C_ A ) -> Word J C_ A )
38	37	ex 450	. 2 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> ( ( U " J ) C_ A -> Word J C_ A ) )
39		simpl1 1064	. . . . . . 7 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ x e. J ) -> I e. V )
40		simp2 1062	. . . . . . . 8 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> J C_ I )
41	40	sselda 3603	. . . . . . 7 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ x e. J ) -> x e. I )
42	8	vrmdval 17394	. . . . . . 7 |- ( ( I e. V /\ x e. I ) -> ( U ` x ) = <" x "> )
43	39, 41, 42	syl2anc 693	. . . . . 6 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ x e. J ) -> ( U ` x ) = <" x "> )
44		simpr 477	. . . . . . 7 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ x e. J ) -> x e. J )
45	44	s1cld 13383	. . . . . 6 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ x e. J ) -> <" x "> e. Word J )
46	43, 45	eqeltrd 2701	. . . . 5 |- ( ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) /\ x e. J ) -> ( U ` x ) e. Word J )
47	46	ralrimiva 2966	. . . 4 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> A. x e. J ( U ` x ) e. Word J )
48		fnfun 5988	. . . . . 6 |- ( U Fn I -> Fun U )
49	21, 48	syl 17	. . . . 5 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> Fun U )
50		fndm 5990	. . . . . . 7 |- ( U Fn I -> dom U = I )
51	21, 50	syl 17	. . . . . 6 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> dom U = I )
52	40, 51	sseqtr4d 3642	. . . . 5 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> J C_ dom U )
53		funimass4 6247	. . . . 5 |- ( ( Fun U /\ J C_ dom U ) -> ( ( U " J ) C_ Word J <-> A. x e. J ( U ` x ) e. Word J ) )
54	49, 52, 53	syl2anc 693	. . . 4 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> ( ( U " J ) C_ Word J <-> A. x e. J ( U ` x ) e. Word J ) )
55	47, 54	mpbird 247	. . 3 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> ( U " J ) C_ Word J )
56		sstr2 3610	. . 3 |- ( ( U " J ) C_ Word J -> (Word J C_ A -> ( U " J ) C_ A ) )
57	55, 56	syl 17	. 2 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> (Word J C_ A -> ( U " J ) C_ A ) )
58	38, 57	impbid 202	1 |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> ( ( U " J ) C_ A <-> Word J C_ A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs1 13294   gsum_g cgsu 16101  SubMndcsubmnd 17334  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: (None)