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Theorem frmdss2 14737
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.
StepHypRef Expression
1 simpl1 960 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  I  e.  V )
2 simpl2 961 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  J  C_  I
)
3 sswrd 11666 . . . . . . . . 9  |-  ( J 
C_  I  -> Word  J  C_ Word  I )
42, 3syl 16 . . . . . . . 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 734 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  x  e. Word  J )
64, 5sseldd 3294 . . . . . . 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 )
97, 8frmdgsum 14736 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
101, 6, 9syl2anc 643 . . . . . 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 962 . . . . . . 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 11662 . . . . . . . . . . 11  |-  ( x  e. Word  J  ->  x : ( 0..^ (
# `  x )
) --> J )
1312ad2antll 710 . . . . . . . . . 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 5539 . . . . . . . . . 10  |-  ( x : ( 0..^ (
# `  x )
) --> J  ->  ran  x  C_  J )
1513, 14syl 16 . . . . . . . . 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 5315 . . . . . . . . 9  |-  ( ran  x  C_  J  ->  ( ( U  |`  J )  o.  x )  =  ( U  o.  x
) )
1715, 16syl 16 . . . . . . . 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 ) )
188vrmdf 14732 . . . . . . . . . . . . . 14  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant1 978 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  U :
I -->Word  I )
20 ffn 5533 . . . . . . . . . . . . 13  |-  ( U : I -->Word  I  ->  U  Fn  I )
2119, 20syl 16 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  U  Fn  I )
2221adantr 452 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  U  Fn  I )
23 fnssres 5500 . . . . . . . . . . 11  |-  ( ( U  Fn  I  /\  J  C_  I )  -> 
( U  |`  J )  Fn  J )
2422, 2, 23syl2anc 643 . . . . . . . . . 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 4833 . . . . . . . . . . 11  |-  ( U
" J )  =  ran  ( U  |`  J )
26 simprl 733 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  ( U " J )  C_  A
)
2725, 26syl5eqssr 3338 . . . . . . . . . 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 5400 . . . . . . . . . 10  |-  ( ( U  |`  J ) : J --> A  <->  ( ( U  |`  J )  Fn  J  /\  ran  ( U  |`  J )  C_  A ) )
2924, 27, 28sylanbrc 646 . . . . . . . . 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 11729 . . . . . . . . 9  |-  ( ( x  e. Word  J  /\  ( U  |`  J ) : J --> A )  ->  ( ( U  |`  J )  o.  x
)  e. Word  A )
315, 29, 30syl2anc 643 . . . . . . . 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
)
3217, 31eqeltrrd 2464 . . . . . . 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 14713 . . . . . . 7  |-  ( ( A  e.  (SubMnd `  M )  /\  ( U  o.  x )  e. Word  A )  ->  ( M  gsumg  ( U  o.  x
) )  e.  A
)
3411, 32, 33syl2anc 643 . . . . . 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 )
3510, 34eqeltrrd 2464 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  x  e.  A )
3635expr 599 . . . 4  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  ( U " J )  C_  A )  ->  (
x  e. Word  J  ->  x  e.  A ) )
3736ssrdv 3299 . . 3  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  ( U " J )  C_  A )  -> Word  J  C_  A )
3837ex 424 . 2  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( ( U " J )  C_  A  -> Word  J  C_  A ) )
39 simpl1 960 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  I  e.  V )
40 simp2 958 . . . . . . . 8  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  I
)
4140sselda 3293 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  I )
428vrmdval 14731 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e.  I )  ->  ( U `  x
)  =  <" x "> )
4339, 41, 42syl2anc 643 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  =  <" x "> )
44 simpr 448 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  J )
4544s1cld 11685 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  <" x ">  e. Word  J )
4643, 45eqeltrd 2463 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  e. Word  J )
4746ralrimiva 2734 . . . 4  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  A. x  e.  J  ( U `  x )  e. Word  J
)
48 fnfun 5484 . . . . . 6  |-  ( U  Fn  I  ->  Fun  U )
4921, 48syl 16 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  Fun  U )
50 fndm 5486 . . . . . . 7  |-  ( U  Fn  I  ->  dom  U  =  I )
5121, 50syl 16 . . . . . 6  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  dom  U  =  I )
5240, 51sseqtr4d 3330 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  dom  U )
53 funimass4 5718 . . . . 5  |-  ( ( Fun  U  /\  J  C_ 
dom  U )  -> 
( ( U " J )  C_ Word  J  <->  A. x  e.  J  ( U `  x )  e. Word  J
) )
5449, 52, 53syl2anc 643 . . . 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 ) )
5547, 54mpbird 224 . . 3  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( U " J )  C_ Word  J )
56 sstr2 3300 . . 3  |-  ( ( U " J ) 
C_ Word  J  ->  (Word  J  C_  A  ->  ( U " J )  C_  A
) )
5755, 56syl 16 . 2  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  (Word  J  C_  A  ->  ( U " J )  C_  A
) )
5838, 57impbid 184 1  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( ( U " J )  C_  A 
<-> Word 
J  C_  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651    C_ wss 3265   dom cdm 4820   ran crn 4821    |` cres 4822   "cima 4823    o. ccom 4824   Fun wfun 5390    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   0cc0 8925  ..^cfzo 11067   #chash 11547  Word cword 11646   <"cs1 11648    gsumg cgsu 13653  SubMndcsubmnd 14666  freeMndcfrmd 14721  varFMndcvrmd 14722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-word 11652  df-concat 11653  df-s1 11654  df-substr 11655  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-gsum 13657  df-mnd 14619  df-submnd 14668  df-frmd 14723  df-vrmd 14724
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