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Theorem frmdss2 14800
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 11729 . . . . . . . . 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 3341 . . . . . . 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 14799 . . . . . . 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 11725 . . . . . . . . . . 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 5589 . . . . . . . . . 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 5365 . . . . . . . . 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 14795 . . . . . . . . . . . . . 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 5583 . . . . . . . . . . . . 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 5550 . . . . . . . . . . 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 4883 . . . . . . . . . . 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 3385 . . . . . . . . . 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 5450 . . . . . . . . . 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 11792 . . . . . . . . 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 2510 . . . . . . 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 14776 . . . . . . 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 2510 . . . . 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 3346 . . 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 3340 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  I )
428vrmdval 14794 . . . . . . 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 11748 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  <" x ">  e. Word  J )
4643, 45eqeltrd 2509 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  e. Word  J )
4746ralrimiva 2781 . . . 4  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  A. x  e.  J  ( U `  x )  e. Word  J
)
48 fnfun 5534 . . . . . 6  |-  ( U  Fn  I  ->  Fun  U )
4921, 48syl 16 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  Fun  U )
50 fndm 5536 . . . . . . 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 3377 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  dom  U )
53 funimass4 5769 . . . . 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 3347 . . 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 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873    o. ccom 4874   Fun wfun 5440    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   0cc0 8982  ..^cfzo 11127   #chash 11610  Word cword 11709   <"cs1 11711    gsumg cgsu 13716  SubMndcsubmnd 14729  freeMndcfrmd 14784  varFMndcvrmd 14785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-word 11715  df-concat 11716  df-s1 11717  df-substr 11718  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mnd 14682  df-submnd 14731  df-frmd 14786  df-vrmd 14787
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