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Theorem frmdss2 14485
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 958 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  I  e.  V )
2 simpl2 959 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  J  C_  I
)
3 sswrd 11423 . . . . . . . . 9  |-  ( J 
C_  I  -> Word  J  C_ Word  I )
42, 3syl 15 . . . . . . . 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 733 . . . . . . . 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 3181 . . . . . . 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 14484 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
101, 6, 9syl2anc 642 . . . . . 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 960 . . . . . . 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 11419 . . . . . . . . . . 11  |-  ( x  e. Word  J  ->  x : ( 0..^ (
# `  x )
) --> J )
1312ad2antll 709 . . . . . . . . . 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 5395 . . . . . . . . . 10  |-  ( x : ( 0..^ (
# `  x )
) --> J  ->  ran  x  C_  J )
1513, 14syl 15 . . . . . . . . 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 5176 . . . . . . . . 9  |-  ( ran  x  C_  J  ->  ( ( U  |`  J )  o.  x )  =  ( U  o.  x
) )
1715, 16syl 15 . . . . . . . 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 14480 . . . . . . . . . . . . . 14  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant1 976 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  U :
I -->Word  I )
20 ffn 5389 . . . . . . . . . . . . 13  |-  ( U : I -->Word  I  ->  U  Fn  I )
2119, 20syl 15 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  U  Fn  I )
2221adantr 451 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  U  Fn  I )
23 fnssres 5357 . . . . . . . . . . 11  |-  ( ( U  Fn  I  /\  J  C_  I )  -> 
( U  |`  J )  Fn  J )
2422, 2, 23syl2anc 642 . . . . . . . . . 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 4702 . . . . . . . . . . 11  |-  ( U
" J )  =  ran  ( U  |`  J )
26 simprl 732 . . . . . . . . . . 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 3223 . . . . . . . . . 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 5259 . . . . . . . . . 10  |-  ( ( U  |`  J ) : J --> A  <->  ( ( U  |`  J )  Fn  J  /\  ran  ( U  |`  J )  C_  A ) )
2924, 27, 28sylanbrc 645 . . . . . . . . 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 11486 . . . . . . . . 9  |-  ( ( x  e. Word  J  /\  ( U  |`  J ) : J --> A )  ->  ( ( U  |`  J )  o.  x
)  e. Word  A )
315, 29, 30syl2anc 642 . . . . . . . 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 2358 . . . . . . 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 14461 . . . . . . 7  |-  ( ( A  e.  (SubMnd `  M )  /\  ( U  o.  x )  e. Word  A )  ->  ( M  gsumg  ( U  o.  x
) )  e.  A
)
3411, 32, 33syl2anc 642 . . . . . 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 2358 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  x  e.  A )
3635expr 598 . . . 4  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  ( U " J )  C_  A )  ->  (
x  e. Word  J  ->  x  e.  A ) )
3736ssrdv 3185 . . 3  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  ( U " J )  C_  A )  -> Word  J  C_  A )
3837ex 423 . 2  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( ( U " J )  C_  A  -> Word  J  C_  A ) )
39 simpl1 958 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  I  e.  V )
40 simp2 956 . . . . . . . 8  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  I
)
4140sselda 3180 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  I )
428vrmdval 14479 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e.  I )  ->  ( U `  x
)  =  <" x "> )
4339, 41, 42syl2anc 642 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  =  <" x "> )
44 simpr 447 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  J )
4544s1cld 11442 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  <" x ">  e. Word  J )
4643, 45eqeltrd 2357 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  e. Word  J )
4746ralrimiva 2626 . . . 4  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  A. x  e.  J  ( U `  x )  e. Word  J
)
48 fnfun 5341 . . . . . 6  |-  ( U  Fn  I  ->  Fun  U )
4921, 48syl 15 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  Fun  U )
50 fndm 5343 . . . . . . 7  |-  ( U  Fn  I  ->  dom  U  =  I )
5121, 50syl 15 . . . . . 6  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  dom  U  =  I )
5240, 51sseqtr4d 3215 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  dom  U )
53 funimass4 5573 . . . . 5  |-  ( ( Fun  U  /\  J  C_ 
dom  U )  -> 
( ( U " J )  C_ Word  J  <->  A. x  e.  J  ( U `  x )  e. Word  J
) )
5449, 52, 53syl2anc 642 . . . 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 223 . . 3  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( U " J )  C_ Word  J )
56 sstr2 3186 . . 3  |-  ( ( U " J ) 
C_ Word  J  ->  (Word  J  C_  A  ->  ( U " J )  C_  A
) )
5755, 56syl 15 . 2  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  (Word  J  C_  A  ->  ( U " J )  C_  A
) )
5838, 57impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   0cc0 8737  ..^cfzo 10870   #chash 11337  Word cword 11403   <"cs1 11405    gsumg cgsu 13401  SubMndcsubmnd 14414  freeMndcfrmd 14469  varFMndcvrmd 14470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-frmd 14471  df-vrmd 14472
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