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Theorem frmdss2 15534
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 986 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  I  e.  V )
2 simpl2 987 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  J  C_  I
)
3 sswrd 12238 . . . . . . . . 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 751 . . . . . . . 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 3354 . . . . . . 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 15533 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
101, 6, 9syl2anc 656 . . . . . 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 988 . . . . . . 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 12236 . . . . . . . . . . 11  |-  ( x  e. Word  J  ->  x : ( 0..^ (
# `  x )
) --> J )
1312ad2antll 723 . . . . . . . . . 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 5562 . . . . . . . . . 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 5338 . . . . . . . . 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 15529 . . . . . . . . . . . . . 14  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant1 1004 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  U :
I -->Word  I )
20 ffn 5556 . . . . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  U  Fn  I )
23 fnssres 5521 . . . . . . . . . . 11  |-  ( ( U  Fn  I  /\  J  C_  I )  -> 
( U  |`  J )  Fn  J )
2422, 2, 23syl2anc 656 . . . . . . . . . 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 4849 . . . . . . . . . . 11  |-  ( U
" J )  =  ran  ( U  |`  J )
26 simprl 750 . . . . . . . . . . 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 3398 . . . . . . . . . 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 5419 . . . . . . . . . 10  |-  ( ( U  |`  J ) : J --> A  <->  ( ( U  |`  J )  Fn  J  /\  ran  ( U  |`  J )  C_  A ) )
2924, 27, 28sylanbrc 659 . . . . . . . . 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 12455 . . . . . . . . 9  |-  ( ( x  e. Word  J  /\  ( U  |`  J ) : J --> A )  ->  ( ( U  |`  J )  o.  x
)  e. Word  A )
315, 29, 30syl2anc 656 . . . . . . . 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 2516 . . . . . . 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 15509 . . . . . . 7  |-  ( ( A  e.  (SubMnd `  M )  /\  ( U  o.  x )  e. Word  A )  ->  ( M  gsumg  ( U  o.  x
) )  e.  A
)
3411, 32, 33syl2anc 656 . . . . . 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 2516 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  x  e.  A )
3635expr 612 . . . 4  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  ( U " J )  C_  A )  ->  (
x  e. Word  J  ->  x  e.  A ) )
3736ssrdv 3359 . . 3  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  ( U " J )  C_  A )  -> Word  J  C_  A )
3837ex 434 . 2  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( ( U " J )  C_  A  -> Word  J  C_  A ) )
39 simpl1 986 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  I  e.  V )
40 simp2 984 . . . . . . . 8  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  I
)
4140sselda 3353 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  I )
428vrmdval 15528 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e.  I )  ->  ( U `  x
)  =  <" x "> )
4339, 41, 42syl2anc 656 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  =  <" x "> )
44 simpr 458 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  J )
4544s1cld 12290 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  <" x ">  e. Word  J )
4643, 45eqeltrd 2515 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  e. Word  J )
4746ralrimiva 2797 . . . 4  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  A. x  e.  J  ( U `  x )  e. Word  J
)
48 fnfun 5505 . . . . . 6  |-  ( U  Fn  I  ->  Fun  U )
4921, 48syl 16 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  Fun  U )
50 fndm 5507 . . . . . . 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 3390 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  dom  U )
53 funimass4 5739 . . . . 5  |-  ( ( Fun  U  /\  J  C_ 
dom  U )  -> 
( ( U " J )  C_ Word  J  <->  A. x  e.  J  ( U `  x )  e. Word  J
) )
5449, 52, 53syl2anc 656 . . . 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 232 . . 3  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( U " J )  C_ Word  J )
56 sstr2 3360 . . 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 191 1  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  ( ( U " J )  C_  A 
<-> Word 
J  C_  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713    C_ wss 3325   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839    o. ccom 4840   Fun wfun 5409    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278  ..^cfzo 11544   #chash 12099  Word cword 12217   <"cs1 12220    gsumg cgsu 14375  SubMndcsubmnd 15459  freeMndcfrmd 15518  varFMndcvrmd 15519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-substr 12229  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-gsum 14377  df-mnd 15411  df-submnd 15461  df-frmd 15520  df-vrmd 15521
This theorem is referenced by: (None)
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