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Theorem frmdss2 15847
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 994 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  I  e.  V )
2 simpl2 995 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  J  C_  I
)
3 sswrd 12508 . . . . . . . . 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 756 . . . . . . . 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 3498 . . . . . . 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 15846 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
101, 6, 9syl2anc 661 . . . . . 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 996 . . . . . . 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 12506 . . . . . . . . . . 11  |-  ( x  e. Word  J  ->  x : ( 0..^ (
# `  x )
) --> J )
1312ad2antll 728 . . . . . . . . . 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 5728 . . . . . . . . . 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 5501 . . . . . . . . 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 15842 . . . . . . . . . . . . . 14  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant1 1012 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  U :
I -->Word  I )
20 ffn 5722 . . . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  U  Fn  I )
23 fnssres 5685 . . . . . . . . . . 11  |-  ( ( U  Fn  I  /\  J  C_  I )  -> 
( U  |`  J )  Fn  J )
2422, 2, 23syl2anc 661 . . . . . . . . . 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 5005 . . . . . . . . . . 11  |-  ( U
" J )  =  ran  ( U  |`  J )
26 simprl 755 . . . . . . . . . . 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 3542 . . . . . . . . . 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 5583 . . . . . . . . . 10  |-  ( ( U  |`  J ) : J --> A  <->  ( ( U  |`  J )  Fn  J  /\  ran  ( U  |`  J )  C_  A ) )
2924, 27, 28sylanbrc 664 . . . . . . . . 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 12747 . . . . . . . . 9  |-  ( ( x  e. Word  J  /\  ( U  |`  J ) : J --> A )  ->  ( ( U  |`  J )  o.  x
)  e. Word  A )
315, 29, 30syl2anc 661 . . . . . . . 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 2549 . . . . . . 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 15822 . . . . . . 7  |-  ( ( A  e.  (SubMnd `  M )  /\  ( U  o.  x )  e. Word  A )  ->  ( M  gsumg  ( U  o.  x
) )  e.  A
)
3411, 32, 33syl2anc 661 . . . . . 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 2549 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  (
( U " J
)  C_  A  /\  x  e. Word  J )
)  ->  x  e.  A )
3635expr 615 . . . 4  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  ( U " J )  C_  A )  ->  (
x  e. Word  J  ->  x  e.  A ) )
3736ssrdv 3503 . . 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 994 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  I  e.  V )
40 simp2 992 . . . . . . . 8  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  I
)
4140sselda 3497 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  I )
428vrmdval 15841 . . . . . . 7  |-  ( ( I  e.  V  /\  x  e.  I )  ->  ( U `  x
)  =  <" x "> )
4339, 41, 42syl2anc 661 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  =  <" x "> )
44 simpr 461 . . . . . . 7  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  x  e.  J )
4544s1cld 12565 . . . . . 6  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  <" x ">  e. Word  J )
4643, 45eqeltrd 2548 . . . . 5  |-  ( ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M
) )  /\  x  e.  J )  ->  ( U `  x )  e. Word  J )
4746ralrimiva 2871 . . . 4  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  A. x  e.  J  ( U `  x )  e. Word  J
)
48 fnfun 5669 . . . . . 6  |-  ( U  Fn  I  ->  Fun  U )
4921, 48syl 16 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  Fun  U )
50 fndm 5671 . . . . . . 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 3534 . . . . 5  |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M )
)  ->  J  C_  dom  U )
53 funimass4 5909 . . . . 5  |-  ( ( Fun  U  /\  J  C_ 
dom  U )  -> 
( ( U " J )  C_ Word  J  <->  A. x  e.  J  ( U `  x )  e. Word  J
) )
5449, 52, 53syl2anc 661 . . . 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 3504 . . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995    o. ccom 4996   Fun wfun 5573    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481  ..^cfzo 11781   #chash 12360  Word cword 12487   <"cs1 12490    gsumg cgsu 14685  SubMndcsubmnd 15769  freeMndcfrmd 15831  varFMndcvrmd 15832
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-substr 12499  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-gsum 14687  df-mnd 15721  df-submnd 15771  df-frmd 15833  df-vrmd 15834
This theorem is referenced by: (None)
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