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Theorem frmdup3 14816
Description: Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
frmdup3.m  |-  M  =  (freeMnd `  I )
frmdup3.b  |-  B  =  ( Base `  G
)
frmdup3.u  |-  U  =  (varFMnd `  I )
Assertion
Ref Expression
frmdup3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  E! m  e.  ( M MndHom  G ) ( m  o.  U
)  =  A )
Distinct variable groups:    A, m    B, m    m, G    m, I    U, m    m, M   
m, V

Proof of Theorem frmdup3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frmdup3.m . . 3  |-  M  =  (freeMnd `  I )
2 frmdup3.b . . 3  |-  B  =  ( Base `  G
)
3 eqid 2438 . . 3  |-  ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )
4 simp1 958 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  G  e.  Mnd )
5 simp2 959 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  I  e.  V
)
6 simp3 960 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A : I --> B )
71, 2, 3, 4, 5, 6frmdup1 14814 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) )  e.  ( M MndHom  G ) )
84adantr 453 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  G  e.  Mnd )
95adantr 453 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  I  e.  V )
106adantr 453 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  A :
I --> B )
11 frmdup3.u . . . . 5  |-  U  =  (varFMnd `  I )
12 simpr 449 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  y  e.  I )
131, 2, 3, 8, 9, 10, 11, 12frmdup2 14815 . . . 4  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  ( (
x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) ) `  ( U `  y ) )  =  ( A `
 y ) )
1413mpteq2dva 4298 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( y  e.  I  |->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) ) `  ( U `  y ) ) )  =  ( y  e.  I  |->  ( A `  y ) ) )
15 eqid 2438 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
1615, 2mhmf 14748 . . . . 5  |-  ( ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )  e.  ( M MndHom  G )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) ) : ( Base `  M ) --> B )
177, 16syl 16 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) ) : ( Base `  M ) --> B )
1811vrmdf 14808 . . . . . 6  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant2 980 . . . . 5  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I -->Word  I )
201, 15frmdbas 14802 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
21203ad2ant2 980 . . . . . 6  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( Base `  M
)  = Word  I )
22 feq3 5581 . . . . . 6  |-  ( (
Base `  M )  = Word  I  ->  ( U : I --> ( Base `  M )  <->  U :
I -->Word  I ) )
2321, 22syl 16 . . . . 5  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( U :
I --> ( Base `  M
)  <->  U : I -->Word  I )
)
2419, 23mpbird 225 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I --> ( Base `  M
) )
25 fcompt 5907 . . . 4  |-  ( ( ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) ) : ( Base `  M
) --> B  /\  U : I --> ( Base `  M ) )  -> 
( ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) )  o.  U )  =  ( y  e.  I  |->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) ) `  ( U `  y ) ) ) )
2617, 24, 25syl2anc 644 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  o.  U )  =  ( y  e.  I  |->  ( ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) ) `  ( U `  y ) ) ) )
276feqmptd 5782 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A  =  ( y  e.  I  |->  ( A `  y ) ) )
2814, 26, 273eqtr4d 2480 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  o.  U )  =  A )
2915, 2mhmf 14748 . . . . . . . 8  |-  ( m  e.  ( M MndHom  G
)  ->  m :
( Base `  M ) --> B )
3029ad2antrl 710 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m : ( Base `  M
) --> B )
3121adantr 453 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  ( Base `  M )  = Word 
I )
3231feq2d 5584 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  (
m : ( Base `  M ) --> B  <->  m :Word  I
--> B ) )
3330, 32mpbid 203 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m :Word  I --> B )
3433feqmptd 5782 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m  =  ( x  e. Word 
I  |->  ( m `  x ) ) )
35 simplrl 738 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  m  e.  ( M MndHom  G ) )
36 simpr 449 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  x  e. Word  I )
3724ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  U :
I --> ( Base `  M
) )
38 wrdco 11805 . . . . . . . . 9  |-  ( ( x  e. Word  I  /\  U : I --> ( Base `  M ) )  -> 
( U  o.  x
)  e. Word  ( Base `  M ) )
3936, 37, 38syl2anc 644 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( U  o.  x )  e. Word  ( Base `  M ) )
4015gsumwmhm 14795 . . . . . . . 8  |-  ( ( m  e.  ( M MndHom  G )  /\  ( U  o.  x )  e. Word  ( Base `  M
) )  ->  (
m `  ( M  gsumg  ( U  o.  x ) ) )  =  ( G  gsumg  ( m  o.  ( U  o.  x )
) ) )
4135, 39, 40syl2anc 644 . . . . . . 7  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( m `  ( M  gsumg  ( U  o.  x
) ) )  =  ( G  gsumg  ( m  o.  ( U  o.  x )
) ) )
425ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  I  e.  V )
431, 11frmdgsum 14812 . . . . . . . . 9  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
4442, 36, 43syl2anc 644 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( M  gsumg  ( U  o.  x ) )  =  x )
4544fveq2d 5735 . . . . . . 7  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( m `  ( M  gsumg  ( U  o.  x
) ) )  =  ( m `  x
) )
46 coass 5391 . . . . . . . . 9  |-  ( ( m  o.  U )  o.  x )  =  ( m  o.  ( U  o.  x )
)
47 simplrr 739 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( m  o.  U )  =  A )
4847coeq1d 5037 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( (
m  o.  U )  o.  x )  =  ( A  o.  x
) )
4946, 48syl5eqr 2484 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( m  o.  ( U  o.  x
) )  =  ( A  o.  x ) )
5049oveq2d 6100 . . . . . . 7  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( G  gsumg  ( m  o.  ( U  o.  x ) ) )  =  ( G 
gsumg  ( A  o.  x
) ) )
5141, 45, 503eqtr3d 2478 . . . . . 6  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  (
m  e.  ( M MndHom  G )  /\  (
m  o.  U )  =  A ) )  /\  x  e. Word  I
)  ->  ( m `  x )  =  ( G  gsumg  ( A  o.  x
) ) )
5251mpteq2dva 4298 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  (
x  e. Word  I  |->  ( m `  x ) )  =  ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) ) )
5334, 52eqtrd 2470 . . . 4  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m  =  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) ) )
5453expr 600 . . 3  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  m  e.  ( M MndHom  G ) )  ->  ( ( m  o.  U )  =  A  ->  m  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) ) ) )
5554ralrimiva 2791 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A. m  e.  ( M MndHom  G ) ( ( m  o.  U
)  =  A  ->  m  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x ) ) ) ) )
56 coeq1 5033 . . . 4  |-  ( m  =  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) )  ->  ( m  o.  U )  =  ( ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )  o.  U ) )
5756eqeq1d 2446 . . 3  |-  ( m  =  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) )  ->  ( (
m  o.  U )  =  A  <->  ( (
x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )  o.  U )  =  A ) )
5857eqreu 3128 . 2  |-  ( ( ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )  e.  ( M MndHom  G )  /\  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  o.  U )  =  A  /\  A. m  e.  ( M MndHom  G ) ( ( m  o.  U )  =  A  ->  m  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) ) ) )  ->  E! m  e.  ( M MndHom  G ) ( m  o.  U
)  =  A )
597, 28, 55, 58syl3anc 1185 1  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  E! m  e.  ( M MndHom  G ) ( m  o.  U
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E!wreu 2709    e. cmpt 4269    o. ccom 4885   -->wf 5453   ` cfv 5457  (class class class)co 6084  Word cword 11722   Basecbs 13474    gsumg cgsu 13729   Mndcmnd 14689   MndHom cmhm 14741  freeMndcfrmd 14797  varFMndcvrmd 14798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-0g 13732  df-gsum 13733  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-frmd 14799  df-vrmd 14800
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