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Theorem frmdup3 14740
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 2389 . . 3  |-  ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )
4 simp1 957 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  G  e.  Mnd )
5 simp2 958 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  I  e.  V
)
6 simp3 959 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A : I --> B )
71, 2, 3, 4, 5, 6frmdup1 14738 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) )  e.  ( M MndHom  G ) )
84adantr 452 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  G  e.  Mnd )
95adantr 452 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  I  e.  V )
106adantr 452 . . . . 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 448 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  y  e.  I )
131, 2, 3, 8, 9, 10, 11, 12frmdup2 14739 . . . 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 4238 . . 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 2389 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
1615, 2mhmf 14672 . . . . 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 14732 . . . . . 6  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant2 979 . . . . 5  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I -->Word  I )
201, 15frmdbas 14726 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
21203ad2ant2 979 . . . . . 6  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( Base `  M
)  = Word  I )
22 feq3 5520 . . . . . 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 224 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I --> ( Base `  M
) )
25 fcompt 5845 . . . 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 643 . . 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 5720 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A  =  ( y  e.  I  |->  ( A `  y ) ) )
2814, 26, 273eqtr4d 2431 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  o.  U )  =  A )
2915, 2mhmf 14672 . . . . . . . 8  |-  ( m  e.  ( M MndHom  G
)  ->  m :
( Base `  M ) --> B )
3029ad2antrl 709 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m : ( Base `  M
) --> B )
3121adantr 452 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  ( Base `  M )  = Word 
I )
3231feq2d 5523 . . . . . . 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 202 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m :Word  I --> B )
3433feqmptd 5720 . . . . 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 737 . . . . . . . 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 448 . . . . . . . . 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 707 . . . . . . . . 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 11729 . . . . . . . . 9  |-  ( ( x  e. Word  I  /\  U : I --> ( Base `  M ) )  -> 
( U  o.  x
)  e. Word  ( Base `  M ) )
3936, 37, 38syl2anc 643 . . . . . . . 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 14719 . . . . . . . 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 643 . . . . . . 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 707 . . . . . . . . 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 14736 . . . . . . . . 9  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
4442, 36, 43syl2anc 643 . . . . . . . 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 5674 . . . . . . 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 5330 . . . . . . . . 9  |-  ( ( m  o.  U )  o.  x )  =  ( m  o.  ( U  o.  x )
)
47 simplrr 738 . . . . . . . . . 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 4976 . . . . . . . . 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 2435 . . . . . . . 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 6038 . . . . . . 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 2429 . . . . . 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 4238 . . . . 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 2421 . . . 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 599 . . 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 2734 . 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 4972 . . . 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 2397 . . 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 3071 . 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 1184 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   E!wreu 2653    e. cmpt 4209    o. ccom 4824   -->wf 5392   ` cfv 5396  (class class class)co 6022  Word cword 11646   Basecbs 13398    gsumg cgsu 13653   Mndcmnd 14613   MndHom cmhm 14665  freeMndcfrmd 14721  varFMndcvrmd 14722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-word 11652  df-concat 11653  df-s1 11654  df-substr 11655  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-gsum 13657  df-mnd 14619  df-mhm 14667  df-submnd 14668  df-frmd 14723  df-vrmd 14724
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