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Theorem frmdup3 14801
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 2435 . . 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 14799 . 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 14800 . . . 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 4287 . . 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 2435 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
1615, 2mhmf 14733 . . . . 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 14793 . . . . . 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 14787 . . . . . . 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 5570 . . . . . 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 5896 . . . 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 5771 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A  =  ( y  e.  I  |->  ( A `  y ) ) )
2814, 26, 273eqtr4d 2477 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  o.  U )  =  A )
2915, 2mhmf 14733 . . . . . . . 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 5573 . . . . . . 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 5771 . . . . 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 11790 . . . . . . . . 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 14780 . . . . . . . 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 14797 . . . . . . . . 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 5724 . . . . . . 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 5380 . . . . . . . . 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 5026 . . . . . . . . 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 2481 . . . . . . . 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 6089 . . . . . . 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 2475 . . . . . 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 4287 . . . . 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 2467 . . . 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 2781 . 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 5022 . . . 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 2443 . . 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 3118 . 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 1652    e. wcel 1725   A.wral 2697   E!wreu 2699    e. cmpt 4258    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073  Word cword 11707   Basecbs 13459    gsumg cgsu 13714   Mndcmnd 14674   MndHom cmhm 14726  freeMndcfrmd 14782  varFMndcvrmd 14783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-fzo 11126  df-seq 11314  df-hash 11609  df-word 11713  df-concat 11714  df-s1 11715  df-substr 11716  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-0g 13717  df-gsum 13718  df-mnd 14680  df-mhm 14728  df-submnd 14729  df-frmd 14784  df-vrmd 14785
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