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Theorem frmdup3 14504
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 2296 . . 3  |-  ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )
4 simp1 955 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  G  e.  Mnd )
5 simp2 956 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  I  e.  V
)
6 simp3 957 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A : I --> B )
71, 2, 3, 4, 5, 6frmdup1 14502 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) )  e.  ( M MndHom  G ) )
84adantr 451 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  G  e.  Mnd )
95adantr 451 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  I  e.  V )
106adantr 451 . . . . 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 447 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  y  e.  I )
131, 2, 3, 8, 9, 10, 11, 12frmdup2 14503 . . . 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 4122 . . 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 2296 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
1615, 2mhmf 14436 . . . . 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 15 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) ) : ( Base `  M ) --> B )
1811vrmdf 14496 . . . . . 6  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant2 977 . . . . 5  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I -->Word  I )
201, 15frmdbas 14490 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
21203ad2ant2 977 . . . . . 6  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( Base `  M
)  = Word  I )
22 feq3 5393 . . . . . 6  |-  ( (
Base `  M )  = Word  I  ->  ( U : I --> ( Base `  M )  <->  U :
I -->Word  I ) )
2321, 22syl 15 . . . . 5  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( U :
I --> ( Base `  M
)  <->  U : I -->Word  I )
)
2419, 23mpbird 223 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I --> ( Base `  M
) )
25 fcompt 5710 . . . 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 642 . . 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 5591 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A  =  ( y  e.  I  |->  ( A `  y ) ) )
2814, 26, 273eqtr4d 2338 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  o.  U )  =  A )
2915, 2mhmf 14436 . . . . . . . 8  |-  ( m  e.  ( M MndHom  G
)  ->  m :
( Base `  M ) --> B )
3029ad2antrl 708 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m : ( Base `  M
) --> B )
3121adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  ( Base `  M )  = Word 
I )
3231feq2d 5396 . . . . . . 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 201 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( m  e.  ( M MndHom  G )  /\  ( m  o.  U )  =  A ) )  ->  m :Word  I --> B )
3433feqmptd 5591 . . . . 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 736 . . . . . . . 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 447 . . . . . . . . 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 706 . . . . . . . . 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 11502 . . . . . . . . 9  |-  ( ( x  e. Word  I  /\  U : I --> ( Base `  M ) )  -> 
( U  o.  x
)  e. Word  ( Base `  M ) )
3936, 37, 38syl2anc 642 . . . . . . . 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 14483 . . . . . . . 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 642 . . . . . . 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 706 . . . . . . . . 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 14500 . . . . . . . . 9  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
4442, 36, 43syl2anc 642 . . . . . . . 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 5545 . . . . . . 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 5207 . . . . . . . . 9  |-  ( ( m  o.  U )  o.  x )  =  ( m  o.  ( U  o.  x )
)
47 simplrr 737 . . . . . . . . . 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 4861 . . . . . . . . 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 2342 . . . . . . . 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 5890 . . . . . . 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 2336 . . . . . 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 4122 . . . . 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 2328 . . . 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 598 . . 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 2639 . 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 4857 . . . 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 2304 . . 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 2970 . 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 1182 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E!wreu 2558    e. cmpt 4093    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874  Word cword 11419   Basecbs 13164    gsumg cgsu 13417   Mndcmnd 14377   MndHom cmhm 14429  freeMndcfrmd 14485  varFMndcvrmd 14486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-frmd 14487  df-vrmd 14488
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