MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frmdup3 Structured version   Unicode version

Theorem frmdup3 16357
Description: Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-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    m, M    U, 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 2402 . . 3  |-  ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )
4 simp1 997 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  G  e.  Mnd )
5 simp2 998 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  I  e.  V
)
6 simp3 999 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A : I --> B )
71, 2, 3, 4, 5, 6frmdup1 16354 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) )  e.  ( M MndHom  G ) )
84adantr 463 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  G  e.  Mnd )
95adantr 463 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  I  e.  V )
106adantr 463 . . . . 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 459 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  y  e.  I
)  ->  y  e.  I )
131, 2, 3, 8, 9, 10, 11, 12frmdup2 16355 . . . 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 4480 . . 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 2402 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
1615, 2mhmf 16293 . . . . 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 17 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) ) : ( Base `  M ) --> B )
1811vrmdf 16348 . . . . . 6  |-  ( I  e.  V  ->  U : I -->Word  I )
19183ad2ant2 1019 . . . . 5  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I -->Word  I )
201, 15frmdbas 16342 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
21203ad2ant2 1019 . . . . . 6  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( Base `  M
)  = Word  I )
2221feq3d 5701 . . . . 5  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( U :
I --> ( Base `  M
)  <->  U : I -->Word  I )
)
2319, 22mpbird 232 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I --> ( Base `  M
) )
24 fcompt 6045 . . . 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 ) ) ) )
2517, 23, 24syl2anc 659 . . 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 ) ) ) )
266feqmptd 5901 . . 3  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  A  =  ( y  e.  I  |->  ( A `  y ) ) )
2714, 25, 263eqtr4d 2453 . 2  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) )  o.  U )  =  A )
281, 2, 11frmdup3lem 16356 . . . 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 ) ) ) )
2928expr 613 . . 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
) ) ) ) )
3029ralrimiva 2817 . 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 ) ) ) ) )
31 coeq1 4980 . . . 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 ) )
3231eqeq1d 2404 . . 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 ) )
3332eqreu 3240 . 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 )
347, 27, 30, 33syl3anc 1230 1  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  E! m  e.  ( M MndHom  G ) ( m  o.  U
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   E!wreu 2755    |-> cmpt 4452    o. ccom 4826   -->wf 5564   ` cfv 5568  (class class class)co 6277  Word cword 12581   Basecbs 14839    gsumg cgsu 15053   Mndcmnd 16241   MndHom cmhm 16286  freeMndcfrmd 16337  varFMndcvrmd 16338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-seq 12150  df-hash 12451  df-word 12589  df-lsw 12590  df-concat 12591  df-s1 12592  df-substr 12593  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-0g 15054  df-gsum 15055  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-submnd 16289  df-frmd 16339  df-vrmd 16340
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator