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

Theorem frgpuptf 15178
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpup.b  |-  B  =  ( Base `  H
)
frgpup.n  |-  N  =  ( inv g `  H )
frgpup.t  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
frgpup.h  |-  ( ph  ->  H  e.  Grp )
frgpup.i  |-  ( ph  ->  I  e.  V )
frgpup.a  |-  ( ph  ->  F : I --> B )
Assertion
Ref Expression
frgpuptf  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
Distinct variable groups:    y, z, F    y, N, z    y, B, z    ph, y, z   
y, I, z
Allowed substitution hints:    T( y, z)    H( y, z)    V( y, z)

Proof of Theorem frgpuptf
StepHypRef Expression
1 frgpup.a . . . . . 6  |-  ( ph  ->  F : I --> B )
2 ffvelrn 5746 . . . . . 6  |-  ( ( F : I --> B  /\  y  e.  I )  ->  ( F `  y
)  e.  B )
31, 2sylan 457 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  B )
43adantrr 697 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( F `  y
)  e.  B )
5 frgpup.h . . . . . 6  |-  ( ph  ->  H  e.  Grp )
65adantr 451 . . . . 5  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  ->  H  e.  Grp )
7 frgpup.b . . . . . 6  |-  B  =  ( Base `  H
)
8 frgpup.n . . . . . 6  |-  N  =  ( inv g `  H )
97, 8grpinvcl 14626 . . . . 5  |-  ( ( H  e.  Grp  /\  ( F `  y )  e.  B )  -> 
( N `  ( F `  y )
)  e.  B )
106, 4, 9syl2anc 642 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( N `  ( F `  y )
)  e.  B )
11 ifcl 3677 . . . 4  |-  ( ( ( F `  y
)  e.  B  /\  ( N `  ( F `
 y ) )  e.  B )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
124, 10, 11syl2anc 642 . . 3  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
1312ralrimivva 2711 . 2  |-  ( ph  ->  A. y  e.  I  A. z  e.  2o  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
14 frgpup.t . . 3  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
1514fmpt2 6278 . 2  |-  ( A. y  e.  I  A. z  e.  2o  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B  <->  T :
( I  X.  2o )
--> B )
1613, 15sylib 188 1  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   (/)c0 3531   ifcif 3641    X. cxp 4769   -->wf 5333   ` cfv 5337    e. cmpt2 5947   2oc2o 6560   Basecbs 13245   Grpcgrp 14461   inv gcminusg 14462
This theorem is referenced by:  frgpuplem  15180  frgpupf  15181  frgpup1  15183  frgpup2  15184  frgpup3lem  15185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589
  Copyright terms: Public domain W3C validator