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Theorem frgpuptf 15357
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 )
21ffvelrnda 5829 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  B )
32adantrr 698 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( F `  y
)  e.  B )
4 frgpup.h . . . . . 6  |-  ( ph  ->  H  e.  Grp )
54adantr 452 . . . . 5  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  ->  H  e.  Grp )
6 frgpup.b . . . . . 6  |-  B  =  ( Base `  H
)
7 frgpup.n . . . . . 6  |-  N  =  ( inv g `  H )
86, 7grpinvcl 14805 . . . . 5  |-  ( ( H  e.  Grp  /\  ( F `  y )  e.  B )  -> 
( N `  ( F `  y )
)  e.  B )
95, 3, 8syl2anc 643 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( N `  ( F `  y )
)  e.  B )
10 ifcl 3735 . . . 4  |-  ( ( ( F `  y
)  e.  B  /\  ( N `  ( F `
 y ) )  e.  B )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
113, 9, 10syl2anc 643 . . 3  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
1211ralrimivva 2758 . 2  |-  ( ph  ->  A. y  e.  I  A. z  e.  2o  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
13 frgpup.t . . 3  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
1413fmpt2 6377 . 2  |-  ( A. y  e.  I  A. z  e.  2o  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B  <->  T :
( I  X.  2o )
--> B )
1512, 14sylib 189 1  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   (/)c0 3588   ifcif 3699    X. cxp 4835   -->wf 5409   ` cfv 5413    e. cmpt2 6042   2oc2o 6677   Basecbs 13424   Grpcgrp 14640   inv gcminusg 14641
This theorem is referenced by:  frgpuplem  15359  frgpupf  15360  frgpup1  15362  frgpup2  15363  frgpup3lem  15364
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768
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