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Theorem frgpuptf 15407
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 5873 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  B )
32adantrr 699 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( F `  y
)  e.  B )
4 frgpup.h . . . . . 6  |-  ( ph  ->  H  e.  Grp )
54adantr 453 . . . . 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 14855 . . . . 5  |-  ( ( H  e.  Grp  /\  ( F `  y )  e.  B )  -> 
( N `  ( F `  y )
)  e.  B )
95, 3, 8syl2anc 644 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( N `  ( F `  y )
)  e.  B )
10 ifcl 3777 . . . 4  |-  ( ( ( F `  y
)  e.  B  /\  ( N `  ( F `
 y ) )  e.  B )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
113, 9, 10syl2anc 644 . . 3  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
1211ralrimivva 2800 . 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 6421 . 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 190 1  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   (/)c0 3630   ifcif 3741    X. cxp 4879   -->wf 5453   ` cfv 5457    e. cmpt2 6086   2oc2o 6721   Basecbs 13474   Grpcgrp 14690   inv gcminusg 14691
This theorem is referenced by:  frgpuplem  15409  frgpupf  15410  frgpup1  15412  frgpup2  15413  frgpup3lem  15414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818
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