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Theorem frgpuptf 16389
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  =  ( invg `  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 5953 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  B )
32adantrr 716 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( F `  y
)  e.  B )
4 frgpup.h . . . . . 6  |-  ( ph  ->  H  e.  Grp )
54adantr 465 . . . . 5  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  ->  H  e.  Grp )
6 frgpup.b . . . . . 6  |-  B  =  ( Base `  H
)
7 frgpup.n . . . . . 6  |-  N  =  ( invg `  H )
86, 7grpinvcl 15703 . . . . 5  |-  ( ( H  e.  Grp  /\  ( F `  y )  e.  B )  -> 
( N `  ( F `  y )
)  e.  B )
95, 3, 8syl2anc 661 . . . 4  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  -> 
( N `  ( F `  y )
)  e.  B )
10 ifcl 3940 . . . 4  |-  ( ( ( F `  y
)  e.  B  /\  ( N `  ( F `
 y ) )  e.  B )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
113, 9, 10syl2anc 661 . . 3  |-  ( (
ph  /\  ( y  e.  I  /\  z  e.  2o ) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  e.  B )
1211ralrimivva 2914 . 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 6752 . 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 196 1  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   (/)c0 3746   ifcif 3900    X. cxp 4947   -->wf 5523   ` cfv 5527    |-> cmpt2 6203   2oc2o 7025   Basecbs 14293   Grpcgrp 15530   invgcminusg 15531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666
This theorem is referenced by:  frgpuplem  16391  frgpupf  16392  frgpup1  16394  frgpup2  16395  frgpup3lem  16396
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