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Theorem frgpup1 15362
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.) (Revised by Mario Carneiro, 28-Feb-2016.)
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 )
frgpup.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpup.r  |-  .~  =  ( ~FG  `  I )
frgpup.g  |-  G  =  (freeGrp `  I )
frgpup.x  |-  X  =  ( Base `  G
)
frgpup.e  |-  E  =  ran  ( g  e.  W  |->  <. [ g ]  .~  ,  ( H 
gsumg  ( T  o.  g
) ) >. )
Assertion
Ref Expression
frgpup1  |-  ( ph  ->  E  e.  ( G 
GrpHom  H ) )
Distinct variable groups:    y, g,
z    g, H    y, F, z    y, N, z    B, g, y, z    T, g    .~ , g    ph, g, y, z    y, I, z   
g, W
Allowed substitution hints:    .~ ( y, z)    T( y, z)    E( y, z, g)    F( g)    G( y, z, g)    H( y, z)    I( g)    N( g)    V( y, z, g)    W( y, z)    X( y, z, g)

Proof of Theorem frgpup1
Dummy variables  a  u  c  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup.x . 2  |-  X  =  ( Base `  G
)
2 frgpup.b . 2  |-  B  =  ( Base `  H
)
3 eqid 2404 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2404 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 frgpup.i . . 3  |-  ( ph  ->  I  e.  V )
6 frgpup.g . . . 4  |-  G  =  (freeGrp `  I )
76frgpgrp 15349 . . 3  |-  ( I  e.  V  ->  G  e.  Grp )
85, 7syl 16 . 2  |-  ( ph  ->  G  e.  Grp )
9 frgpup.h . 2  |-  ( ph  ->  H  e.  Grp )
10 frgpup.n . . 3  |-  N  =  ( inv g `  H )
11 frgpup.t . . 3  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
12 frgpup.a . . 3  |-  ( ph  ->  F : I --> B )
13 frgpup.w . . 3  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
14 frgpup.r . . 3  |-  .~  =  ( ~FG  `  I )
15 frgpup.e . . 3  |-  E  =  ran  ( g  e.  W  |->  <. [ g ]  .~  ,  ( H 
gsumg  ( T  o.  g
) ) >. )
162, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupf 15360 . 2  |-  ( ph  ->  E : X --> B )
17 eqid 2404 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
186, 17, 14frgpval 15345 . . . . . . . . . 10  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
195, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
20 2on 6691 . . . . . . . . . . . . 13  |-  2o  e.  On
21 xpexg 4948 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
225, 20, 21sylancl 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
23 wrdexg 11694 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
24 fvi 5742 . . . . . . . . . . . 12  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
2522, 23, 243syl 19 . . . . . . . . . . 11  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
2613, 25syl5eq 2448 . . . . . . . . . 10  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
27 eqid 2404 . . . . . . . . . . . 12  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
2817, 27frmdbas 14752 . . . . . . . . . . 11  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
2922, 28syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
3026, 29eqtr4d 2439 . . . . . . . . 9  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
31 fvex 5701 . . . . . . . . . . 11  |-  ( ~FG  `  I
)  e.  _V
3214, 31eqeltri 2474 . . . . . . . . . 10  |-  .~  e.  _V
3332a1i 11 . . . . . . . . 9  |-  ( ph  ->  .~  e.  _V )
34 fvex 5701 . . . . . . . . . 10  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
3534a1i 11 . . . . . . . . 9  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
3619, 30, 33, 35divsbas 13725 . . . . . . . 8  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
3736, 1syl6reqr 2455 . . . . . . 7  |-  ( ph  ->  X  =  ( W /.  .~  ) )
38 eqimss 3360 . . . . . . 7  |-  ( X  =  ( W /.  .~  )  ->  X  C_  ( W /.  .~  ) )
3937, 38syl 16 . . . . . 6  |-  ( ph  ->  X  C_  ( W /.  .~  ) )
4039adantr 452 . . . . 5  |-  ( (
ph  /\  a  e.  X )  ->  X  C_  ( W /.  .~  ) )
4140sselda 3308 . . . 4  |-  ( ( ( ph  /\  a  e.  X )  /\  c  e.  X )  ->  c  e.  ( W /.  .~  ) )
42 eqid 2404 . . . . 5  |-  ( W /.  .~  )  =  ( W /.  .~  )
43 oveq2 6048 . . . . . . 7  |-  ( [ u ]  .~  =  c  ->  ( a ( +g  `  G ) [ u ]  .~  )  =  ( a
( +g  `  G ) c ) )
4443fveq2d 5691 . . . . . 6  |-  ( [ u ]  .~  =  c  ->  ( E `  ( a ( +g  `  G ) [ u ]  .~  ) )  =  ( E `  (
a ( +g  `  G
) c ) ) )
45 fveq2 5687 . . . . . . 7  |-  ( [ u ]  .~  =  c  ->  ( E `  [ u ]  .~  )  =  ( E `  c ) )
4645oveq2d 6056 . . . . . 6  |-  ( [ u ]  .~  =  c  ->  ( ( E `
 a ) ( +g  `  H ) ( E `  [
u ]  .~  )
)  =  ( ( E `  a ) ( +g  `  H
) ( E `  c ) ) )
4744, 46eqeq12d 2418 . . . . 5  |-  ( [ u ]  .~  =  c  ->  ( ( E `
 ( a ( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) )  <->  ( E `  ( a ( +g  `  G ) c ) )  =  ( ( E `  a ) ( +g  `  H
) ( E `  c ) ) ) )
4839sselda 3308 . . . . . . . 8  |-  ( (
ph  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
4948adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  W )  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
50 oveq1 6047 . . . . . . . . . 10  |-  ( [ t ]  .~  =  a  ->  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  )  =  ( a ( +g  `  G
) [ u ]  .~  ) )
5150fveq2d 5691 . . . . . . . . 9  |-  ( [ t ]  .~  =  a  ->  ( E `  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  ) )  =  ( E `  ( a ( +g  `  G
) [ u ]  .~  ) ) )
52 fveq2 5687 . . . . . . . . . 10  |-  ( [ t ]  .~  =  a  ->  ( E `  [ t ]  .~  )  =  ( E `  a ) )
5352oveq1d 6055 . . . . . . . . 9  |-  ( [ t ]  .~  =  a  ->  ( ( E `
 [ t ]  .~  ) ( +g  `  H ) ( E `
 [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
5451, 53eqeq12d 2418 . . . . . . . 8  |-  ( [ t ]  .~  =  a  ->  ( ( E `
 ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  [ t ]  .~  ) ( +g  `  H
) ( E `  [ u ]  .~  ) )  <->  ( E `  ( a ( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a ) ( +g  `  H ) ( E `
 [ u ]  .~  ) ) ) )
55 fviss 5743 . . . . . . . . . . . . . . . 16  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
5613, 55eqsstri 3338 . . . . . . . . . . . . . . 15  |-  W  C_ Word  ( I  X.  2o )
5756sseli 3304 . . . . . . . . . . . . . 14  |-  ( t  e.  W  ->  t  e. Word  ( I  X.  2o ) )
5856sseli 3304 . . . . . . . . . . . . . 14  |-  ( u  e.  W  ->  u  e. Word  ( I  X.  2o ) )
59 ccatcl 11698 . . . . . . . . . . . . . 14  |-  ( ( t  e. Word  ( I  X.  2o )  /\  u  e. Word  ( I  X.  2o ) )  -> 
( t concat  u )  e. Word  ( I  X.  2o ) )
6057, 58, 59syl2an 464 . . . . . . . . . . . . 13  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( t concat  u )  e. Word  ( I  X.  2o ) )
6113efgrcl 15302 . . . . . . . . . . . . . . 15  |-  ( t  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
6261adantr 452 . . . . . . . . . . . . . 14  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
6362simprd 450 . . . . . . . . . . . . 13  |-  ( ( t  e.  W  /\  u  e.  W )  ->  W  = Word  ( I  X.  2o ) )
6460, 63eleqtrrd 2481 . . . . . . . . . . . 12  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( t concat  u )  e.  W )
652, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 15361 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t concat  u )  e.  W )  ->  ( E `  [ ( t concat  u
) ]  .~  )  =  ( H  gsumg  ( T  o.  ( t concat  u
) ) ) )
6664, 65sylan2 461 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  [
( t concat  u ) ]  .~  )  =  ( H  gsumg  ( T  o.  (
t concat  u ) ) ) )
6757ad2antrl 709 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
t  e. Word  ( I  X.  2o ) )
6858ad2antll 710 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  ->  u  e. Word  ( I  X.  2o ) )
692, 10, 11, 9, 5, 12frgpuptf 15357 . . . . . . . . . . . . . 14  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
7069adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  ->  T : ( I  X.  2o ) --> B )
71 ccatco 11759 . . . . . . . . . . . . 13  |-  ( ( t  e. Word  ( I  X.  2o )  /\  u  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  (
t concat  u ) )  =  ( ( T  o.  t ) concat  ( T  o.  u ) ) )
7267, 68, 70, 71syl3anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( T  o.  (
t concat  u ) )  =  ( ( T  o.  t ) concat  ( T  o.  u ) ) )
7372oveq2d 6056 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( H  gsumg  ( T  o.  (
t concat  u ) ) )  =  ( H  gsumg  ( ( T  o.  t ) concat 
( T  o.  u
) ) ) )
74 grpmnd 14772 . . . . . . . . . . . . . 14  |-  ( H  e.  Grp  ->  H  e.  Mnd )
759, 74syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  H  e.  Mnd )
7675adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  ->  H  e.  Mnd )
77 wrdco 11755 . . . . . . . . . . . . . 14  |-  ( ( t  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  t
)  e. Word  B )
7857, 69, 77syl2anr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  e. Word  B )
7978adantrr 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( T  o.  t
)  e. Word  B )
80 wrdco 11755 . . . . . . . . . . . . 13  |-  ( ( u  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  u
)  e. Word  B )
8168, 70, 80syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( T  o.  u
)  e. Word  B )
822, 4gsumccat 14742 . . . . . . . . . . . 12  |-  ( ( H  e.  Mnd  /\  ( T  o.  t
)  e. Word  B  /\  ( T  o.  u
)  e. Word  B )  ->  ( H  gsumg  ( ( T  o.  t ) concat  ( T  o.  u ) ) )  =  ( ( H 
gsumg  ( T  o.  t
) ) ( +g  `  H ) ( H 
gsumg  ( T  o.  u
) ) ) )
8376, 79, 81, 82syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( H  gsumg  ( ( T  o.  t ) concat  ( T  o.  u ) ) )  =  ( ( H 
gsumg  ( T  o.  t
) ) ( +g  `  H ) ( H 
gsumg  ( T  o.  u
) ) ) )
8466, 73, 833eqtrd 2440 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  [
( t concat  u ) ]  .~  )  =  ( ( H  gsumg  ( T  o.  t
) ) ( +g  `  H ) ( H 
gsumg  ( T  o.  u
) ) ) )
8513, 6, 14, 3frgpadd 15350 . . . . . . . . . . . 12  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( [ t ]  .~  ( +g  `  G
) [ u ]  .~  )  =  [
( t concat  u ) ]  .~  )
8685adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( [ t ]  .~  ( +g  `  G
) [ u ]  .~  )  =  [
( t concat  u ) ]  .~  )
8786fveq2d 5691 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  ) )  =  ( E `  [ ( t concat  u ) ]  .~  ) )
882, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 15361 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( H  gsumg  ( T  o.  t
) ) )
8988adantrr 698 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  [
t ]  .~  )  =  ( H  gsumg  ( T  o.  t ) ) )
902, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 15361 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  W )  ->  ( E `  [ u ]  .~  )  =  ( H  gsumg  ( T  o.  u
) ) )
9190adantrl 697 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  [
u ]  .~  )  =  ( H  gsumg  ( T  o.  u ) ) )
9289, 91oveq12d 6058 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( ( E `  [ t ]  .~  ) ( +g  `  H
) ( E `  [ u ]  .~  ) )  =  ( ( H  gsumg  ( T  o.  t
) ) ( +g  `  H ) ( H 
gsumg  ( T  o.  u
) ) ) )
9384, 87, 923eqtr4d 2446 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  [
t ]  .~  )
( +g  `  H ) ( E `  [
u ]  .~  )
) )
9493anass1rs 783 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  W )  /\  t  e.  W )  ->  ( E `  ( [
t ]  .~  ( +g  `  G ) [ u ]  .~  )
)  =  ( ( E `  [ t ]  .~  ) ( +g  `  H ) ( E `  [
u ]  .~  )
) )
9542, 54, 94ectocld 6930 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  W )  /\  a  e.  ( W /.  .~  ) )  ->  ( E `  ( a
( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
9649, 95syldan 457 . . . . . 6  |-  ( ( ( ph  /\  u  e.  W )  /\  a  e.  X )  ->  ( E `  ( a
( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
9796an32s 780 . . . . 5  |-  ( ( ( ph  /\  a  e.  X )  /\  u  e.  W )  ->  ( E `  ( a
( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
9842, 47, 97ectocld 6930 . . . 4  |-  ( ( ( ph  /\  a  e.  X )  /\  c  e.  ( W /.  .~  ) )  ->  ( E `  ( a
( +g  `  G ) c ) )  =  ( ( E `  a ) ( +g  `  H ) ( E `
 c ) ) )
9941, 98syldan 457 . . 3  |-  ( ( ( ph  /\  a  e.  X )  /\  c  e.  X )  ->  ( E `  ( a
( +g  `  G ) c ) )  =  ( ( E `  a ) ( +g  `  H ) ( E `
 c ) ) )
10099anasss 629 . 2  |-  ( (
ph  /\  ( a  e.  X  /\  c  e.  X ) )  -> 
( E `  (
a ( +g  `  G
) c ) )  =  ( ( E `
 a ) ( +g  `  H ) ( E `  c
) ) )
1011, 2, 3, 4, 8, 9, 16, 100isghmd 14970 1  |-  ( ph  ->  E  e.  ( G 
GrpHom  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   (/)c0 3588   ifcif 3699   <.cop 3777    e. cmpt 4226    _I cid 4453   Oncon0 4541    X. cxp 4835   ran crn 4838    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   2oc2o 6677   [cec 6862   /.cqs 6863  Word cword 11672   concat cconcat 11673   Basecbs 13424   +g cplusg 13484    gsumg cgsu 13679    /.s cqus 13686   Mndcmnd 14639   Grpcgrp 14640   inv gcminusg 14641  freeMndcfrmd 14747    GrpHom cghm 14958   ~FG cefg 15293  freeGrpcfrgp 15294
This theorem is referenced by:  frgpup3lem  15364  frgpup3  15365
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  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2570  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-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-reverse 11683  df-s2 11767  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-0g 13682  df-gsum 13683  df-imas 13689  df-divs 13690  df-mnd 14645  df-submnd 14694  df-frmd 14749  df-grp 14767  df-minusg 14768  df-ghm 14959  df-efg 15296  df-frgp 15297
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