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Theorem frgpup1 15366
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 2408 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2408 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 frgpup.i . . 3  |-  ( ph  ->  I  e.  V )
6 frgpup.g . . . 4  |-  G  =  (freeGrp `  I )
76frgpgrp 15353 . . 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 15364 . 2  |-  ( ph  ->  E : X --> B )
17 eqid 2408 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
186, 17, 14frgpval 15349 . . . . . . . . . 10  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
195, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
20 2on 6695 . . . . . . . . . . . . 13  |-  2o  e.  On
21 xpexg 4952 . . . . . . . . . . . . 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 11698 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
24 fvi 5746 . . . . . . . . . . . 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 2452 . . . . . . . . . 10  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
27 eqid 2408 . . . . . . . . . . . 12  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
2817, 27frmdbas 14756 . . . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
31 fvex 5705 . . . . . . . . . . 11  |-  ( ~FG  `  I
)  e.  _V
3214, 31eqeltri 2478 . . . . . . . . . 10  |-  .~  e.  _V
3332a1i 11 . . . . . . . . 9  |-  ( ph  ->  .~  e.  _V )
34 fvex 5705 . . . . . . . . . 10  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
3534a1i 11 . . . . . . . . 9  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
3619, 30, 33, 35divsbas 13729 . . . . . . . 8  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
3736, 1syl6reqr 2459 . . . . . . 7  |-  ( ph  ->  X  =  ( W /.  .~  ) )
38 eqimss 3364 . . . . . . 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 3312 . . . 4  |-  ( ( ( ph  /\  a  e.  X )  /\  c  e.  X )  ->  c  e.  ( W /.  .~  ) )
42 eqid 2408 . . . . 5  |-  ( W /.  .~  )  =  ( W /.  .~  )
43 oveq2 6052 . . . . . . 7  |-  ( [ u ]  .~  =  c  ->  ( a ( +g  `  G ) [ u ]  .~  )  =  ( a
( +g  `  G ) c ) )
4443fveq2d 5695 . . . . . 6  |-  ( [ u ]  .~  =  c  ->  ( E `  ( a ( +g  `  G ) [ u ]  .~  ) )  =  ( E `  (
a ( +g  `  G
) c ) ) )
45 fveq2 5691 . . . . . . 7  |-  ( [ u ]  .~  =  c  ->  ( E `  [ u ]  .~  )  =  ( E `  c ) )
4645oveq2d 6060 . . . . . 6  |-  ( [ u ]  .~  =  c  ->  ( ( E `
 a ) ( +g  `  H ) ( E `  [
u ]  .~  )
)  =  ( ( E `  a ) ( +g  `  H
) ( E `  c ) ) )
4744, 46eqeq12d 2422 . . . . 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 3312 . . . . . . . 8  |-  ( (
ph  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
4948adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  W )  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
50 oveq1 6051 . . . . . . . . . 10  |-  ( [ t ]  .~  =  a  ->  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  )  =  ( a ( +g  `  G
) [ u ]  .~  ) )
5150fveq2d 5695 . . . . . . . . 9  |-  ( [ t ]  .~  =  a  ->  ( E `  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  ) )  =  ( E `  ( a ( +g  `  G
) [ u ]  .~  ) ) )
52 fveq2 5691 . . . . . . . . . 10  |-  ( [ t ]  .~  =  a  ->  ( E `  [ t ]  .~  )  =  ( E `  a ) )
5352oveq1d 6059 . . . . . . . . 9  |-  ( [ t ]  .~  =  a  ->  ( ( E `
 [ t ]  .~  ) ( +g  `  H ) ( E `
 [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
5451, 53eqeq12d 2422 . . . . . . . 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 5747 . . . . . . . . . . . . . . . 16  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
5613, 55eqsstri 3342 . . . . . . . . . . . . . . 15  |-  W  C_ Word  ( I  X.  2o )
5756sseli 3308 . . . . . . . . . . . . . 14  |-  ( t  e.  W  ->  t  e. Word  ( I  X.  2o ) )
5856sseli 3308 . . . . . . . . . . . . . 14  |-  ( u  e.  W  ->  u  e. Word  ( I  X.  2o ) )
59 ccatcl 11702 . . . . . . . . . . . . . 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 15306 . . . . . . . . . . . . . . 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 2485 . . . . . . . . . . . 12  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( t concat  u )  e.  W )
652, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 15365 . . . . . . . . . . . 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 15361 . . . . . . . . . . . . . 14  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
7069adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  ->  T : ( I  X.  2o ) --> B )
71 ccatco 11763 . . . . . . . . . . . . 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 6060 . . . . . . . . . . 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 14776 . . . . . . . . . . . . . 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 11759 . . . . . . . . . . . . . 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 11759 . . . . . . . . . . . . 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 14746 . . . . . . . . . . . 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 2444 . . . . . . . . . 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 15354 . . . . . . . . . . . 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 5695 . . . . . . . . . 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 15365 . . . . . . . . . . . 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 15365 . . . . . . . . . . . 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 6062 . . . . . . . . . 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 2450 . . . . . . . . 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 6934 . . . . . . 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 6934 . . . 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 14974 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 2920    C_ wss 3284   (/)c0 3592   ifcif 3703   <.cop 3781    e. cmpt 4230    _I cid 4457   Oncon0 4545    X. cxp 4839   ran crn 4842    o. ccom 4845   -->wf 5413   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   2oc2o 6681   [cec 6866   /.cqs 6867  Word cword 11676   concat cconcat 11677   Basecbs 13428   +g cplusg 13488    gsumg cgsu 13683    /.s cqus 13690   Mndcmnd 14643   Grpcgrp 14644   inv gcminusg 14645  freeMndcfrmd 14751    GrpHom cghm 14962   ~FG cefg 15297  freeGrpcfrgp 15298
This theorem is referenced by:  frgpup3lem  15368  frgpup3  15369
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-ot 3788  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-ec 6870  df-qs 6874  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-fz 11004  df-fzo 11095  df-seq 11283  df-hash 11578  df-word 11682  df-concat 11683  df-s1 11684  df-substr 11685  df-splice 11686  df-reverse 11687  df-s2 11771  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-tset 13507  df-ple 13508  df-ds 13510  df-0g 13686  df-gsum 13687  df-imas 13693  df-divs 13694  df-mnd 14649  df-submnd 14698  df-frmd 14753  df-grp 14771  df-minusg 14772  df-ghm 14963  df-efg 15300  df-frgp 15301
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