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Theorem frgpup1 15412
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 2438 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2438 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 frgpup.i . . 3  |-  ( ph  ->  I  e.  V )
6 frgpup.g . . . 4  |-  G  =  (freeGrp `  I )
76frgpgrp 15399 . . 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 15410 . 2  |-  ( ph  ->  E : X --> B )
17 eqid 2438 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
186, 17, 14frgpval 15395 . . . . . . . . . 10  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
195, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
20 2on 6735 . . . . . . . . . . . . 13  |-  2o  e.  On
21 xpexg 4992 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
225, 20, 21sylancl 645 . . . . . . . . . . . 12  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
23 wrdexg 11744 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
24 fvi 5786 . . . . . . . . . . . 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 2482 . . . . . . . . . 10  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
27 eqid 2438 . . . . . . . . . . . 12  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
2817, 27frmdbas 14802 . . . . . . . . . . 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 2473 . . . . . . . . 9  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
31 fvex 5745 . . . . . . . . . . 11  |-  ( ~FG  `  I
)  e.  _V
3214, 31eqeltri 2508 . . . . . . . . . 10  |-  .~  e.  _V
3332a1i 11 . . . . . . . . 9  |-  ( ph  ->  .~  e.  _V )
34 fvex 5745 . . . . . . . . . 10  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
3534a1i 11 . . . . . . . . 9  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
3619, 30, 33, 35divsbas 13775 . . . . . . . 8  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
3736, 1syl6reqr 2489 . . . . . . 7  |-  ( ph  ->  X  =  ( W /.  .~  ) )
38 eqimss 3402 . . . . . . 7  |-  ( X  =  ( W /.  .~  )  ->  X  C_  ( W /.  .~  ) )
3937, 38syl 16 . . . . . 6  |-  ( ph  ->  X  C_  ( W /.  .~  ) )
4039adantr 453 . . . . 5  |-  ( (
ph  /\  a  e.  X )  ->  X  C_  ( W /.  .~  ) )
4140sselda 3350 . . . 4  |-  ( ( ( ph  /\  a  e.  X )  /\  c  e.  X )  ->  c  e.  ( W /.  .~  ) )
42 eqid 2438 . . . . 5  |-  ( W /.  .~  )  =  ( W /.  .~  )
43 oveq2 6092 . . . . . . 7  |-  ( [ u ]  .~  =  c  ->  ( a ( +g  `  G ) [ u ]  .~  )  =  ( a
( +g  `  G ) c ) )
4443fveq2d 5735 . . . . . 6  |-  ( [ u ]  .~  =  c  ->  ( E `  ( a ( +g  `  G ) [ u ]  .~  ) )  =  ( E `  (
a ( +g  `  G
) c ) ) )
45 fveq2 5731 . . . . . . 7  |-  ( [ u ]  .~  =  c  ->  ( E `  [ u ]  .~  )  =  ( E `  c ) )
4645oveq2d 6100 . . . . . 6  |-  ( [ u ]  .~  =  c  ->  ( ( E `
 a ) ( +g  `  H ) ( E `  [
u ]  .~  )
)  =  ( ( E `  a ) ( +g  `  H
) ( E `  c ) ) )
4744, 46eqeq12d 2452 . . . . 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 3350 . . . . . . . 8  |-  ( (
ph  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
4948adantlr 697 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  W )  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
50 oveq1 6091 . . . . . . . . . 10  |-  ( [ t ]  .~  =  a  ->  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  )  =  ( a ( +g  `  G
) [ u ]  .~  ) )
5150fveq2d 5735 . . . . . . . . 9  |-  ( [ t ]  .~  =  a  ->  ( E `  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  ) )  =  ( E `  ( a ( +g  `  G
) [ u ]  .~  ) ) )
52 fveq2 5731 . . . . . . . . . 10  |-  ( [ t ]  .~  =  a  ->  ( E `  [ t ]  .~  )  =  ( E `  a ) )
5352oveq1d 6099 . . . . . . . . 9  |-  ( [ t ]  .~  =  a  ->  ( ( E `
 [ t ]  .~  ) ( +g  `  H ) ( E `
 [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
5451, 53eqeq12d 2452 . . . . . . . 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 5787 . . . . . . . . . . . . . . . 16  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
5613, 55eqsstri 3380 . . . . . . . . . . . . . . 15  |-  W  C_ Word  ( I  X.  2o )
5756sseli 3346 . . . . . . . . . . . . . 14  |-  ( t  e.  W  ->  t  e. Word  ( I  X.  2o ) )
5856sseli 3346 . . . . . . . . . . . . . 14  |-  ( u  e.  W  ->  u  e. Word  ( I  X.  2o ) )
59 ccatcl 11748 . . . . . . . . . . . . . 14  |-  ( ( t  e. Word  ( I  X.  2o )  /\  u  e. Word  ( I  X.  2o ) )  -> 
( t concat  u )  e. Word  ( I  X.  2o ) )
6057, 58, 59syl2an 465 . . . . . . . . . . . . 13  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( t concat  u )  e. Word  ( I  X.  2o ) )
6113efgrcl 15352 . . . . . . . . . . . . . . 15  |-  ( t  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
6261adantr 453 . . . . . . . . . . . . . 14  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
6362simprd 451 . . . . . . . . . . . . 13  |-  ( ( t  e.  W  /\  u  e.  W )  ->  W  = Word  ( I  X.  2o ) )
6460, 63eleqtrrd 2515 . . . . . . . . . . . 12  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( t concat  u )  e.  W )
652, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 15411 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t concat  u )  e.  W )  ->  ( E `  [ ( t concat  u
) ]  .~  )  =  ( H  gsumg  ( T  o.  ( t concat  u
) ) ) )
6664, 65sylan2 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  [
( t concat  u ) ]  .~  )  =  ( H  gsumg  ( T  o.  (
t concat  u ) ) ) )
6757ad2antrl 710 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
t  e. Word  ( I  X.  2o ) )
6858ad2antll 711 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  ->  u  e. Word  ( I  X.  2o ) )
692, 10, 11, 9, 5, 12frgpuptf 15407 . . . . . . . . . . . . . 14  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
7069adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  ->  T : ( I  X.  2o ) --> B )
71 ccatco 11809 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( T  o.  (
t concat  u ) )  =  ( ( T  o.  t ) concat  ( T  o.  u ) ) )
7372oveq2d 6100 . . . . . . . . . . 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 14822 . . . . . . . . . . . . . 14  |-  ( H  e.  Grp  ->  H  e.  Mnd )
759, 74syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  H  e.  Mnd )
7675adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  ->  H  e.  Mnd )
77 wrdco 11805 . . . . . . . . . . . . . 14  |-  ( ( t  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  t
)  e. Word  B )
7857, 69, 77syl2anr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  e. Word  B )
7978adantrr 699 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( T  o.  t
)  e. Word  B )
80 wrdco 11805 . . . . . . . . . . . . 13  |-  ( ( u  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  u
)  e. Word  B )
8168, 70, 80syl2anc 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( T  o.  u
)  e. Word  B )
822, 4gsumccat 14792 . . . . . . . . . . . 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 1185 . . . . . . . . . . 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 2474 . . . . . . . . . 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 15400 . . . . . . . . . . . 12  |-  ( ( t  e.  W  /\  u  e.  W )  ->  ( [ t ]  .~  ( +g  `  G
) [ u ]  .~  )  =  [
( t concat  u ) ]  .~  )
8685adantl 454 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( [ t ]  .~  ( +g  `  G
) [ u ]  .~  )  =  [
( t concat  u ) ]  .~  )
8786fveq2d 5735 . . . . . . . . . 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 15411 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( H  gsumg  ( T  o.  t
) ) )
8988adantrr 699 . . . . . . . . . . 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 15411 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  W )  ->  ( E `  [ u ]  .~  )  =  ( H  gsumg  ( T  o.  u
) ) )
9190adantrl 698 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  [
u ]  .~  )  =  ( H  gsumg  ( T  o.  u ) ) )
9289, 91oveq12d 6102 . . . . . . . . . 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 2480 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  W  /\  u  e.  W ) )  -> 
( E `  ( [ t ]  .~  ( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  [
t ]  .~  )
( +g  `  H ) ( E `  [
u ]  .~  )
) )
9493anass1rs 784 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  W )  /\  t  e.  W )  ->  ( E `  ( [
t ]  .~  ( +g  `  G ) [ u ]  .~  )
)  =  ( ( E `  [ t ]  .~  ) ( +g  `  H ) ( E `  [
u ]  .~  )
) )
9542, 54, 94ectocld 6974 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  W )  /\  a  e.  ( W /.  .~  ) )  ->  ( E `  ( a
( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
9649, 95syldan 458 . . . . . 6  |-  ( ( ( ph  /\  u  e.  W )  /\  a  e.  X )  ->  ( E `  ( a
( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
9796an32s 781 . . . . 5  |-  ( ( ( ph  /\  a  e.  X )  /\  u  e.  W )  ->  ( E `  ( a
( +g  `  G ) [ u ]  .~  ) )  =  ( ( E `  a
) ( +g  `  H
) ( E `  [ u ]  .~  ) ) )
9842, 47, 97ectocld 6974 . . . 4  |-  ( ( ( ph  /\  a  e.  X )  /\  c  e.  ( W /.  .~  ) )  ->  ( E `  ( a
( +g  `  G ) c ) )  =  ( ( E `  a ) ( +g  `  H ) ( E `
 c ) ) )
9941, 98syldan 458 . . 3  |-  ( ( ( ph  /\  a  e.  X )  /\  c  e.  X )  ->  ( E `  ( a
( +g  `  G ) c ) )  =  ( ( E `  a ) ( +g  `  H ) ( E `
 c ) ) )
10099anasss 630 . 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 15020 1  |-  ( ph  ->  E  e.  ( G 
GrpHom  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ifcif 3741   <.cop 3819    e. cmpt 4269    _I cid 4496   Oncon0 4584    X. cxp 4879   ran crn 4882    o. ccom 4885   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   2oc2o 6721   [cec 6906   /.cqs 6907  Word cword 11722   concat cconcat 11723   Basecbs 13474   +g cplusg 13534    gsumg cgsu 13729    /.s cqus 13736   Mndcmnd 14689   Grpcgrp 14690   inv gcminusg 14691  freeMndcfrmd 14797    GrpHom cghm 15008   ~FG cefg 15343  freeGrpcfrgp 15344
This theorem is referenced by:  frgpup3lem  15414  frgpup3  15415
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  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-nel 2604  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-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  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-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-splice 11732  df-reverse 11733  df-s2 11817  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-0g 13732  df-gsum 13733  df-imas 13739  df-divs 13740  df-mnd 14695  df-submnd 14744  df-frmd 14799  df-grp 14817  df-minusg 14818  df-ghm 15009  df-efg 15346  df-frgp 15347
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