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Theorem frgpnabllem2 15178
Description: Lemma for frgpnabl 15179. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
frgpnabl.g  |-  G  =  (freeGrp `  I )
frgpnabl.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpnabl.r  |-  .~  =  ( ~FG  `  I )
frgpnabl.p  |-  .+  =  ( +g  `  G )
frgpnabl.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
frgpnabl.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
frgpnabl.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
frgpnabl.u  |-  U  =  (varFGrp `  I )
frgpnabl.i  |-  ( ph  ->  I  e.  _V )
frgpnabl.a  |-  ( ph  ->  A  e.  I )
frgpnabl.b  |-  ( ph  ->  B  e.  I )
frgpnabl.n  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
Assertion
Ref Expression
frgpnabllem2  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, A    v, n, w, x, y, z, I    ph, x    x, 
.~ , y, z    x, B    n, W, v, w, x, y, z    x, G    n, M, v, w, x    x, T
Allowed substitution hints:    ph( y, z, w, v, n)    A( y, z, w, v, n)    B( y, z, w, v, n)    D( x, y, z, w, v, n)    .+ ( x, y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    U( x, y, z, w, v, n)    G( y,
z, w, v, n)    M( y, z)

Proof of Theorem frgpnabllem2
Dummy variables  d  m  t  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpnabl.a . 2  |-  ( ph  ->  A  e.  I )
2 0ex 4166 . . 3  |-  (/)  e.  _V
32a1i 10 . 2  |-  ( ph  -> 
(/)  e.  _V )
4 frgpnabl.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
5 difss 3316 . . . . . . . 8  |-  ( W 
\  U_ x  e.  W  ran  ( T `  x
) )  C_  W
64, 5eqsstri 3221 . . . . . . 7  |-  D  C_  W
7 inss1 3402 . . . . . . . 8  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  D
8 frgpnabl.g . . . . . . . . 9  |-  G  =  (freeGrp `  I )
9 frgpnabl.w . . . . . . . . 9  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
10 frgpnabl.r . . . . . . . . 9  |-  .~  =  ( ~FG  `  I )
11 frgpnabl.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
12 frgpnabl.m . . . . . . . . 9  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
13 frgpnabl.t . . . . . . . . 9  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
14 frgpnabl.u . . . . . . . . 9  |-  U  =  (varFGrp `  I )
15 frgpnabl.i . . . . . . . . 9  |-  ( ph  ->  I  e.  _V )
16 frgpnabl.b . . . . . . . . 9  |-  ( ph  ->  B  e.  I )
178, 9, 10, 11, 12, 13, 4, 14, 15, 16, 1frgpnabllem1 15177 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) ) )
187, 17sseldi 3191 . . . . . . 7  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  D )
196, 18sseldi 3191 . . . . . 6  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  W )
20 eqid 2296 . . . . . . 7  |-  ( m  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )  =  ( m  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
219, 10, 12, 13, 4, 20efgredeu 15077 . . . . . 6  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  W  ->  E! d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
22 reu5 2766 . . . . . . 7  |-  ( E! d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  ( E. d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  /\ 
E* d  e.  D
d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
2322simprbi 450 . . . . . 6  |-  ( E! d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  E* d  e.  D d  .~  <"
<. B ,  (/) >. <. A ,  (/)
>. "> )
2419, 21, 233syl 18 . . . . 5  |-  ( ph  ->  E* d  e.  D
d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
25 inss1 3402 . . . . . 6  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  D
268, 9, 10, 11, 12, 13, 4, 14, 15, 1, 16frgpnabllem1 15177 . . . . . 6  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) ) )
2725, 26sseldi 3191 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D )
289, 10efger 15043 . . . . . . . . 9  |-  .~  Er  W
2928a1i 10 . . . . . . . 8  |-  ( ph  ->  .~  Er  W )
308frgpgrp 15087 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  e.  Grp )
3115, 30syl 15 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
32 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3310, 14, 8, 32vrgpf 15093 . . . . . . . . . . . 12  |-  ( I  e.  _V  ->  U : I --> ( Base `  G ) )
3415, 33syl 15 . . . . . . . . . . 11  |-  ( ph  ->  U : I --> ( Base `  G ) )
35 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( U : I --> ( Base `  G )  /\  A  e.  I )  ->  ( U `  A )  e.  ( Base `  G
) )
3634, 1, 35syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( U `  A
)  e.  ( Base `  G ) )
37 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( U : I --> ( Base `  G )  /\  B  e.  I )  ->  ( U `  B )  e.  ( Base `  G
) )
3834, 16, 37syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( U `  B
)  e.  ( Base `  G ) )
3932, 11grpcl 14511 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( U `  A )  e.  ( Base `  G
)  /\  ( U `  B )  e.  (
Base `  G )
)  ->  ( ( U `  A )  .+  ( U `  B
) )  e.  (
Base `  G )
)
4031, 36, 38, 39syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( Base `  G ) )
41 eqid 2296 . . . . . . . . . . . 12  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
428, 41, 10frgpval 15083 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
4315, 42syl 15 . . . . . . . . . 10  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
44 2on 6503 . . . . . . . . . . . . . 14  |-  2o  e.  On
45 xpexg 4816 . . . . . . . . . . . . . 14  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
4615, 44, 45sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
47 wrdexg 11441 . . . . . . . . . . . . 13  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
48 fvi 5595 . . . . . . . . . . . . 13  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
4946, 47, 483syl 18 . . . . . . . . . . . 12  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
509, 49syl5eq 2340 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
51 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
5241, 51frmdbas 14490 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
5346, 52syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
5450, 53eqtr4d 2331 . . . . . . . . . 10  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
55 fvex 5555 . . . . . . . . . . . 12  |-  ( ~FG  `  I
)  e.  _V
5610, 55eqeltri 2366 . . . . . . . . . . 11  |-  .~  e.  _V
5756a1i 10 . . . . . . . . . 10  |-  ( ph  ->  .~  e.  _V )
58 fvex 5555 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5958a1i 10 . . . . . . . . . 10  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
6043, 54, 57, 59divsbas 13463 . . . . . . . . 9  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
6140, 60eleqtrrd 2373 . . . . . . . 8  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  ) )
62 inss2 3403 . . . . . . . . 9  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  (
( U `  A
)  .+  ( U `  B ) )
6362, 26sseldi 3191 . . . . . . . 8  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
64 qsel 6754 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  )
6529, 61, 63, 64syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. A ,  (/) >. <. B ,  (/)
>. "> ]  .~  )
66 inss2 3403 . . . . . . . . . 10  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  (
( U `  B
)  .+  ( U `  A ) )
6766, 17sseldi 3191 . . . . . . . . 9  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  B ) 
.+  ( U `  A ) ) )
68 frgpnabl.n . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
6967, 68eleqtrrd 2373 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
70 qsel 6754 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. B ,  (/) >. <. A ,  (/) >. "> ]  .~  )
7129, 61, 69, 70syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. B ,  (/) >. <. A ,  (/)
>. "> ]  .~  )
7265, 71eqtr3d 2330 . . . . . 6  |-  ( ph  ->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
736, 27sseldi 3191 . . . . . . 7  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  W )
7429, 73erth 6720 . . . . . 6  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  <->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
)
7572, 74mpbird 223 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7629, 19erref 6696 . . . . 5  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
77 breq1 4042 . . . . . 6  |-  ( d  =  <" <. A ,  (/)
>. <. B ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
78 breq1 4042 . . . . . 6  |-  ( d  =  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
7977, 78rmoi 3093 . . . . 5  |-  ( ( E* d  e.  D
d  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  /\  ( <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  D  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )  /\  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  D  /\  <" <. B ,  (/) >. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )  ->  <" <. A ,  (/) >. <. B ,  (/) >. ">  =  <" <. B ,  (/)
>. <. A ,  (/) >. "> )
8024, 27, 75, 18, 76, 79syl122anc 1191 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" <. B ,  (/) >. <. A ,  (/) >. "> )
8180fveq1d 5543 . . 3  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )
)
82 opex 4253 . . . 4  |-  <. A ,  (/)
>.  e.  _V
83 s2fv0 11551 . . . 4  |-  ( <. A ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
)
8482, 83ax-mp 8 . . 3  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
85 opex 4253 . . . 4  |-  <. B ,  (/)
>.  e.  _V
86 s2fv0 11551 . . . 4  |-  ( <. B ,  (/) >.  e.  _V  ->  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
)
8785, 86ax-mp 8 . . 3  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
8881, 84, 873eqtr3g 2351 . 2  |-  ( ph  -> 
<. A ,  (/) >.  =  <. B ,  (/) >. )
89 opthg 4262 . . 3  |-  ( ( A  e.  I  /\  (/) 
e.  _V )  ->  ( <. A ,  (/) >.  =  <. B ,  (/) >.  <->  ( A  =  B  /\  (/)  =  (/) ) ) )
9089simprbda 606 . 2  |-  ( ( ( A  e.  I  /\  (/)  e.  _V )  /\  <. A ,  (/) >.  =  <. B ,  (/) >.
)  ->  A  =  B )
911, 3, 88, 90syl21anc 1181 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   E*wrmo 2559   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039    e. cmpt 4093    _I cid 4320   Oncon0 4408    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489    Er wer 6673   [cec 6674   /.cqs 6675   0cc0 8753   1c1 8754    - cmin 9053   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   Basecbs 13164   +g cplusg 13224    /.s cqus 13424   Grpcgrp 14378  freeMndcfrmd 14485   ~FG cefg 15031  freeGrpcfrgp 15032  varFGrpcvrgp 15033
This theorem is referenced by:  frgpnabl  15179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-frmd 14487  df-grp 14505  df-efg 15034  df-frgp 15035  df-vrgp 15036
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