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Theorem frgpnabllem2 17077
Description: Lemma for frgpnabl 17078. (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 4569 . . 3  |-  (/)  e.  _V
32a1i 11 . 2  |-  ( ph  -> 
(/)  e.  _V )
4 frgpnabl.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
5 difss 3617 . . . . . . . 8  |-  ( W 
\  U_ x  e.  W  ran  ( T `  x
) )  C_  W
64, 5eqsstri 3519 . . . . . . 7  |-  D  C_  W
7 inss1 3704 . . . . . . . 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 17076 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) ) )
187, 17sseldi 3487 . . . . . . 7  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  D )
196, 18sseldi 3487 . . . . . 6  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  W )
20 eqid 2454 . . . . . . 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 16969 . . . . . 6  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  W  ->  E! d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
22 reurmo 3072 . . . . . 6  |-  ( E! d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  E* d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
2319, 21, 223syl 20 . . . . 5  |-  ( ph  ->  E* d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. "> )
24 inss1 3704 . . . . . 6  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  D
258, 9, 10, 11, 12, 13, 4, 14, 15, 1, 16frgpnabllem1 17076 . . . . . 6  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) ) )
2624, 25sseldi 3487 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D )
279, 10efger 16935 . . . . . . . . 9  |-  .~  Er  W
2827a1i 11 . . . . . . . 8  |-  ( ph  ->  .~  Er  W )
298frgpgrp 16979 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  e.  Grp )
3015, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
31 eqid 2454 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3210, 14, 8, 31vrgpf 16985 . . . . . . . . . . . 12  |-  ( I  e.  _V  ->  U : I --> ( Base `  G ) )
3315, 32syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U : I --> ( Base `  G ) )
3433, 1ffvelrnd 6008 . . . . . . . . . 10  |-  ( ph  ->  ( U `  A
)  e.  ( Base `  G ) )
3533, 16ffvelrnd 6008 . . . . . . . . . 10  |-  ( ph  ->  ( U `  B
)  e.  ( Base `  G ) )
3631, 11grpcl 16262 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( U `  A )  e.  ( Base `  G
)  /\  ( U `  B )  e.  (
Base `  G )
)  ->  ( ( U `  A )  .+  ( U `  B
) )  e.  (
Base `  G )
)
3730, 34, 35, 36syl3anc 1226 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( Base `  G ) )
38 eqid 2454 . . . . . . . . . . . 12  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
398, 38, 10frgpval 16975 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
4015, 39syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
41 2on 7130 . . . . . . . . . . . . . 14  |-  2o  e.  On
42 xpexg 6575 . . . . . . . . . . . . . 14  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
4315, 41, 42sylancl 660 . . . . . . . . . . . . 13  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
44 wrdexg 12544 . . . . . . . . . . . . 13  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
45 fvi 5905 . . . . . . . . . . . . 13  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
4643, 44, 453syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
479, 46syl5eq 2507 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
48 eqid 2454 . . . . . . . . . . . . 13  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
4938, 48frmdbas 16219 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
5043, 49syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
5147, 50eqtr4d 2498 . . . . . . . . . 10  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
52 fvex 5858 . . . . . . . . . . . 12  |-  ( ~FG  `  I
)  e.  _V
5310, 52eqeltri 2538 . . . . . . . . . . 11  |-  .~  e.  _V
5453a1i 11 . . . . . . . . . 10  |-  ( ph  ->  .~  e.  _V )
55 fvex 5858 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5655a1i 11 . . . . . . . . . 10  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
5740, 51, 54, 56qusbas 15034 . . . . . . . . 9  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
5837, 57eleqtrrd 2545 . . . . . . . 8  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  ) )
59 inss2 3705 . . . . . . . . 9  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  (
( U `  A
)  .+  ( U `  B ) )
6059, 25sseldi 3487 . . . . . . . 8  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
61 qsel 7382 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  )
6228, 58, 60, 61syl3anc 1226 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. A ,  (/) >. <. B ,  (/)
>. "> ]  .~  )
63 inss2 3705 . . . . . . . . . 10  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  (
( U `  B
)  .+  ( U `  A ) )
6463, 17sseldi 3487 . . . . . . . . 9  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  B ) 
.+  ( U `  A ) ) )
65 frgpnabl.n . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
6664, 65eleqtrrd 2545 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
67 qsel 7382 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. B ,  (/) >. <. A ,  (/) >. "> ]  .~  )
6828, 58, 66, 67syl3anc 1226 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. B ,  (/) >. <. A ,  (/)
>. "> ]  .~  )
6962, 68eqtr3d 2497 . . . . . 6  |-  ( ph  ->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
706, 26sseldi 3487 . . . . . . 7  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  W )
7128, 70erth 7348 . . . . . 6  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  <->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
)
7269, 71mpbird 232 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7328, 19erref 7323 . . . . 5  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
74 breq1 4442 . . . . . 6  |-  ( d  =  <" <. A ,  (/)
>. <. B ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
75 breq1 4442 . . . . . 6  |-  ( d  =  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
7674, 75rmoi 3417 . . . . 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 ,  (/) >. "> )
7723, 26, 72, 18, 73, 76syl122anc 1235 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7877fveq1d 5850 . . 3  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )
)
79 opex 4701 . . . 4  |-  <. A ,  (/)
>.  e.  _V
80 s2fv0 12841 . . . 4  |-  ( <. A ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
)
8179, 80ax-mp 5 . . 3  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
82 opex 4701 . . . 4  |-  <. B ,  (/)
>.  e.  _V
83 s2fv0 12841 . . . 4  |-  ( <. B ,  (/) >.  e.  _V  ->  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
)
8482, 83ax-mp 5 . . 3  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
8578, 81, 843eqtr3g 2518 . 2  |-  ( ph  -> 
<. A ,  (/) >.  =  <. B ,  (/) >. )
86 opthg 4712 . . 3  |-  ( ( A  e.  I  /\  (/) 
e.  _V )  ->  ( <. A ,  (/) >.  =  <. B ,  (/) >.  <->  ( A  =  B  /\  (/)  =  (/) ) ) )
8786simprbda 621 . 2  |-  ( ( ( A  e.  I  /\  (/)  e.  _V )  /\  <. A ,  (/) >.  =  <. B ,  (/) >.
)  ->  A  =  B )
881, 3, 85, 87syl21anc 1225 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E!wreu 2806   E*wrmo 2807   {crab 2808   _Vcvv 3106    \ cdif 3458    i^i cin 3460   (/)c0 3783   {csn 4016   <.cop 4022   <.cotp 4024   U_ciun 4315   class class class wbr 4439    |-> cmpt 4497    _I cid 4779   Oncon0 4867    X. cxp 4986   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1oc1o 7115   2oc2o 7116    Er wer 7300   [cec 7301   /.cqs 7302   0cc0 9481   1c1 9482    - cmin 9796   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   splice csplice 12523   <"cs2 12797   Basecbs 14716   +g cplusg 14784    /.s cqus 14994  freeMndcfrmd 16214   Grpcgrp 16252   ~FG cefg 16923  freeGrpcfrgp 16924  varFGrpcvrgp 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-ec 7305  df-qs 7309  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-substr 12530  df-splice 12531  df-reverse 12532  df-s2 12804  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-0g 14931  df-imas 14997  df-qus 14998  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-frmd 16216  df-grp 16256  df-efg 16926  df-frgp 16927  df-vrgp 16928
This theorem is referenced by:  frgpnabl  17078
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