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Theorem frgpnabllem2 16681
Description: Lemma for frgpnabl 16682. (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 4577 . . 3  |-  (/)  e.  _V
32a1i 11 . 2  |-  ( ph  -> 
(/)  e.  _V )
4 frgpnabl.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
5 difss 3631 . . . . . . . 8  |-  ( W 
\  U_ x  e.  W  ran  ( T `  x
) )  C_  W
64, 5eqsstri 3534 . . . . . . 7  |-  D  C_  W
7 inss1 3718 . . . . . . . 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 16680 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) ) )
187, 17sseldi 3502 . . . . . . 7  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  D )
196, 18sseldi 3502 . . . . . 6  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  W )
20 eqid 2467 . . . . . . 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 16576 . . . . . 6  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  W  ->  E! d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
22 reurmo 3079 . . . . . 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 3718 . . . . . 6  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  D
258, 9, 10, 11, 12, 13, 4, 14, 15, 1, 16frgpnabllem1 16680 . . . . . 6  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) ) )
2624, 25sseldi 3502 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D )
279, 10efger 16542 . . . . . . . . 9  |-  .~  Er  W
2827a1i 11 . . . . . . . 8  |-  ( ph  ->  .~  Er  W )
298frgpgrp 16586 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  e.  Grp )
3015, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
31 eqid 2467 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3210, 14, 8, 31vrgpf 16592 . . . . . . . . . . . 12  |-  ( I  e.  _V  ->  U : I --> ( Base `  G ) )
3315, 32syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U : I --> ( Base `  G ) )
3433, 1ffvelrnd 6022 . . . . . . . . . 10  |-  ( ph  ->  ( U `  A
)  e.  ( Base `  G ) )
3533, 16ffvelrnd 6022 . . . . . . . . . 10  |-  ( ph  ->  ( U `  B
)  e.  ( Base `  G ) )
3631, 11grpcl 15873 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( Base `  G ) )
38 eqid 2467 . . . . . . . . . . . 12  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
398, 38, 10frgpval 16582 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
4015, 39syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
41 2on 7138 . . . . . . . . . . . . . 14  |-  2o  e.  On
42 xpexg 6586 . . . . . . . . . . . . . 14  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
4315, 41, 42sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
44 wrdexg 12523 . . . . . . . . . . . . 13  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
45 fvi 5924 . . . . . . . . . . . . 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 2520 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
48 eqid 2467 . . . . . . . . . . . . 13  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
4938, 48frmdbas 15852 . . . . . . . . . . . 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 2511 . . . . . . . . . 10  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
52 fvex 5876 . . . . . . . . . . . 12  |-  ( ~FG  `  I
)  e.  _V
5310, 52eqeltri 2551 . . . . . . . . . . 11  |-  .~  e.  _V
5453a1i 11 . . . . . . . . . 10  |-  ( ph  ->  .~  e.  _V )
55 fvex 5876 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5655a1i 11 . . . . . . . . . 10  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
5740, 51, 54, 56divsbas 14800 . . . . . . . . 9  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
5837, 57eleqtrrd 2558 . . . . . . . 8  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  ) )
59 inss2 3719 . . . . . . . . 9  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  (
( U `  A
)  .+  ( U `  B ) )
6059, 25sseldi 3502 . . . . . . . 8  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
61 qsel 7390 . . . . . . . 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 1228 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. A ,  (/) >. <. B ,  (/)
>. "> ]  .~  )
63 inss2 3719 . . . . . . . . . 10  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  (
( U `  B
)  .+  ( U `  A ) )
6463, 17sseldi 3502 . . . . . . . . 9  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  B ) 
.+  ( U `  A ) ) )
65 frgpnabl.n . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
6664, 65eleqtrrd 2558 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
67 qsel 7390 . . . . . . . 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 1228 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. B ,  (/) >. <. A ,  (/)
>. "> ]  .~  )
6962, 68eqtr3d 2510 . . . . . 6  |-  ( ph  ->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
706, 26sseldi 3502 . . . . . . 7  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  W )
7128, 70erth 7356 . . . . . 6  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  <->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
)
7269, 71mpbird 232 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7328, 19erref 7331 . . . . 5  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
74 breq1 4450 . . . . . 6  |-  ( d  =  <" <. A ,  (/)
>. <. B ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
75 breq1 4450 . . . . . 6  |-  ( d  =  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
7674, 75rmoi 3432 . . . . 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 1237 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7877fveq1d 5868 . . 3  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )
)
79 opex 4711 . . . 4  |-  <. A ,  (/)
>.  e.  _V
80 s2fv0 12813 . . . 4  |-  ( <. A ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
)
8179, 80ax-mp 5 . . 3  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
82 opex 4711 . . . 4  |-  <. B ,  (/)
>.  e.  _V
83 s2fv0 12813 . . . 4  |-  ( <. B ,  (/) >.  e.  _V  ->  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
)
8482, 83ax-mp 5 . . 3  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
8578, 81, 843eqtr3g 2531 . 2  |-  ( ph  -> 
<. A ,  (/) >.  =  <. B ,  (/) >. )
86 opthg 4722 . . 3  |-  ( ( A  e.  I  /\  (/) 
e.  _V )  ->  ( <. A ,  (/) >.  =  <. B ,  (/) >.  <->  ( A  =  B  /\  (/)  =  (/) ) ) )
8786simprbda 623 . 2  |-  ( ( ( A  e.  I  /\  (/)  e.  _V )  /\  <. A ,  (/) >.  =  <. B ,  (/) >.
)  ->  A  =  B )
881, 3, 85, 87syl21anc 1227 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E!wreu 2816   E*wrmo 2817   {crab 2818   _Vcvv 3113    \ cdif 3473    i^i cin 3475   (/)c0 3785   {csn 4027   <.cop 4033   <.cotp 4035   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    _I cid 4790   Oncon0 4878    X. cxp 4997   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1oc1o 7123   2oc2o 7124    Er wer 7308   [cec 7309   /.cqs 7310   0cc0 9492   1c1 9493    - cmin 9805   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   splice csplice 12505   <"cs2 12769   Basecbs 14490   +g cplusg 14555    /.s cqus 14760   Grpcgrp 15727  freeMndcfrmd 15847   ~FG cefg 16530  freeGrpcfrgp 16531  varFGrpcvrgp 16532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512  df-splice 12513  df-reverse 12514  df-s2 12776  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-0g 14697  df-imas 14763  df-divs 14764  df-mnd 15732  df-frmd 15849  df-grp 15867  df-efg 16533  df-frgp 16534  df-vrgp 16535
This theorem is referenced by:  frgpnabl  16682
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