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Theorem fin23lem27 8699
Description: The mapping constructed in fin23lem22 8698 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
fin23lem22.b  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  i )
)
Assertion
Ref Expression
fin23lem27  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
Distinct variable group:    i, j, S
Allowed substitution hints:    C( i, j)

Proof of Theorem fin23lem27
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordom 6682 . . . 4  |-  Ord  om
2 ordwe 4880 . . . 4  |-  ( Ord 
om  ->  _E  We  om )
3 weso 4859 . . . 4  |-  (  _E  We  om  ->  _E  Or  om )
41, 2, 3mp2b 10 . . 3  |-  _E  Or  om
54a1i 11 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  _E  Or  om )
6 sopo 4806 . . . . 5  |-  (  _E  Or  om  ->  _E  Po  om )
74, 6ax-mp 5 . . . 4  |-  _E  Po  om
8 poss 4791 . . . 4  |-  ( S 
C_  om  ->  (  _E  Po  om  ->  _E  Po  S ) )
97, 8mpi 17 . . 3  |-  ( S 
C_  om  ->  _E  Po  S )
109adantr 463 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  _E  Po  S )
11 fin23lem22.b . . . 4  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  i )
)
1211fin23lem22 8698 . . 3  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
13 f1ofo 5805 . . 3  |-  ( C : om -1-1-onto-> S  ->  C : om -onto-> S )
1412, 13syl 16 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -onto-> S
)
15 nnsdomel 8362 . . . . . . . 8  |-  ( ( a  e.  om  /\  b  e.  om )  ->  ( a  e.  b  <-> 
a  ~<  b ) )
1615adantl 464 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  <->  a  ~<  b ) )
1716biimpd 207 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
a  ~<  b ) )
18 fin23lem23 8697 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  a
)
1918adantrr 714 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  a )
20 ineq1 3679 . . . . . . . . . . . . . 14  |-  ( j  =  i  ->  (
j  i^i  S )  =  ( i  i^i 
S ) )
2120breq1d 4449 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  a  <->  ( i  i^i  S )  ~~  a
) )
2221cbvreuv 3083 . . . . . . . . . . . 12  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  a  <->  E! i  e.  S  ( i  i^i  S )  ~~  a
)
2319, 22sylib 196 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  a )
24 nfv 1712 . . . . . . . . . . . 12  |-  F/ i ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a
2521cbvriotav 6243 . . . . . . . . . . . 12  |-  ( iota_ j  e.  S  ( j  i^i  S )  ~~  a )  =  (
iota_ i  e.  S  ( i  i^i  S
)  ~~  a )
26 ineq1 3679 . . . . . . . . . . . . 13  |-  ( i  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  ->  ( i  i^i  S )  =  ( ( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  i^i  S ) )
2726breq1d 4449 . . . . . . . . . . . 12  |-  ( i  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  ->  ( (
i  i^i  S )  ~~  a  <->  ( ( iota_ j  e.  S  ( j  i^i  S )  ~~  a )  i^i  S
)  ~~  a )
)
2824, 25, 27riotaprop 6255 . . . . . . . . . . 11  |-  ( E! i  e.  S  ( i  i^i  S ) 
~~  a  ->  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  e.  S  /\  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
) )
2923, 28syl 16 . . . . . . . . . 10  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  e.  S  /\  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
) )
3029simprd 461 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
)
3130adantrr 714 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a )
32 simprr 755 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  b )
33 fin23lem23 8697 . . . . . . . . . . . . . . 15  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  b  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  b
)
3433adantrl 713 . . . . . . . . . . . . . 14  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  b )
3520breq1d 4449 . . . . . . . . . . . . . . 15  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  b  <->  ( i  i^i  S )  ~~  b
) )
3635cbvreuv 3083 . . . . . . . . . . . . . 14  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  b  <->  E! i  e.  S  ( i  i^i  S )  ~~  b
)
3734, 36sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  b )
38 nfv 1712 . . . . . . . . . . . . . 14  |-  F/ i ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )  ~~  b
3935cbvriotav 6243 . . . . . . . . . . . . . 14  |-  ( iota_ j  e.  S  ( j  i^i  S )  ~~  b )  =  (
iota_ i  e.  S  ( i  i^i  S
)  ~~  b )
40 ineq1 3679 . . . . . . . . . . . . . . 15  |-  ( i  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  ->  ( i  i^i  S )  =  ( ( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  i^i  S ) )
4140breq1d 4449 . . . . . . . . . . . . . 14  |-  ( i  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  ->  ( (
i  i^i  S )  ~~  b  <->  ( ( iota_ j  e.  S  ( j  i^i  S )  ~~  b )  i^i  S
)  ~~  b )
)
4238, 39, 41riotaprop 6255 . . . . . . . . . . . . 13  |-  ( E! i  e.  S  ( i  i^i  S ) 
~~  b  ->  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  e.  S  /\  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
) )
4337, 42syl 16 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  e.  S  /\  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
) )
4443simprd 461 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
)
4544ensymd 7559 . . . . . . . . . 10  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  ~~  ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )
)
4645adantrr 714 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
b  ~~  ( ( iota_ j  e.  S  ( j  i^i  S ) 
~~  b )  i^i 
S ) )
47 sdomentr 7644 . . . . . . . . 9  |-  ( ( a  ~<  b  /\  b  ~~  ( ( iota_ j  e.  S  ( j  i^i  S )  ~~  b )  i^i  S
) )  ->  a  ~<  ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )
)
4832, 46, 47syl2anc 659 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  ( ( iota_ j  e.  S  ( j  i^i  S )  ~~  b )  i^i  S
) )
49 ensdomtr 7646 . . . . . . . 8  |-  ( ( ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a  /\  a  ~<  ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )
)  ->  ( ( iota_ j  e.  S  ( j  i^i  S ) 
~~  a )  i^i 
S )  ~<  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  i^i  S ) )
5031, 48, 49syl2anc 659 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )
)
5150expr 613 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  ~<  b  ->  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  i^i  S )  ~<  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  i^i  S ) ) )
52 simpll 751 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_ 
om )
53 omsson 6677 . . . . . . . . 9  |-  om  C_  On
5452, 53syl6ss 3501 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_  On )
5529simpld 457 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  a )  e.  S )
5654, 55sseldd 3490 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  a )  e.  On )
5743simpld 457 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  b )  e.  S )
5854, 57sseldd 3490 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  b )  e.  On )
59 onsdominel 7659 . . . . . . . 8  |-  ( ( ( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  e.  On  /\  ( iota_ j  e.  S  ( j  i^i  S )  ~~  b )  e.  On  /\  ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )
)  ->  ( iota_ j  e.  S  ( j  i^i  S )  ~~  a )  e.  (
iota_ j  e.  S  ( j  i^i  S
)  ~~  b )
)
60593expia 1196 . . . . . . 7  |-  ( ( ( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  e.  On  /\  ( iota_ j  e.  S  ( j  i^i  S )  ~~  b )  e.  On )  ->  ( ( (
iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  i^i  S )  ~<  (
( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  i^i  S )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  a )  e.  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )
) )
6156, 58, 60syl2anc 659 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )  ->  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  e.  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )
) )
6217, 51, 613syld 55 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  e.  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )
) )
63 simprl 754 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  a  e.  om )
64 breq2 4443 . . . . . . . . 9  |-  ( i  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  a
) )
6564riotabidv 6234 . . . . . . . 8  |-  ( i  =  a  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )
)
6665, 11fvmptg 5929 . . . . . . 7  |-  ( ( a  e.  om  /\  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )  e.  S )  ->  ( C `  a )  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
) )
6763, 55, 66syl2anc 659 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  a )  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
) )
68 simprr 755 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  e.  om )
69 breq2 4443 . . . . . . . . 9  |-  ( i  =  b  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  b
) )
7069riotabidv 6234 . . . . . . . 8  |-  ( i  =  b  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )
)
7170, 11fvmptg 5929 . . . . . . 7  |-  ( ( b  e.  om  /\  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )  e.  S )  ->  ( C `  b )  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
) )
7268, 57, 71syl2anc 659 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  b )  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
) )
7367, 72eleq12d 2536 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( C `  a
)  e.  ( C `
 b )  <->  ( iota_ j  e.  S  ( j  i^i  S )  ~~  a )  e.  (
iota_ j  e.  S  ( j  i^i  S
)  ~~  b )
) )
7462, 73sylibrd 234 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( C `  a
)  e.  ( C `
 b ) ) )
75 epel 4783 . . . 4  |-  ( a  _E  b  <->  a  e.  b )
76 fvex 5858 . . . . 5  |-  ( C `
 b )  e. 
_V
7776epelc 4782 . . . 4  |-  ( ( C `  a )  _E  ( C `  b )  <->  ( C `  a )  e.  ( C `  b ) )
7874, 75, 773imtr4g 270 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
7978ralrimivva 2875 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  A. a  e.  om  A. b  e.  om  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
80 soisoi 6199 . 2  |-  ( ( (  _E  Or  om  /\  _E  Po  S )  /\  ( C : om -onto-> S  /\  A. a  e.  om  A. b  e. 
om  ( a  _E  b  ->  ( C `  a )  _E  ( C `  b )
) ) )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
815, 10, 14, 79, 80syl22anc 1227 1  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E!wreu 2806    i^i cin 3460    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497    _E cep 4778    Po wpo 4787    Or wor 4788    We wwe 4826   Ord word 4866   Oncon0 4867   -onto->wfo 5568   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571   iota_crio 6231   omcom 6673    ~~ cen 7506    ~< csdm 7508   Fincfn 7509
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
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-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-uni 4236  df-int 4272  df-iun 4317  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-se 4828  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-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311
This theorem is referenced by: (None)
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