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Theorem fucco 13935
Description: Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucco.q  |-  Q  =  ( C FuncCat  D )
fucco.n  |-  N  =  ( C Nat  D )
fucco.a  |-  A  =  ( Base `  C
)
fucco.o  |-  .x.  =  (comp `  D )
fucco.x  |-  .xb  =  (comp `  Q )
fucco.f  |-  ( ph  ->  R  e.  ( F N G ) )
fucco.g  |-  ( ph  ->  S  e.  ( G N H ) )
Assertion
Ref Expression
fucco  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  =  ( x  e.  A  |->  ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( R `  x ) ) ) )
Distinct variable groups:    x, A    ph, x    x, R    x, S    x, C    x, D    x, 
.x.    x, F    x, G    x, H
Allowed substitution hints:    Q( x)    .xb ( x)    N( x)

Proof of Theorem fucco
Dummy variables  a 
b  f  g  h  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucco.q . . . 4  |-  Q  =  ( C FuncCat  D )
2 eqid 2358 . . . 4  |-  ( C 
Func  D )  =  ( C  Func  D )
3 fucco.n . . . 4  |-  N  =  ( C Nat  D )
4 fucco.a . . . 4  |-  A  =  ( Base `  C
)
5 fucco.o . . . 4  |-  .x.  =  (comp `  D )
6 fucco.f . . . . . . . 8  |-  ( ph  ->  R  e.  ( F N G ) )
73natrcl 13923 . . . . . . . 8  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
86, 7syl 15 . . . . . . 7  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
98simpld 445 . . . . . 6  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 funcrcl 13836 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
119, 10syl 15 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1211simpld 445 . . . 4  |-  ( ph  ->  C  e.  Cat )
1311simprd 449 . . . 4  |-  ( ph  ->  D  e.  Cat )
14 fucco.x . . . 4  |-  .xb  =  (comp `  Q )
151, 2, 3, 4, 5, 12, 13, 14fuccofval 13932 . . 3  |-  ( ph  -> 
.xb  =  ( v  e.  ( ( C 
Func  D )  X.  ( C  Func  D ) ) ,  h  e.  ( C  Func  D )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) ) )
16 fvex 5622 . . . . 5  |-  ( 1st `  v )  e.  _V
1716a1i 10 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  e. 
_V )
18 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  v  =  <. F ,  G >. )
1918fveq2d 5612 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  ( 1st `  <. F ,  G >. )
)
20 op1stg 6219 . . . . . . 7  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  <. F ,  G >. )  =  F )
218, 20syl 15 . . . . . 6  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
2221adantr 451 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  <. F ,  G >. )  =  F )
2319, 22eqtrd 2390 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  F )
24 fvex 5622 . . . . . 6  |-  ( 2nd `  v )  e.  _V
2524a1i 10 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  e.  _V )
2618adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  v  =  <. F ,  G >. )
2726fveq2d 5612 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  ( 2nd `  <. F ,  G >. ) )
28 op2ndg 6220 . . . . . . . 8  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 2nd `  <. F ,  G >. )  =  G )
298, 28syl 15 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
3029ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  <. F ,  G >. )  =  G )
3127, 30eqtrd 2390 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  G )
32 simpr 447 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  g  =  G )
33 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  h  =  H )
3433ad2antrr 706 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  h  =  H )
3532, 34oveq12d 5963 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( g N h )  =  ( G N H ) )
36 simplr 731 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  f  =  F )
3736, 32oveq12d 5963 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( f N g )  =  ( F N G ) )
38 eqidd 2359 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  A  =  A )
3936fveq2d 5612 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
4039fveq1d 5610 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  f ) `  x )  =  ( ( 1st `  F
) `  x )
)
4132fveq2d 5612 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  g )  =  ( 1st `  G ) )
4241fveq1d 5610 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  g ) `  x )  =  ( ( 1st `  G
) `  x )
)
4340, 42opeq12d 3885 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  <. ( ( 1st `  f ) `
 x ) ,  ( ( 1st `  g
) `  x ) >.  =  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
4434fveq2d 5612 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  h )  =  ( 1st `  H ) )
4544fveq1d 5610 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  h ) `  x )  =  ( ( 1st `  H
) `  x )
)
4643, 45oveq12d 5963 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( <. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) )  =  ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.  .x.  ( ( 1st `  H
) `  x )
) )
4746oveqd 5962 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( (
b `  x )
( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.  .x.  ( ( 1st `  h
) `  x )
) ( a `  x ) )  =  ( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) )
4838, 47mpteq12dv 4179 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( x  e.  A  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.  .x.  ( ( 1st `  h
) `  x )
) ( a `  x ) ) )  =  ( x  e.  A  |->  ( ( b `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) ) )
4935, 37, 48mpt2eq123dv 5997 . . . . 5  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) )  =  ( b  e.  ( G N H ) ,  a  e.  ( F N G ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) ) ) )
5025, 31, 49csbied2 3200 . . . 4  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) )  =  ( b  e.  ( G N H ) ,  a  e.  ( F N G ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) ) ) )
5117, 23, 50csbied2 3200 . . 3  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) )  =  ( b  e.  ( G N H ) ,  a  e.  ( F N G ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) ) ) )
528simprd 449 . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
53 opelxpi 4803 . . . 4  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  <. F ,  G >.  e.  ( ( C  Func  D )  X.  ( C  Func  D
) ) )
549, 52, 53syl2anc 642 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( ( C  Func  D )  X.  ( C 
Func  D ) ) )
55 fucco.g . . . . 5  |-  ( ph  ->  S  e.  ( G N H ) )
563natrcl 13923 . . . . 5  |-  ( S  e.  ( G N H )  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D
) ) )
5755, 56syl 15 . . . 4  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D ) ) )
5857simprd 449 . . 3  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
59 ovex 5970 . . . . 5  |-  ( G N H )  e. 
_V
60 ovex 5970 . . . . 5  |-  ( F N G )  e. 
_V
6159, 60mpt2ex 6285 . . . 4  |-  ( b  e.  ( G N H ) ,  a  e.  ( F N G )  |->  ( x  e.  A  |->  ( ( b `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.  .x.  ( ( 1st `  H
) `  x )
) ( a `  x ) ) ) )  e.  _V
6261a1i 10 . . 3  |-  ( ph  ->  ( b  e.  ( G N H ) ,  a  e.  ( F N G ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) ) )  e. 
_V )
6315, 51, 54, 58, 62ovmpt2d 6062 . 2  |-  ( ph  ->  ( <. F ,  G >. 
.xb  H )  =  ( b  e.  ( G N H ) ,  a  e.  ( F N G ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) ) ) )
64 eqidd 2359 . . 3  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  ->  A  =  A )
65 simprl 732 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
b  =  S )
6665fveq1d 5610 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( b `  x
)  =  ( S `
 x ) )
67 simprr 733 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
a  =  R )
6867fveq1d 5610 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( a `  x
)  =  ( R `
 x ) )
6966, 68oveq12d 5963 . . 3  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) )  =  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.  .x.  ( ( 1st `  H
) `  x )
) ( R `  x ) ) )
7064, 69mpteq12dv 4179 . 2  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( a `  x ) ) )  =  ( x  e.  A  |->  ( ( S `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( R `  x ) ) ) )
71 fvex 5622 . . . . 5  |-  ( Base `  C )  e.  _V
724, 71eqeltri 2428 . . . 4  |-  A  e. 
_V
7372mptex 5832 . . 3  |-  ( x  e.  A  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.  .x.  ( ( 1st `  H
) `  x )
) ( R `  x ) ) )  e.  _V
7473a1i 10 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( S `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( R `  x ) ) )  e.  _V )
7563, 70, 55, 6, 74ovmpt2d 6062 1  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  =  ( x  e.  A  |->  ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( R `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   [_csb 3157   <.cop 3719    e. cmpt 4158    X. cxp 4769   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   1stc1st 6207   2ndc2nd 6208   Basecbs 13245  compcco 13317   Catccat 13665    Func cfunc 13827   Nat cnat 13914   FuncCat cfuc 13915
This theorem is referenced by:  fuccoval  13936  fuccocl  13937  fuclid  13939  fucrid  13940  fucass  13941  fucsect  13945  curfcl  14105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-hom 13329  df-cco 13330  df-func 13831  df-nat 13916  df-fuc 13917
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