MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fucco Structured version   Unicode version

Theorem fucco 15206
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 2467 . . . 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 15194 . . . . . . . 8  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
86, 7syl 16 . . . . . . 7  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
98simpld 459 . . . . . 6  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 funcrcl 15107 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
119, 10syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1211simpld 459 . . . 4  |-  ( ph  ->  C  e.  Cat )
1311simprd 463 . . . 4  |-  ( ph  ->  D  e.  Cat )
14 fucco.x . . . 4  |-  .xb  =  (comp `  Q )
151, 2, 3, 4, 5, 12, 13, 14fuccofval 15203 . . 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 5882 . . . . 5  |-  ( 1st `  v )  e.  _V
1716a1i 11 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  e. 
_V )
18 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  v  =  <. F ,  G >. )
1918fveq2d 5876 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  ( 1st `  <. F ,  G >. )
)
20 op1stg 6807 . . . . . . 7  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  <. F ,  G >. )  =  F )
218, 20syl 16 . . . . . 6  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
2221adantr 465 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  <. F ,  G >. )  =  F )
2319, 22eqtrd 2508 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  F )
24 fvex 5882 . . . . . 6  |-  ( 2nd `  v )  e.  _V
2524a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  e.  _V )
2618adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  v  =  <. F ,  G >. )
2726fveq2d 5876 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  ( 2nd `  <. F ,  G >. ) )
28 op2ndg 6808 . . . . . . . 8  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 2nd `  <. F ,  G >. )  =  G )
298, 28syl 16 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
3029ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  <. F ,  G >. )  =  G )
3127, 30eqtrd 2508 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  G )
32 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  g  =  G )
33 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  h  =  H )
3433ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  h  =  H )
3532, 34oveq12d 6313 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( g N h )  =  ( G N H ) )
36 simplr 754 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  f  =  F )
3736, 32oveq12d 6313 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( f N g )  =  ( F N G ) )
3836fveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
3938fveq1d 5874 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  f ) `  x )  =  ( ( 1st `  F
) `  x )
)
4032fveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  g )  =  ( 1st `  G ) )
4140fveq1d 5874 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  g ) `  x )  =  ( ( 1st `  G
) `  x )
)
4239, 41opeq12d 4227 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  <. ( ( 1st `  f ) `
 x ) ,  ( ( 1st `  g
) `  x ) >.  =  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
4334fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  h )  =  ( 1st `  H ) )
4443fveq1d 5874 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  h ) `  x )  =  ( ( 1st `  H
) `  x )
)
4542, 44oveq12d 6313 . . . . . . . 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 )
) )
4645oveqd 6312 . . . . . . 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 ) ) )
4746mpteq2dv 4540 . . . . . 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 ) ) ) )
4835, 37, 47mpt2eq123dv 6354 . . . . 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 ) ) ) ) )
4925, 31, 48csbied2 3468 . . . 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 ) ) ) ) )
5017, 23, 49csbied2 3468 . . 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 ) ) ) ) )
51 opelxpi 5037 . . . 4  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  <. F ,  G >.  e.  ( ( C  Func  D )  X.  ( C  Func  D
) ) )
528, 51syl 16 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( ( C  Func  D )  X.  ( C 
Func  D ) ) )
53 fucco.g . . . . 5  |-  ( ph  ->  S  e.  ( G N H ) )
543natrcl 15194 . . . . 5  |-  ( S  e.  ( G N H )  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D
) ) )
5553, 54syl 16 . . . 4  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D ) ) )
5655simprd 463 . . 3  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
57 ovex 6320 . . . . 5  |-  ( G N H )  e. 
_V
58 ovex 6320 . . . . 5  |-  ( F N G )  e. 
_V
5957, 58mpt2ex 6872 . . . 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
6059a1i 11 . . 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 )
6115, 50, 52, 56, 60ovmpt2d 6425 . 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 ) ) ) ) )
62 simprl 755 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
b  =  S )
6362fveq1d 5874 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( b `  x
)  =  ( S `
 x ) )
64 simprr 756 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
a  =  R )
6564fveq1d 5874 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( a `  x
)  =  ( R `
 x ) )
6663, 65oveq12d 6313 . . 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 ) ) )
6766mpteq2dv 4540 . 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 ) ) ) )
68 fvex 5882 . . . . 5  |-  ( Base `  C )  e.  _V
694, 68eqeltri 2551 . . . 4  |-  A  e. 
_V
7069mptex 6142 . . 3  |-  ( x  e.  A  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.  .x.  ( ( 1st `  H
) `  x )
) ( R `  x ) ) )  e.  _V
7170a1i 11 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( S `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( R `  x ) ) )  e.  _V )
7261, 67, 53, 6, 71ovmpt2d 6425 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   [_csb 3440   <.cop 4039    |-> cmpt 4511    X. cxp 5003   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   Basecbs 14507  compcco 14584   Catccat 14936    Func cfunc 15098   Nat cnat 15185   FuncCat cfuc 15186
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-hom 14596  df-cco 14597  df-func 15102  df-nat 15187  df-fuc 15188
This theorem is referenced by:  fuccoval  15207  fuccocl  15208  fuclid  15210  fucrid  15211  fucass  15212  fucsect  15216  curfcl  15376
  Copyright terms: Public domain W3C validator