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Theorem fucco 14190
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 2442 . . . 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 14178 . . . . . . . 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 447 . . . . . 6  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 funcrcl 14091 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
119, 10syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1211simpld 447 . . . 4  |-  ( ph  ->  C  e.  Cat )
1311simprd 451 . . . 4  |-  ( ph  ->  D  e.  Cat )
14 fucco.x . . . 4  |-  .xb  =  (comp `  Q )
151, 2, 3, 4, 5, 12, 13, 14fuccofval 14187 . . 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 5771 . . . . 5  |-  ( 1st `  v )  e.  _V
1716a1i 11 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  e. 
_V )
18 simprl 734 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  v  =  <. F ,  G >. )
1918fveq2d 5761 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  ( 1st `  <. F ,  G >. )
)
20 op1stg 6388 . . . . . . 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 453 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  <. F ,  G >. )  =  F )
2319, 22eqtrd 2474 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  F )
24 fvex 5771 . . . . . 6  |-  ( 2nd `  v )  e.  _V
2524a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  e.  _V )
2618adantr 453 . . . . . . 7  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  v  =  <. F ,  G >. )
2726fveq2d 5761 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  ( 2nd `  <. F ,  G >. ) )
28 op2ndg 6389 . . . . . . . 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 708 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  <. F ,  G >. )  =  G )
3127, 30eqtrd 2474 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  G )
32 simpr 449 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  g  =  G )
33 simprr 735 . . . . . . . 8  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  h  =  H )
3433ad2antrr 708 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  h  =  H )
3532, 34oveq12d 6128 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( g N h )  =  ( G N H ) )
36 simplr 733 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  f  =  F )
3736, 32oveq12d 6128 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( f N g )  =  ( F N G ) )
3836fveq2d 5761 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
3938fveq1d 5759 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  f ) `  x )  =  ( ( 1st `  F
) `  x )
)
4032fveq2d 5761 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  g )  =  ( 1st `  G ) )
4140fveq1d 5759 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  g ) `  x )  =  ( ( 1st `  G
) `  x )
)
4239, 41opeq12d 4016 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  <. ( ( 1st `  f ) `
 x ) ,  ( ( 1st `  g
) `  x ) >.  =  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
4334fveq2d 5761 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  h )  =  ( 1st `  H ) )
4443fveq1d 5759 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  h ) `  x )  =  ( ( 1st `  H
) `  x )
)
4542, 44oveq12d 6128 . . . . . . . 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 6127 . . . . . . 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 4321 . . . . . 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 6165 . . . . 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 3293 . . . 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 3293 . . 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 4939 . . . 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 14178 . . . . 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 451 . . 3  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
57 ovex 6135 . . . . 5  |-  ( G N H )  e. 
_V
58 ovex 6135 . . . . 5  |-  ( F N G )  e. 
_V
5957, 58mpt2ex 6454 . . . 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 6230 . 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 734 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
b  =  S )
6362fveq1d 5759 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( b `  x
)  =  ( S `
 x ) )
64 simprr 735 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
a  =  R )
6564fveq1d 5759 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( a `  x
)  =  ( R `
 x ) )
6663, 65oveq12d 6128 . . 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 4321 . 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 5771 . . . . 5  |-  ( Base `  C )  e.  _V
694, 68eqeltri 2512 . . . 4  |-  A  e. 
_V
7069mptex 5995 . . 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 6230 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 360    = wceq 1653    e. wcel 1727   _Vcvv 2962   [_csb 3267   <.cop 3841    e. cmpt 4291    X. cxp 4905   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112   1stc1st 6376   2ndc2nd 6377   Basecbs 13500  compcco 13572   Catccat 13920    Func cfunc 14082   Nat cnat 14169   FuncCat cfuc 14170
This theorem is referenced by:  fuccoval  14191  fuccocl  14192  fuclid  14194  fucrid  14195  fucass  14196  fucsect  14200  curfcl  14360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-hom 13584  df-cco 13585  df-func 14086  df-nat 14171  df-fuc 14172
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