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Theorem fucco 15818
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 2429 . . . 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 15806 . . . . . . . 8  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
86, 7syl 17 . . . . . . 7  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
98simpld 460 . . . . . 6  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 funcrcl 15719 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
119, 10syl 17 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1211simpld 460 . . . 4  |-  ( ph  ->  C  e.  Cat )
1311simprd 464 . . . 4  |-  ( ph  ->  D  e.  Cat )
14 fucco.x . . . 4  |-  .xb  =  (comp `  Q )
151, 2, 3, 4, 5, 12, 13, 14fuccofval 15815 . . 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 5891 . . . . 5  |-  ( 1st `  v )  e.  _V
1716a1i 11 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  e. 
_V )
18 simprl 762 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  v  =  <. F ,  G >. )
1918fveq2d 5885 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  ( 1st `  <. F ,  G >. )
)
20 op1stg 6819 . . . . . . 7  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  <. F ,  G >. )  =  F )
218, 20syl 17 . . . . . 6  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
2221adantr 466 . . . . 5  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  <. F ,  G >. )  =  F )
2319, 22eqtrd 2470 . . . 4  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  ( 1st `  v )  =  F )
24 fvex 5891 . . . . . 6  |-  ( 2nd `  v )  e.  _V
2524a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  e.  _V )
2618adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  v  =  <. F ,  G >. )
2726fveq2d 5885 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  ( 2nd `  <. F ,  G >. ) )
28 op2ndg 6820 . . . . . . . 8  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 2nd `  <. F ,  G >. )  =  G )
298, 28syl 17 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
3029ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  <. F ,  G >. )  =  G )
3127, 30eqtrd 2470 . . . . 5  |-  ( ( ( ph  /\  (
v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  ->  ( 2nd `  v
)  =  G )
32 simpr 462 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  g  =  G )
33 simprr 764 . . . . . . . 8  |-  ( (
ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  ->  h  =  H )
3433ad2antrr 730 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  h  =  H )
3532, 34oveq12d 6323 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( g N h )  =  ( G N H ) )
36 simplr 760 . . . . . . 7  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  f  =  F )
3736, 32oveq12d 6323 . . . . . 6  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( f N g )  =  ( F N G ) )
3836fveq2d 5885 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
3938fveq1d 5883 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  f ) `  x )  =  ( ( 1st `  F
) `  x )
)
4032fveq2d 5885 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  g )  =  ( 1st `  G ) )
4140fveq1d 5883 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  g ) `  x )  =  ( ( 1st `  G
) `  x )
)
4239, 41opeq12d 4198 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  <. ( ( 1st `  f ) `
 x ) ,  ( ( 1st `  g
) `  x ) >.  =  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
4334fveq2d 5885 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( 1st `  h )  =  ( 1st `  H ) )
4443fveq1d 5883 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  =  <. F ,  G >.  /\  h  =  H ) )  /\  f  =  F )  /\  g  =  G
)  ->  ( ( 1st `  h ) `  x )  =  ( ( 1st `  H
) `  x )
)
4542, 44oveq12d 6323 . . . . . . . 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 6322 . . . . . . 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 4513 . . . . . 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 6367 . . . . 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 3429 . . . 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 3429 . . 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 4886 . . . 4  |-  ( ( F  e.  ( C 
Func  D )  /\  G  e.  ( C  Func  D
) )  ->  <. F ,  G >.  e.  ( ( C  Func  D )  X.  ( C  Func  D
) ) )
528, 51syl 17 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( ( C  Func  D )  X.  ( C 
Func  D ) ) )
53 fucco.g . . . . 5  |-  ( ph  ->  S  e.  ( G N H ) )
543natrcl 15806 . . . . 5  |-  ( S  e.  ( G N H )  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D
) ) )
5553, 54syl 17 . . . 4  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D ) ) )
5655simprd 464 . . 3  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
57 ovex 6333 . . . . 5  |-  ( G N H )  e. 
_V
58 ovex 6333 . . . . 5  |-  ( F N G )  e. 
_V
5957, 58mpt2ex 6884 . . . 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 6438 . 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 762 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
b  =  S )
6362fveq1d 5883 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( b `  x
)  =  ( S `
 x ) )
64 simprr 764 . . . . 5  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
a  =  R )
6564fveq1d 5883 . . . 4  |-  ( (
ph  /\  ( b  =  S  /\  a  =  R ) )  -> 
( a `  x
)  =  ( R `
 x ) )
6663, 65oveq12d 6323 . . 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 4513 . 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 5891 . . . . 5  |-  ( Base `  C )  e.  _V
694, 68eqeltri 2513 . . . 4  |-  A  e. 
_V
7069mptex 6151 . . 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 6438 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 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   [_csb 3401   <.cop 4008    |-> cmpt 4484    X. cxp 4852   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   Basecbs 15084  compcco 15164   Catccat 15521    Func cfunc 15710   Nat cnat 15797   FuncCat cfuc 15798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-hom 15176  df-cco 15177  df-func 15714  df-nat 15799  df-fuc 15800
This theorem is referenced by:  fuccoval  15819  fuccocl  15820  fuclid  15822  fucrid  15823  fucass  15824  fucsect  15828  curfcl  16068
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