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Theorem funcco 14068
Description: A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcco.b  |-  B  =  ( Base `  D
)
funcco.h  |-  H  =  (  Hom  `  D
)
funcco.o  |-  .x.  =  (comp `  D )
funcco.O  |-  O  =  (comp `  E )
funcco.f  |-  ( ph  ->  F ( D  Func  E ) G )
funcco.x  |-  ( ph  ->  X  e.  B )
funcco.y  |-  ( ph  ->  Y  e.  B )
funcco.z  |-  ( ph  ->  Z  e.  B )
funcco.m  |-  ( ph  ->  M  e.  ( X H Y ) )
funcco.n  |-  ( ph  ->  N  e.  ( Y H Z ) )
Assertion
Ref Expression
funcco  |-  ( ph  ->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) )

Proof of Theorem funcco
Dummy variables  m  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcco.f . . . 4  |-  ( ph  ->  F ( D  Func  E ) G )
2 funcco.b . . . . 5  |-  B  =  ( Base `  D
)
3 eqid 2436 . . . . 5  |-  ( Base `  E )  =  (
Base `  E )
4 funcco.h . . . . 5  |-  H  =  (  Hom  `  D
)
5 eqid 2436 . . . . 5  |-  (  Hom  `  E )  =  (  Hom  `  E )
6 eqid 2436 . . . . 5  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2436 . . . . 5  |-  ( Id
`  E )  =  ( Id `  E
)
8 funcco.o . . . . 5  |-  .x.  =  (comp `  D )
9 funcco.O . . . . 5  |-  O  =  (comp `  E )
10 df-br 4213 . . . . . . . 8  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
111, 10sylib 189 . . . . . . 7  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
12 funcrcl 14060 . . . . . . 7  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 16 . . . . . 6  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1413simpld 446 . . . . 5  |-  ( ph  ->  D  e.  Cat )
1513simprd 450 . . . . 5  |-  ( ph  ->  E  e.  Cat )
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 14061 . . . 4  |-  ( ph  ->  ( F ( D 
Func  E ) G  <->  ( F : B --> ( Base `  E
)  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z ) ) (  Hom  `  E
) ( F `  ( 2nd `  z ) ) )  ^m  ( H `  z )
)  /\  A. x  e.  B  ( (
( x G x ) `  ( ( Id `  D ) `
 x ) )  =  ( ( Id
`  E ) `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <.
x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. O ( F `  z ) ) ( ( x G y ) `  m ) ) ) ) ) )
171, 16mpbid 202 . . 3  |-  ( ph  ->  ( F : B --> ( Base `  E )  /\  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) (  Hom  `  E )
( F `  ( 2nd `  z ) ) )  ^m  ( H `
 z ) )  /\  A. x  e.  B  ( ( ( x G x ) `
 ( ( Id
`  D ) `  x ) )  =  ( ( Id `  E ) `  ( F `  x )
)  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <.
x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. O ( F `  z ) ) ( ( x G y ) `  m ) ) ) ) )
1817simp3d 971 . 2  |-  ( ph  ->  A. x  e.  B  ( ( ( x G x ) `  ( ( Id `  D ) `  x
) )  =  ( ( Id `  E
) `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <. x ,  y
>.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n
) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) ) )
19 funcco.x . . 3  |-  ( ph  ->  X  e.  B )
20 funcco.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
2120adantr 452 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
22 funcco.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
2322ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  Z  e.  B )
24 funcco.m . . . . . . . . 9  |-  ( ph  ->  M  e.  ( X H Y ) )
2524ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  M  e.  ( X H Y ) )
26 simpllr 736 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  x  =  X )
27 simplr 732 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  y  =  Y )
2826, 27oveq12d 6099 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  (
x H y )  =  ( X H Y ) )
2925, 28eleqtrrd 2513 . . . . . . 7  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  M  e.  ( x H y ) )
30 funcco.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  ( Y H Z ) )
3130ad4antr 713 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  N  e.  ( Y H Z ) )
32 simpllr 736 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  y  =  Y )
33 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  z  =  Z )
3432, 33oveq12d 6099 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  (
y H z )  =  ( Y H Z ) )
3531, 34eleqtrrd 2513 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  N  e.  ( y H z ) )
36 simp-5r 746 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  x  =  X )
37 simpllr 736 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  z  =  Z )
3836, 37oveq12d 6099 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
x G z )  =  ( X G Z ) )
39 simp-4r 744 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  y  =  Y )
4036, 39opeq12d 3992 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  <. x ,  y >.  =  <. X ,  Y >. )
4140, 37oveq12d 6099 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( <. x ,  y >.  .x.  z )  =  (
<. X ,  Y >.  .x. 
Z ) )
42 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  n  =  N )
43 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  m  =  M )
4441, 42, 43oveq123d 6102 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
n ( <. x ,  y >.  .x.  z
) m )  =  ( N ( <. X ,  Y >.  .x. 
Z ) M ) )
4538, 44fveq12d 5734 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
( x G z ) `  ( n ( <. x ,  y
>.  .x.  z ) m ) )  =  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) ) )
4636fveq2d 5732 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( F `  x )  =  ( F `  X ) )
4739fveq2d 5732 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( F `  y )  =  ( F `  Y ) )
4846, 47opeq12d 3992 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  <. ( F `  x ) ,  ( F `  y ) >.  =  <. ( F `  X ) ,  ( F `  Y ) >. )
4937fveq2d 5732 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( F `  z )  =  ( F `  Z ) )
5048, 49oveq12d 6099 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( <. ( F `  x
) ,  ( F `
 y ) >. O ( F `  z ) )  =  ( <. ( F `  X ) ,  ( F `  Y )
>. O ( F `  Z ) ) )
5139, 37oveq12d 6099 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
y G z )  =  ( Y G Z ) )
5251, 42fveq12d 5734 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
( y G z ) `  n )  =  ( ( Y G Z ) `  N ) )
5336, 39oveq12d 6099 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
x G y )  =  ( X G Y ) )
5453, 43fveq12d 5734 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
( x G y ) `  m )  =  ( ( X G Y ) `  M ) )
5550, 52, 54oveq123d 6102 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
( ( y G z ) `  n
) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) )
5645, 55eqeq12d 2450 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
( ( x G z ) `  (
n ( <. x ,  y >.  .x.  z
) m ) )  =  ( ( ( y G z ) `
 n ) (
<. ( F `  x
) ,  ( F `
 y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) )  <->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) ) )
5735, 56rspcdv 3055 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  ( A. n  e.  (
y H z ) ( ( x G z ) `  (
n ( <. x ,  y >.  .x.  z
) m ) )  =  ( ( ( y G z ) `
 n ) (
<. ( F `  x
) ,  ( F `
 y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) )  ->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) ) )
5829, 57rspcimdv 3053 . . . . . 6  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  ( A. m  e.  (
x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  (
n ( <. x ,  y >.  .x.  z
) m ) )  =  ( ( ( y G z ) `
 n ) (
<. ( F `  x
) ,  ( F `
 y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) )  ->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) ) )
5923, 58rspcimdv 3053 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( A. z  e.  B  A. m  e.  (
x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  (
n ( <. x ,  y >.  .x.  z
) m ) )  =  ( ( ( y G z ) `
 n ) (
<. ( F `  x
) ,  ( F `
 y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) )  ->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) ) )
6021, 59rspcimdv 3053 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  B  A. z  e.  B  A. m  e.  (
x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  (
n ( <. x ,  y >.  .x.  z
) m ) )  =  ( ( ( y G z ) `
 n ) (
<. ( F `  x
) ,  ( F `
 y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) )  ->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) ) )
6160adantld 454 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( ( ( x G x ) `  ( ( Id `  D ) `  x
) )  =  ( ( Id `  E
) `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <. x ,  y
>.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n
) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) )  ->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) ) )
6219, 61rspcimdv 3053 . 2  |-  ( ph  ->  ( A. x  e.  B  ( ( ( x G x ) `
 ( ( Id
`  D ) `  x ) )  =  ( ( Id `  E ) `  ( F `  x )
)  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <.
x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. O ( F `  z ) ) ( ( x G y ) `  m ) ) )  ->  (
( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N
) ( <. ( F `  X ) ,  ( F `  Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) ) )
6318, 62mpd 15 1  |-  ( ph  ->  ( ( X G Z ) `  ( N ( <. X ,  Y >.  .x.  Z ) M ) )  =  ( ( ( Y G Z ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817   class class class wbr 4212    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348    ^m cmap 7018   X_cixp 7063   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889   Idccid 13890    Func cfunc 14051
This theorem is referenced by:  funcsect  14069  funcoppc  14072  cofucl  14085  funcres  14093  fthsect  14122  fthmon  14124  catcisolem  14261  prfcl  14300  evlfcllem  14318  curf1cl  14325  curf2cl  14328  curfcl  14329  uncfcurf  14336  yonedalem4c  14374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-ixp 7064  df-func 14055
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