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

Theorem funcco 13745
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 2283 . . . . 5  |-  ( Base `  E )  =  (
Base `  E )
4 funcco.h . . . . 5  |-  H  =  (  Hom  `  D
)
5 eqid 2283 . . . . 5  |-  (  Hom  `  E )  =  (  Hom  `  E )
6 eqid 2283 . . . . 5  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2283 . . . . 5  |-  ( Id
`  E )  =  ( Id `  E
)
8 funcco.o . . . . 5  |-  .x.  =  (comp `  D )
9 funcco.O . . . . 5  |-  O  =  (comp `  E )
10 df-br 4024 . . . . . . . 8  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
111, 10sylib 188 . . . . . . 7  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
12 funcrcl 13737 . . . . . . 7  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 15 . . . . . 6  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1413simpld 445 . . . . 5  |-  ( ph  ->  D  e.  Cat )
1513simprd 449 . . . . 5  |-  ( ph  ->  E  e.  Cat )
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 13738 . . . 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 201 . . 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 969 . 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 451 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
22 funcco.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
2322ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  Z  e.  B )
24 funcco.m . . . . . . . . 9  |-  ( ph  ->  M  e.  ( X H Y ) )
2524ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  M  e.  ( X H Y ) )
26 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  x  =  X )
27 simplr 731 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  y  =  Y )
2826, 27oveq12d 5876 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  (
x H y )  =  ( X H Y ) )
2925, 28eleqtrrd 2360 . . . . . . 7  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  M  e.  ( x H y ) )
30 funcco.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  ( Y H Z ) )
3130ad4antr 712 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  N  e.  ( Y H Z ) )
32 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  y  =  Y )
33 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  z  =  Z )
3432, 33oveq12d 5876 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  (
y H z )  =  ( Y H Z ) )
3531, 34eleqtrrd 2360 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  ->  N  e.  ( y H z ) )
36 simp-5r 745 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  x  =  X )
37 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  z  =  Z )
3836, 37oveq12d 5876 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
x G z )  =  ( X G Z ) )
39 simp-4r 743 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  y  =  Y )
4036, 39opeq12d 3804 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  <. x ,  y >.  =  <. X ,  Y >. )
4140, 37oveq12d 5876 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( <. x ,  y >.  .x.  z )  =  (
<. X ,  Y >.  .x. 
Z ) )
42 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  n  =  N )
43 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  m  =  M )
4441, 42, 43oveq123d 5879 . . . . . . . . . 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 5531 . . . . . . . . 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 5529 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( F `  x )  =  ( F `  X ) )
4739fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( F `  y )  =  ( F `  Y ) )
4846, 47opeq12d 3804 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  <. ( F `  x ) ,  ( F `  y ) >.  =  <. ( F `  X ) ,  ( F `  Y ) >. )
4937fveq2d 5529 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  ( F `  z )  =  ( F `  Z ) )
5048, 49oveq12d 5876 . . . . . . . . . 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 5876 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
y G z )  =  ( Y G Z ) )
5251, 42fveq12d 5531 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
( y G z ) `  n )  =  ( ( Y G Z ) `  N ) )
5336, 39oveq12d 5876 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
x G y )  =  ( X G Y ) )
5453, 43fveq12d 5531 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  m  =  M )  /\  n  =  N )  ->  (
( x G y ) `  m )  =  ( ( X G Y ) `  M ) )
5550, 52, 54oveq123d 5879 . . . . . . . . 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 2297 . . . . . . . 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 2887 . . . . . . 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 2885 . . . . . 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 2885 . . . . 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 2885 . . . 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 453 . . 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 2885 . 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 14 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   X_cixp 6817   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    Func cfunc 13728
This theorem is referenced by:  funcsect  13746  funcoppc  13749  cofucl  13762  funcres  13770  fthsect  13799  fthmon  13801  catcisolem  13938  prfcl  13977  evlfcllem  13995  curf1cl  14002  curf2cl  14005  curfcl  14006  uncfcurf  14013  yonedalem4c  14051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-ixp 6818  df-func 13732
  Copyright terms: Public domain W3C validator