MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofuval2 Structured version   Unicode version

Theorem cofuval2 15105
Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval2.b  |-  B  =  ( Base `  C
)
cofuval2.f  |-  ( ph  ->  F ( C  Func  D ) G )
cofuval2.x  |-  ( ph  ->  H ( D  Func  E ) K )
Assertion
Ref Expression
cofuval2  |-  ( ph  ->  ( <. H ,  K >.  o.func 
<. F ,  G >. )  =  <. ( H  o.  F ) ,  ( x  e.  B , 
y  e.  B  |->  ( ( ( F `  x ) K ( F `  y ) )  o.  ( x G y ) ) ) >. )
Distinct variable groups:    x, y, B    x, F, y    x, G, y    x, H, y    ph, x, y    x, K, y
Allowed substitution hints:    C( x, y)    D( x, y)    E( x, y)

Proof of Theorem cofuval2
StepHypRef Expression
1 cofuval2.b . . 3  |-  B  =  ( Base `  C
)
2 cofuval2.f . . . 4  |-  ( ph  ->  F ( C  Func  D ) G )
3 df-br 4443 . . . 4  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
42, 3sylib 196 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
5 cofuval2.x . . . 4  |-  ( ph  ->  H ( D  Func  E ) K )
6 df-br 4443 . . . 4  |-  ( H ( D  Func  E
) K  <->  <. H ,  K >.  e.  ( D 
Func  E ) )
75, 6sylib 196 . . 3  |-  ( ph  -> 
<. H ,  K >.  e.  ( D  Func  E
) )
81, 4, 7cofuval 15100 . 2  |-  ( ph  ->  ( <. H ,  K >.  o.func 
<. F ,  G >. )  =  <. ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. ) ) ,  ( x  e.  B , 
y  e.  B  |->  ( ( ( ( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. )
( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. ) y ) ) ) >. )
9 relfunc 15080 . . . . . 6  |-  Rel  ( D  Func  E )
10 brrelex12 5031 . . . . . 6  |-  ( ( Rel  ( D  Func  E )  /\  H ( D  Func  E ) K )  ->  ( H  e.  _V  /\  K  e.  _V ) )
119, 5, 10sylancr 663 . . . . 5  |-  ( ph  ->  ( H  e.  _V  /\  K  e.  _V )
)
12 op1stg 6788 . . . . 5  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 1st `  <. H ,  K >. )  =  H )
1311, 12syl 16 . . . 4  |-  ( ph  ->  ( 1st `  <. H ,  K >. )  =  H )
14 relfunc 15080 . . . . . 6  |-  Rel  ( C  Func  D )
15 brrelex12 5031 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F ( C  Func  D ) G )  ->  ( F  e.  _V  /\  G  e.  _V ) )
1614, 2, 15sylancr 663 . . . . 5  |-  ( ph  ->  ( F  e.  _V  /\  G  e.  _V )
)
17 op1stg 6788 . . . . 5  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 1st `  <. F ,  G >. )  =  F )
1816, 17syl 16 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
1913, 18coeq12d 5160 . . 3  |-  ( ph  ->  ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. )
)  =  ( H  o.  F ) )
20 op2ndg 6789 . . . . . . . 8  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 2nd `  <. H ,  K >. )  =  K )
2111, 20syl 16 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. H ,  K >. )  =  K )
22213ad2ant1 1012 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. H ,  K >. )  =  K )
23183ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 1st ` 
<. F ,  G >. )  =  F )
2423fveq1d 5861 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  x )  =  ( F `  x ) )
2523fveq1d 5861 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  y )  =  ( F `  y ) )
2622, 24, 25oveq123d 6298 . . . . 5  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( (
( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. ) ( ( 1st `  <. F ,  G >. ) `  y ) )  =  ( ( F `  x ) K ( F `  y ) ) )
27 op2ndg 6789 . . . . . . . 8  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 2nd `  <. F ,  G >. )  =  G )
2816, 27syl 16 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
29283ad2ant1 1012 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. F ,  G >. )  =  G )
3029oveqd 6294 . . . . 5  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x
( 2nd `  <. F ,  G >. )
y )  =  ( x G y ) )
3126, 30coeq12d 5160 . . . 4  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( (
( ( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. ) ( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. )
y ) )  =  ( ( ( F `
 x ) K ( F `  y
) )  o.  (
x G y ) ) )
3231mpt2eq3dva 6338 . . 3  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  <. F ,  G >. ) `  x
) ( 2nd `  <. H ,  K >. )
( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. ) y ) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( ( ( F `  x
) K ( F `
 y ) )  o.  ( x G y ) ) ) )
3319, 32opeq12d 4216 . 2  |-  ( ph  -> 
<. ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. )
) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. ) ( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. )
y ) ) )
>.  =  <. ( H  o.  F ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( F `
 x ) K ( F `  y
) )  o.  (
x G y ) ) ) >. )
348, 33eqtrd 2503 1  |-  ( ph  ->  ( <. H ,  K >.  o.func 
<. F ,  G >. )  =  <. ( H  o.  F ) ,  ( x  e.  B , 
y  e.  B  |->  ( ( ( F `  x ) K ( F `  y ) )  o.  ( x G y ) ) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3108   <.cop 4028   class class class wbr 4442    o. ccom 4998   Rel wrel 4999   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   Basecbs 14481    Func cfunc 15072    o.func ccofu 15074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-ixp 7462  df-func 15076  df-cofu 15078
This theorem is referenced by:  catcisolem  15282
  Copyright terms: Public domain W3C validator