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

Theorem natixp 15368
Description: A natural transformation is a function from the objects of 
C to homomorphisms from  F ( x ) to  G ( x ). (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1  |-  N  =  ( C Nat  D )
natixp.2  |-  ( ph  ->  A  e.  ( <. F ,  G >. N
<. K ,  L >. ) )
natixp.b  |-  B  =  ( Base `  C
)
natixp.j  |-  J  =  ( Hom  `  D
)
Assertion
Ref Expression
natixp  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
Distinct variable groups:    x, A    x, F    x, G    x, C    x, K    ph, x    x, D    x, L    x, B    x, J
Allowed substitution hint:    N( x)

Proof of Theorem natixp
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natixp.2 . . 3  |-  ( ph  ->  A  e.  ( <. F ,  G >. N
<. K ,  L >. ) )
2 natrcl.1 . . . 4  |-  N  =  ( C Nat  D )
3 natixp.b . . . 4  |-  B  =  ( Base `  C
)
4 eqid 2457 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 natixp.j . . . 4  |-  J  =  ( Hom  `  D
)
6 eqid 2457 . . . 4  |-  (comp `  D )  =  (comp `  D )
72natrcl 15366 . . . . . . 7  |-  ( A  e.  ( <. F ,  G >. N <. K ,  L >. )  ->  ( <. F ,  G >.  e.  ( C  Func  D
)  /\  <. K ,  L >.  e.  ( C 
Func  D ) ) )
81, 7syl 16 . . . . . 6  |-  ( ph  ->  ( <. F ,  G >.  e.  ( C  Func  D )  /\  <. K ,  L >.  e.  ( C 
Func  D ) ) )
98simpld 459 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
10 df-br 4457 . . . . 5  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylibr 212 . . . 4  |-  ( ph  ->  F ( C  Func  D ) G )
128simprd 463 . . . . 5  |-  ( ph  -> 
<. K ,  L >.  e.  ( C  Func  D
) )
13 df-br 4457 . . . . 5  |-  ( K ( C  Func  D
) L  <->  <. K ,  L >.  e.  ( C 
Func  D ) )
1412, 13sylibr 212 . . . 4  |-  ( ph  ->  K ( C  Func  D ) L )
152, 3, 4, 5, 6, 11, 14isnat 15363 . . 3  |-  ( ph  ->  ( A  e.  (
<. F ,  G >. N
<. K ,  L >. )  <-> 
( A  e.  X_ x  e.  B  (
( F `  x
) J ( K `
 x ) )  /\  A. x  e.  B  A. y  e.  B  A. z  e.  ( x ( Hom  `  C ) y ) ( ( A `  y ) ( <.
( F `  x
) ,  ( F `
 y ) >.
(comp `  D )
( K `  y
) ) ( ( x G y ) `
 z ) )  =  ( ( ( x L y ) `
 z ) (
<. ( F `  x
) ,  ( K `
 x ) >.
(comp `  D )
( K `  y
) ) ( A `
 x ) ) ) ) )
161, 15mpbid 210 . 2  |-  ( ph  ->  ( A  e.  X_ x  e.  B  (
( F `  x
) J ( K `
 x ) )  /\  A. x  e.  B  A. y  e.  B  A. z  e.  ( x ( Hom  `  C ) y ) ( ( A `  y ) ( <.
( F `  x
) ,  ( F `
 y ) >.
(comp `  D )
( K `  y
) ) ( ( x G y ) `
 z ) )  =  ( ( ( x L y ) `
 z ) (
<. ( F `  x
) ,  ( K `
 x ) >.
(comp `  D )
( K `  y
) ) ( A `
 x ) ) ) )
1716simpld 459 1  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   X_cixp 7488   Basecbs 14644   Hom chom 14723  compcco 14724    Func cfunc 15270   Nat cnat 15357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-ixp 7489  df-func 15274  df-nat 15359
This theorem is referenced by:  natcl  15369  natfn  15370
  Copyright terms: Public domain W3C validator