MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0catg Structured version   Unicode version

Theorem 0catg 14933
Description: Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
0catg  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  Cat )

Proof of Theorem 0catg
Dummy variables  f 
g  h  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (/)  =  (
Base `  C )
)
2 eqidd 2463 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  ( Hom  `  C )  =  ( Hom  `  C
) )
3 eqidd 2463 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (comp `  C )  =  (comp `  C ) )
4 simpl 457 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  V )
5 noel 3784 . . . 4  |-  -.  x  e.  (/)
65pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  (/)  e.  ( x ( Hom  `  C
) x ) )
76adantl 466 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  x  e.  (/) )  ->  (/) 
e.  ( x ( Hom  `  C )
x ) )
8 simpr1 997 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( y ( Hom  `  C ) x ) ) )  ->  x  e.  (/) )
95pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( (/) ( <.
y ,  x >. (comp `  C ) x ) f )  =  f )
108, 9syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( y ( Hom  `  C ) x ) ) )  ->  ( (/) ( <. y ,  x >. (comp `  C )
x ) f )  =  f )
11 simpr1 997 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( x ( Hom  `  C ) y ) ) )  ->  x  e.  (/) )
125pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( f
( <. x ,  x >. (comp `  C )
y ) (/) )  =  f )
1311, 12syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( x ( Hom  `  C ) y ) ) )  ->  (
f ( <. x ,  x >. (comp `  C
) y ) (/) )  =  f )
14 simp21 1024 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  z  e.  (/) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z ) ) )  ->  x  e.  (/) )
155pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x ( Hom  `  C )
z ) )
1614, 15syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  z  e.  (/) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  C )
z ) f )  e.  ( x ( Hom  `  C )
z ) )
17 simp2ll 1058 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( ( x  e.  (/)  /\  y  e.  (/) )  /\  ( z  e.  (/)  /\  w  e.  (/) ) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z )  /\  h  e.  ( z ( Hom  `  C ) w ) ) )  ->  x  e.  (/) )
185pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( (
h ( <. y ,  z >. (comp `  C ) w ) g ) ( <.
x ,  y >.
(comp `  C )
w ) f )  =  ( h (
<. x ,  z >.
(comp `  C )
w ) ( g ( <. x ,  y
>. (comp `  C )
z ) f ) ) )
1917, 18syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( ( x  e.  (/)  /\  y  e.  (/) )  /\  ( z  e.  (/)  /\  w  e.  (/) ) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z )  /\  h  e.  ( z ( Hom  `  C ) w ) ) )  ->  (
( h ( <.
y ,  z >.
(comp `  C )
w ) g ) ( <. x ,  y
>. (comp `  C )
w ) f )  =  ( h (
<. x ,  z >.
(comp `  C )
w ) ( g ( <. x ,  y
>. (comp `  C )
z ) f ) ) )
201, 2, 3, 4, 7, 10, 13, 16, 19iscatd 14919 1  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   (/)c0 3780   <.cop 4028   ` cfv 5581  (class class class)co 6277   Basecbs 14481   Hom chom 14557  compcco 14558   Catccat 14910
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-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571
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-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-cat 14914
This theorem is referenced by:  0cat  14934
  Copyright terms: Public domain W3C validator