Metamath Proof Explorer

Theorem arwlid

Description: Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses arwlid.h 
|- H = ( HomA ` C )
arwlid.o 
|- .x. = ( compA ` C )
arwlid.a 
|- .1. = ( IdA ` C )
arwlid.f 
|- ( ph -> F e. ( X H Y ) )
Assertion arwlid 
|- ( ph -> ( ( .1. ` Y ) .x. F ) = F )

Proof

Step Hyp Ref Expression
1 arwlid.h 
 |-  H = ( HomA ` C )
2 arwlid.o 
 |-  .x. = ( compA ` C )
3 arwlid.a 
 |-  .1. = ( IdA ` C )
4 arwlid.f 
 |-  ( ph -> F e. ( X H Y ) )
5 eqid 
 |-  ( Base ` C ) = ( Base ` C )
6 1 homarcl 
 |-  ( F e. ( X H Y ) -> C e. Cat )
7 4 6 syl 
 |-  ( ph -> C e. Cat )
8 eqid 
 |-  ( Id ` C ) = ( Id ` C )
9 1 5 homarcl2 
 |-  ( F e. ( X H Y ) -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
10 4 9 syl 
 |-  ( ph -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
11 10 simprd 
 |-  ( ph -> Y e. ( Base ` C ) )
12 3 5 7 8 11 ida2 
 |-  ( ph -> ( 2nd ` ( .1. ` Y ) ) = ( ( Id ` C ) ` Y ) )
13 12 oveq1d 
 |-  ( ph -> ( ( 2nd ` ( .1. ` Y ) ) ( <. X , Y >. ( comp ` C ) Y ) ( 2nd ` F ) ) = ( ( ( Id ` C ) ` Y ) ( <. X , Y >. ( comp ` C ) Y ) ( 2nd ` F ) ) )
14 eqid 
 |-  ( Hom ` C ) = ( Hom ` C )
15 10 simpld 
 |-  ( ph -> X e. ( Base ` C ) )
16 eqid 
 |-  ( comp ` C ) = ( comp ` C )
17 1 14 homahom 
 |-  ( F e. ( X H Y ) -> ( 2nd ` F ) e. ( X ( Hom ` C ) Y ) )
18 4 17 syl 
 |-  ( ph -> ( 2nd ` F ) e. ( X ( Hom ` C ) Y ) )
19 5 14 8 7 15 16 11 18 catlid 
 |-  ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. ( comp ` C ) Y ) ( 2nd ` F ) ) = ( 2nd ` F ) )
20 13 19 eqtrd 
 |-  ( ph -> ( ( 2nd ` ( .1. ` Y ) ) ( <. X , Y >. ( comp ` C ) Y ) ( 2nd ` F ) ) = ( 2nd ` F ) )
21 20 oteq3d 
 |-  ( ph -> <. X , Y , ( ( 2nd ` ( .1. ` Y ) ) ( <. X , Y >. ( comp ` C ) Y ) ( 2nd ` F ) ) >. = <. X , Y , ( 2nd ` F ) >. )
22 3 5 7 11 1 idahom 
 |-  ( ph -> ( .1. ` Y ) e. ( Y H Y ) )
23 2 1 4 22 16 coaval 
 |-  ( ph -> ( ( .1. ` Y ) .x. F ) = <. X , Y , ( ( 2nd ` ( .1. ` Y ) ) ( <. X , Y >. ( comp ` C ) Y ) ( 2nd ` F ) ) >. )
24 1 homadmcd 
 |-  ( F e. ( X H Y ) -> F = <. X , Y , ( 2nd ` F ) >. )
25 4 24 syl 
 |-  ( ph -> F = <. X , Y , ( 2nd ` F ) >. )
26 21 23 25 3eqtr4d 
 |-  ( ph -> ( ( .1. ` Y ) .x. F ) = F )