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Theorem arwlid 14217
Description: Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwlid.h  |-  H  =  (Homa
`  C )
arwlid.o  |-  .x.  =  (compa `  C )
arwlid.a  |-  .1.  =  (Ida `  C )
arwlid.f  |-  ( ph  ->  F  e.  ( X H Y ) )
Assertion
Ref Expression
arwlid  |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F
)  =  F )

Proof of Theorem arwlid
StepHypRef Expression
1 arwlid.a . . . . . 6  |-  .1.  =  (Ida `  C )
2 eqid 2435 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 arwlid.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X H Y ) )
4 arwlid.h . . . . . . . 8  |-  H  =  (Homa
`  C )
54homarcl 14173 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
63, 5syl 16 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
7 eqid 2435 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
84, 2homarcl2 14180 . . . . . . . 8  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
93, 8syl 16 . . . . . . 7  |-  ( ph  ->  ( X  e.  (
Base `  C )  /\  Y  e.  ( Base `  C ) ) )
109simprd 450 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  C ) )
111, 2, 6, 7, 10ida2 14204 . . . . 5  |-  ( ph  ->  ( 2nd `  (  .1.  `  Y ) )  =  ( ( Id
`  C ) `  Y ) )
1211oveq1d 6088 . . . 4  |-  ( ph  ->  ( ( 2nd `  (  .1.  `  Y ) ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F ) )  =  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) ) )
13 eqid 2435 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
149simpld 446 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  C ) )
15 eqid 2435 . . . . 5  |-  (comp `  C )  =  (comp `  C )
164, 13homahom 14184 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X (  Hom  `  C ) Y ) )
173, 16syl 16 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  e.  ( X (  Hom  `  C
) Y ) )
182, 13, 7, 6, 14, 15, 10, 17catlid 13898 . . . 4  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) )  =  ( 2nd `  F ) )
1912, 18eqtrd 2467 . . 3  |-  ( ph  ->  ( ( 2nd `  (  .1.  `  Y ) ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F ) )  =  ( 2nd `  F
) )
2019oteq3d 3990 . 2  |-  ( ph  -> 
<. X ,  Y , 
( ( 2nd `  (  .1.  `  Y ) ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F ) ) >.  =  <. X ,  Y ,  ( 2nd `  F
) >. )
21 arwlid.o . . 3  |-  .x.  =  (compa `  C )
221, 2, 6, 10, 4idahom 14205 . . 3  |-  ( ph  ->  (  .1.  `  Y
)  e.  ( Y H Y ) )
2321, 4, 3, 22, 15coaval 14213 . 2  |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F
)  =  <. X ,  Y ,  ( ( 2nd `  (  .1.  `  Y ) ) (
<. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) ) >. )
244homadmcd 14187 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
253, 24syl 16 . 2  |-  ( ph  ->  F  =  <. X ,  Y ,  ( 2nd `  F ) >. )
2620, 23, 253eqtr4d 2477 1  |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F
)  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809   <.cotp 3810   ` cfv 5446  (class class class)co 6073   2ndc2nd 6340   Basecbs 13459    Hom chom 13530  compcco 13531   Catccat 13879   Idccid 13880  Homachoma 14168  Idacida 14198  compaccoa 14199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-ot 3816  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-cat 13883  df-cid 13884  df-doma 14169  df-coda 14170  df-homa 14171  df-arw 14172  df-ida 14200  df-coa 14201
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