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Theorem arwlid 14940
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 2443 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 arwlid.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X H Y ) )
4 arwlid.h . . . . . . . 8  |-  H  =  (Homa
`  C )
54homarcl 14896 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
63, 5syl 16 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
7 eqid 2443 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
84, 2homarcl2 14903 . . . . . . . 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 463 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  C ) )
111, 2, 6, 7, 10ida2 14927 . . . . 5  |-  ( ph  ->  ( 2nd `  (  .1.  `  Y ) )  =  ( ( Id
`  C ) `  Y ) )
1211oveq1d 6106 . . . 4  |-  ( ph  ->  ( ( 2nd `  (  .1.  `  Y ) ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F ) )  =  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) ) )
13 eqid 2443 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
149simpld 459 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  C ) )
15 eqid 2443 . . . . 5  |-  (comp `  C )  =  (comp `  C )
164, 13homahom 14907 . . . . . 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 14621 . . . 4  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) )  =  ( 2nd `  F ) )
1912, 18eqtrd 2475 . . 3  |-  ( ph  ->  ( ( 2nd `  (  .1.  `  Y ) ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F ) )  =  ( 2nd `  F
) )
2019oteq3d 4073 . 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 14928 . . 3  |-  ( ph  ->  (  .1.  `  Y
)  e.  ( Y H Y ) )
2321, 4, 3, 22, 15coaval 14936 . 2  |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F
)  =  <. X ,  Y ,  ( ( 2nd `  (  .1.  `  Y ) ) (
<. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) ) >. )
244homadmcd 14910 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
253, 24syl 16 . 2  |-  ( ph  ->  F  =  <. X ,  Y ,  ( 2nd `  F ) >. )
2620, 23, 253eqtr4d 2485 1  |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F
)  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3883   <.cotp 3885   ` cfv 5418  (class class class)co 6091   2ndc2nd 6576   Basecbs 14174   Hom chom 14249  compcco 14250   Catccat 14602   Idccid 14603  Homachoma 14891  Idacida 14921  compaccoa 14922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-ot 3886  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-cat 14606  df-cid 14607  df-doma 14892  df-coda 14893  df-homa 14894  df-arw 14895  df-ida 14923  df-coa 14924
This theorem is referenced by: (None)
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