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Theorem arwlid 14156
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 2389 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 arwlid.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X H Y ) )
4 arwlid.h . . . . . . . 8  |-  H  =  (Homa
`  C )
54homarcl 14112 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
63, 5syl 16 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
7 eqid 2389 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
84, 2homarcl2 14119 . . . . . . . 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 14143 . . . . 5  |-  ( ph  ->  ( 2nd `  (  .1.  `  Y ) )  =  ( ( Id
`  C ) `  Y ) )
1211oveq1d 6037 . . . 4  |-  ( ph  ->  ( ( 2nd `  (  .1.  `  Y ) ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F ) )  =  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) ) )
13 eqid 2389 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
149simpld 446 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  C ) )
15 eqid 2389 . . . . 5  |-  (comp `  C )  =  (comp `  C )
164, 13homahom 14123 . . . . . 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 13837 . . . 4  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) )  =  ( 2nd `  F ) )
1912, 18eqtrd 2421 . . 3  |-  ( ph  ->  ( ( 2nd `  (  .1.  `  Y ) ) ( <. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F ) )  =  ( 2nd `  F
) )
2019oteq3d 3942 . 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 14144 . . 3  |-  ( ph  ->  (  .1.  `  Y
)  e.  ( Y H Y ) )
2321, 4, 3, 22, 15coaval 14152 . 2  |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F
)  =  <. X ,  Y ,  ( ( 2nd `  (  .1.  `  Y ) ) (
<. X ,  Y >. (comp `  C ) Y ) ( 2nd `  F
) ) >. )
244homadmcd 14126 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
253, 24syl 16 . 2  |-  ( ph  ->  F  =  <. X ,  Y ,  ( 2nd `  F ) >. )
2620, 23, 253eqtr4d 2431 1  |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F
)  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3762   <.cotp 3763   ` cfv 5396  (class class class)co 6022   2ndc2nd 6289   Basecbs 13398    Hom chom 13469  compcco 13470   Catccat 13818   Idccid 13819  Homachoma 14107  Idacida 14137  compaccoa 14138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-ot 3769  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-cat 13822  df-cid 13823  df-doma 14108  df-coda 14109  df-homa 14110  df-arw 14111  df-ida 14139  df-coa 14140
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