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

Theorem fucrid 15823
Description: Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuclid.q  |-  Q  =  ( C FuncCat  D )
fuclid.n  |-  N  =  ( C Nat  D )
fuclid.x  |-  .xb  =  (comp `  Q )
fuclid.1  |-  .1.  =  ( Id `  D )
fuclid.r  |-  ( ph  ->  R  e.  ( F N G ) )
Assertion
Ref Expression
fucrid  |-  ( ph  ->  ( R ( <. F ,  F >.  .xb 
G ) (  .1. 
o.  ( 1st `  F
) ) )  =  R )

Proof of Theorem fucrid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2429 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 relfunc 15718 . . . . . . . 8  |-  Rel  ( C  Func  D )
4 fuclid.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  ( F N G ) )
5 fuclid.n . . . . . . . . . . 11  |-  N  =  ( C Nat  D )
65natrcl 15806 . . . . . . . . . 10  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
74, 6syl 17 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
87simpld 460 . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
9 1st2ndbr 6856 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
103, 8, 9sylancr 667 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
111, 2, 10funcf1 15722 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
12 fvco3 5958 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  x  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 x )  =  (  .1.  `  (
( 1st `  F
) `  x )
) )
1311, 12sylan 473 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (  .1.  o.  ( 1st `  F
) ) `  x
)  =  (  .1.  `  ( ( 1st `  F
) `  x )
) )
1413oveq2d 6321 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( R `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 x ) >.
(comp `  D )
( ( 1st `  G
) `  x )
) ( (  .1. 
o.  ( 1st `  F
) ) `  x
) )  =  ( ( R `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
15 eqid 2429 . . . . 5  |-  ( Hom  `  D )  =  ( Hom  `  D )
16 fuclid.1 . . . . 5  |-  .1.  =  ( Id `  D )
17 funcrcl 15719 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
188, 17syl 17 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1918simprd 464 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
2019adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2111ffvelrnda 6037 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
22 eqid 2429 . . . . 5  |-  (comp `  D )  =  (comp `  D )
237simprd 464 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
24 1st2ndbr 6856 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
253, 23, 24sylancr 667 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
261, 2, 25funcf1 15722 . . . . . 6  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
2726ffvelrnda 6037 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
285, 4nat1st2nd 15807 . . . . . . 7  |-  ( ph  ->  R  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
2928adantr 466 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
30 simpr 462 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
315, 29, 1, 15, 30natcl 15809 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  G
) `  x )
) )
322, 15, 16, 20, 21, 22, 27, 31catrid 15541 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( R `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 x ) >.
(comp `  D )
( ( 1st `  G
) `  x )
) (  .1.  `  ( ( 1st `  F
) `  x )
) )  =  ( R `  x ) )
3314, 32eqtrd 2470 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( R `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 x ) >.
(comp `  D )
( ( 1st `  G
) `  x )
) ( (  .1. 
o.  ( 1st `  F
) ) `  x
) )  =  ( R `  x ) )
3433mpteq2dva 4512 . 2  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( R `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( R `  x ) ) )
35 fuclid.q . . 3  |-  Q  =  ( C FuncCat  D )
36 fuclid.x . . 3  |-  .xb  =  (comp `  Q )
3735, 5, 16, 8fucidcl 15821 . . 3  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )
3835, 5, 1, 22, 36, 37, 4fucco 15818 . 2  |-  ( ph  ->  ( R ( <. F ,  F >.  .xb 
G ) (  .1. 
o.  ( 1st `  F
) ) )  =  ( x  e.  (
Base `  C )  |->  ( ( R `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) ) ) )
395, 28, 1natfn 15810 . . 3  |-  ( ph  ->  R  Fn  ( Base `  C ) )
40 dffn5 5926 . . 3  |-  ( R  Fn  ( Base `  C
)  <->  R  =  (
x  e.  ( Base `  C )  |->  ( R `
 x ) ) )
4139, 40sylib 199 . 2  |-  ( ph  ->  R  =  ( x  e.  ( Base `  C
)  |->  ( R `  x ) ) )
4234, 38, 413eqtr4d 2480 1  |-  ( ph  ->  ( R ( <. F ,  F >.  .xb 
G ) (  .1. 
o.  ( 1st `  F
) ) )  =  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   <.cop 4008   class class class wbr 4426    |-> cmpt 4484    o. ccom 4858   Rel wrel 4859    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   Basecbs 15084   Hom chom 15163  compcco 15164   Catccat 15521   Idccid 15522    Func cfunc 15710   Nat cnat 15797   FuncCat cfuc 15798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-hom 15176  df-cco 15177  df-cat 15525  df-cid 15526  df-func 15714  df-nat 15799  df-fuc 15800
This theorem is referenced by:  fuccatid  15825
  Copyright terms: Public domain W3C validator