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

Theorem oppccomfpropd 14979
Description: If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
oppchomfpropd.1  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
oppccomfpropd.1  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
Assertion
Ref Expression
oppccomfpropd  |-  ( ph  ->  (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) ) )

Proof of Theorem oppccomfpropd
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2467 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2467 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
4 eqid 2467 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 oppchomfpropd.1 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
65ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( Hom f  `  C
)  =  ( Hom f  `  D ) )
7 oppccomfpropd.1 . . . . . . 7  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
87ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  (compf `  C )  =  (compf `  D ) )
9 simplr3 1040 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  z  e.  ( Base `  C )
)
10 simplr2 1039 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  y  e.  ( Base `  C )
)
11 simplr1 1038 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  x  e.  ( Base `  C )
)
12 simprr 756 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) )
13 eqid 2467 . . . . . . . 8  |-  (oppCat `  C )  =  (oppCat `  C )
142, 13oppchom 14967 . . . . . . 7  |-  ( y ( Hom  `  (oppCat `  C ) ) z )  =  ( z ( Hom  `  C
) y )
1512, 14syl6eleq 2565 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  g  e.  ( z ( Hom  `  C ) y ) )
16 simprl 755 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) )
172, 13oppchom 14967 . . . . . . 7  |-  ( x ( Hom  `  (oppCat `  C ) ) y )  =  ( y ( Hom  `  C
) x )
1816, 17syl6eleq 2565 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  f  e.  ( y ( Hom  `  C ) x ) )
191, 2, 3, 4, 6, 8, 9, 10, 11, 15, 18comfeqval 14960 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( f
( <. z ,  y
>. (comp `  C )
x ) g )  =  ( f (
<. z ,  y >.
(comp `  D )
x ) g ) )
201, 3, 13, 11, 10, 9oppcco 14969 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  C
) ) z ) f )  =  ( f ( <. z ,  y >. (comp `  C ) x ) g ) )
21 eqid 2467 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
22 eqid 2467 . . . . . 6  |-  (oppCat `  D )  =  (oppCat `  D )
235homfeqbas 14948 . . . . . . . 8  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
2423ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( Base `  C )  =  (
Base `  D )
)
2511, 24eleqtrd 2557 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  x  e.  ( Base `  D )
)
2610, 24eleqtrd 2557 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  y  e.  ( Base `  D )
)
279, 24eleqtrd 2557 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  z  e.  ( Base `  D )
)
2821, 4, 22, 25, 26, 27oppcco 14969 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  D
) ) z ) f )  =  ( f ( <. z ,  y >. (comp `  D ) x ) g ) )
2919, 20, 283eqtr4d 2518 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) )
3029ralrimivva 2885 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  A. f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) A. g  e.  ( y
( Hom  `  (oppCat `  C ) ) z ) ( g (
<. x ,  y >.
(comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) )
3130ralrimivvva 2886 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C ) A. f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) A. g  e.  ( y
( Hom  `  (oppCat `  C ) ) z ) ( g (
<. x ,  y >.
(comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) )
32 eqid 2467 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
33 eqid 2467 . . 3  |-  (comp `  (oppCat `  D ) )  =  (comp `  (oppCat `  D ) )
34 eqid 2467 . . 3  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
3513, 1oppcbas 14970 . . . 4  |-  ( Base `  C )  =  (
Base `  (oppCat `  C
) )
3635a1i 11 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  C )
) )
3722, 21oppcbas 14970 . . . 4  |-  ( Base `  D )  =  (
Base `  (oppCat `  D
) )
3823, 37syl6eq 2524 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  D )
) )
395oppchomfpropd 14978 . . 3  |-  ( ph  ->  ( Hom f  `  (oppCat `  C
) )  =  ( Hom f  `  (oppCat `  D )
) )
4032, 33, 34, 36, 38, 39comfeq 14958 . 2  |-  ( ph  ->  ( (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) )  <->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) A. z  e.  ( Base `  C
) A. f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) A. g  e.  ( y
( Hom  `  (oppCat `  C ) ) z ) ( g (
<. x ,  y >.
(comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) ) )
4131, 40mpbird 232 1  |-  ( ph  ->  (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   ` cfv 5586  (class class class)co 6282   Basecbs 14486   Hom chom 14562  compcco 14563   Hom f chomf 14917  compfccomf 14918  oppCatcoppc 14963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-hom 14575  df-cco 14576  df-homf 14921  df-comf 14922  df-oppc 14964
This theorem is referenced by:  yonpropd  15391
  Copyright terms: Public domain W3C validator