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Theorem oppccofval 14968
Description: Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
oppcco.b  |-  B  =  ( Base `  C
)
oppcco.c  |-  .x.  =  (comp `  C )
oppcco.o  |-  O  =  (oppCat `  C )
oppcco.x  |-  ( ph  ->  X  e.  B )
oppcco.y  |-  ( ph  ->  Y  e.  B )
oppcco.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
oppccofval  |-  ( ph  ->  ( <. X ,  Y >. (comp `  O ) Z )  = tpos  ( <. Z ,  Y >.  .x. 
X ) )

Proof of Theorem oppccofval
Dummy variables  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppcco.x . . . . 5  |-  ( ph  ->  X  e.  B )
2 elfvex 5891 . . . . . 6  |-  ( X  e.  ( Base `  C
)  ->  C  e.  _V )
3 oppcco.b . . . . . 6  |-  B  =  ( Base `  C
)
42, 3eleq2s 2575 . . . . 5  |-  ( X  e.  B  ->  C  e.  _V )
5 eqid 2467 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 oppcco.c . . . . . 6  |-  .x.  =  (comp `  C )
7 oppcco.o . . . . . 6  |-  O  =  (oppCat `  C )
83, 5, 6, 7oppcval 14965 . . . . 5  |-  ( C  e.  _V  ->  O  =  ( ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C ) >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
) )
91, 4, 83syl 20 . . . 4  |-  ( ph  ->  O  =  ( ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C ) >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
) )
109fveq2d 5868 . . 3  |-  ( ph  ->  (comp `  O )  =  (comp `  ( ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C ) >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
) ) )
11 ovex 6307 . . . 4  |-  ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C ) >. )  e.  _V
12 fvex 5874 . . . . . . 7  |-  ( Base `  C )  e.  _V
133, 12eqeltri 2551 . . . . . 6  |-  B  e. 
_V
1413, 13xpex 6711 . . . . 5  |-  ( B  X.  B )  e. 
_V
1514, 13mpt2ex 6857 . . . 4  |-  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) )  e.  _V
16 ccoid 14669 . . . . 5  |- comp  = Slot  (comp ` 
ndx )
1716setsid 14527 . . . 4  |-  ( ( ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. )  e.  _V  /\  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )  e. 
_V )  ->  (
u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  (
<. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) )  =  (comp `  ( ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. ) sSet  <. (comp ` 
ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  (
<. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) ) >. ) ) )
1811, 15, 17mp2an 672 . . 3  |-  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) )  =  (comp `  ( ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. ) sSet  <. (comp ` 
ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  (
<. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) ) >. ) )
1910, 18syl6eqr 2526 . 2  |-  ( ph  ->  (comp `  O )  =  ( u  e.  ( B  X.  B
) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) )
20 simprr 756 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
21 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  u  =  <. X ,  Y >. )
2221fveq2d 5868 . . . . . 6  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  u )  =  ( 2nd `  <. X ,  Y >. )
)
231adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  X  e.  B )
24 oppcco.y . . . . . . . 8  |-  ( ph  ->  Y  e.  B )
2524adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  Y  e.  B )
26 op2ndg 6794 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2723, 25, 26syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2822, 27eqtrd 2508 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  u )  =  Y )
2920, 28opeq12d 4221 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  <. z ,  ( 2nd `  u
) >.  =  <. Z ,  Y >. )
3021fveq2d 5868 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  ( 1st `  <. X ,  Y >. )
)
31 op1stg 6793 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
3223, 25, 31syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
3330, 32eqtrd 2508 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  X )
3429, 33oveq12d 6300 . . 3  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( <. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  =  ( <. Z ,  Y >.  .x.  X )
)
3534tposeqd 6955 . 2  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  -> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  = tpos  ( <. Z ,  Y >.  .x.  X )
)
36 opelxpi 5030 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
371, 24, 36syl2anc 661 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
38 oppcco.z . 2  |-  ( ph  ->  Z  e.  B )
39 ovex 6307 . . . 4  |-  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4039tposex 6986 . . 3  |- tpos  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4140a1i 11 . 2  |-  ( ph  -> tpos  ( <. Z ,  Y >.  .x.  X )  e. 
_V )
4219, 35, 37, 38, 41ovmpt2d 6412 1  |-  ( ph  ->  ( <. X ,  Y >. (comp `  O ) Z )  = tpos  ( <. Z ,  Y >.  .x. 
X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033    X. cxp 4997   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780  tpos ctpos 6951   ndxcnx 14483   sSet csts 14484   Basecbs 14486   Hom chom 14562  compcco 14563  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
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-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-ltxr 9629  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-dec 10973  df-ndx 14489  df-slot 14490  df-sets 14492  df-cco 14576  df-oppc 14964
This theorem is referenced by:  oppcco  14969
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