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

Theorem oppccofval 15331
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 5878 . . . . . 6  |-  ( X  e.  ( Base `  C
)  ->  C  e.  _V )
3 oppcco.b . . . . . 6  |-  B  =  ( Base `  C
)
42, 3eleq2s 2512 . . . . 5  |-  ( X  e.  B  ->  C  e.  _V )
5 eqid 2404 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 oppcco.c . . . . . 6  |-  .x.  =  (comp `  C )
7 oppcco.o . . . . . 6  |-  O  =  (oppCat `  C )
83, 5, 6, 7oppcval 15328 . . . . 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 18 . . . 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 5855 . . 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 6308 . . . 4  |-  ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C ) >. )  e.  _V
12 fvex 5861 . . . . . . 7  |-  ( Base `  C )  e.  _V
133, 12eqeltri 2488 . . . . . 6  |-  B  e. 
_V
1413, 13xpex 6588 . . . . 5  |-  ( B  X.  B )  e. 
_V
1514, 13mpt2ex 6863 . . . 4  |-  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) )  e.  _V
16 ccoid 15033 . . . . 5  |- comp  = Slot  (comp ` 
ndx )
1716setsid 14886 . . . 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 2463 . 2  |-  ( ph  ->  (comp `  O )  =  ( u  e.  ( B  X.  B
) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) )
20 simprr 760 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
21 simprl 758 . . . . . . 7  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  u  =  <. X ,  Y >. )
2221fveq2d 5855 . . . . . 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 6799 . . . . . . 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 2445 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  u )  =  Y )
2920, 28opeq12d 4169 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  <. z ,  ( 2nd `  u
) >.  =  <. Z ,  Y >. )
3021fveq2d 5855 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  ( 1st `  <. X ,  Y >. )
)
31 op1stg 6798 . . . . . 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 2445 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  X )
3429, 33oveq12d 6298 . . 3  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( <. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  =  ( <. Z ,  Y >.  .x.  X )
)
3534tposeqd 6963 . 2  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  -> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  = tpos  ( <. Z ,  Y >.  .x.  X )
)
36 opelxpi 4857 . . 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 6308 . . . 4  |-  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4039tposex 6994 . . 3  |- tpos  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4140a1i 11 . 2  |-  ( ph  -> tpos  ( <. Z ,  Y >.  .x.  X )  e. 
_V )
4219, 35, 37, 38, 41ovmpt2d 6413 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 1407    e. wcel 1844   _Vcvv 3061   <.cop 3980    X. cxp 4823   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   1stc1st 6784   2ndc2nd 6785  tpos ctpos 6959   ndxcnx 14840   sSet csts 14841   Basecbs 14843   Hom chom 14922  compcco 14923  oppCatcoppc 15326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-tpos 6960  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-ltxr 9665  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-dec 11022  df-ndx 14846  df-slot 14847  df-sets 14849  df-cco 14936  df-oppc 15327
This theorem is referenced by:  oppcco  15332
  Copyright terms: Public domain W3C validator