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Theorem oppccofval 13619
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 . . . . . 6  |-  ( ph  ->  X  e.  B )
2 elfvex 5555 . . . . . . 7  |-  ( X  e.  ( Base `  C
)  ->  C  e.  _V )
3 oppcco.b . . . . . . 7  |-  B  =  ( Base `  C
)
42, 3eleq2s 2375 . . . . . 6  |-  ( X  e.  B  ->  C  e.  _V )
51, 4syl 15 . . . . 5  |-  ( ph  ->  C  e.  _V )
6 eqid 2283 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
7 oppcco.c . . . . . 6  |-  .x.  =  (comp `  C )
8 oppcco.o . . . . . 6  |-  O  =  (oppCat `  C )
93, 6, 7, 8oppcval 13616 . . . . 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 ) ) ) >. ) )
105, 9syl 15 . . . 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 ) ) ) >. ) )
1110fveq2d 5529 . . 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 ) ) ) >. ) ) )
12 ovex 5883 . . . 4  |-  ( C sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  C
) >. )  e.  _V
13 fvex 5539 . . . . . . 7  |-  ( Base `  C )  e.  _V
143, 13eqeltri 2353 . . . . . 6  |-  B  e. 
_V
1514, 14xpex 4801 . . . . 5  |-  ( B  X.  B )  e. 
_V
1615, 14mpt2ex 6198 . . . 4  |-  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) )  e.  _V
17 df-cco 13233 . . . . . 6  |- comp  = Slot ; 1 5
18 1nn0 9981 . . . . . . 7  |-  1  e.  NN0
19 5nn 9880 . . . . . . 7  |-  5  e.  NN
2018, 19decnncl 10137 . . . . . 6  |- ; 1 5  e.  NN
2117, 20ndxid 13169 . . . . 5  |- comp  = Slot  (comp ` 
ndx )
2221setsid 13187 . . . 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 ) ) ) >. ) ) )
2312, 16, 22mp2an 653 . . 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 ) ) ) >. ) )
2411, 23syl6eqr 2333 . 2  |-  ( ph  ->  (comp `  O )  =  ( u  e.  ( B  X.  B
) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) )
25 simprr 733 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
26 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  u  =  <. X ,  Y >. )
2726fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  u )  =  ( 2nd `  <. X ,  Y >. )
)
281adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  X  e.  B )
29 oppcco.y . . . . . . . 8  |-  ( ph  ->  Y  e.  B )
3029adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  Y  e.  B )
31 op2ndg 6133 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
3228, 30, 31syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
3327, 32eqtrd 2315 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  u )  =  Y )
3425, 33opeq12d 3804 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  <. z ,  ( 2nd `  u
) >.  =  <. Z ,  Y >. )
3526fveq2d 5529 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  ( 1st `  <. X ,  Y >. )
)
36 op1stg 6132 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
3728, 30, 36syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
3835, 37eqtrd 2315 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  X )
3934, 38oveq12d 5876 . . 3  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( <. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  =  ( <. Z ,  Y >.  .x.  X )
)
40 tposeq 6236 . . 3  |-  ( (
<. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  =  ( <. Z ,  Y >.  .x.  X )  -> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) )  = tpos  ( <. Z ,  Y >.  .x. 
X ) )
4139, 40syl 15 . 2  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  -> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  = tpos  ( <. Z ,  Y >.  .x.  X )
)
42 opelxpi 4721 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
431, 29, 42syl2anc 642 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
44 oppcco.z . 2  |-  ( ph  ->  Z  e.  B )
45 ovex 5883 . . . 4  |-  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4645tposex 6268 . . 3  |- tpos  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4746a1i 10 . 2  |-  ( ph  -> tpos  ( <. Z ,  Y >.  .x.  X )  e. 
_V )
4824, 41, 43, 44, 47ovmpt2d 5975 1  |-  ( ph  ->  ( <. X ,  Y >. (comp `  O ) Z )  = tpos  ( <. Z ,  Y >.  .x. 
X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121  tpos ctpos 6233   1c1 8738   5c5 9798  ;cdc 10124   ndxcnx 13145   sSet csts 13146   Basecbs 13148    Hom chom 13219  compcco 13220  oppCatcoppc 13614
This theorem is referenced by:  oppcco  13620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-dec 10125  df-ndx 13151  df-slot 13152  df-sets 13154  df-cco 13233  df-oppc 13615
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