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Theorem oppccofval 13635
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 5571 . . . . . . 7  |-  ( X  e.  ( Base `  C
)  ->  C  e.  _V )
3 oppcco.b . . . . . . 7  |-  B  =  ( Base `  C
)
42, 3eleq2s 2388 . . . . . 6  |-  ( X  e.  B  ->  C  e.  _V )
51, 4syl 15 . . . . 5  |-  ( ph  ->  C  e.  _V )
6 eqid 2296 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
7 oppcco.c . . . . . 6  |-  .x.  =  (comp `  C )
8 oppcco.o . . . . . 6  |-  O  =  (oppCat `  C )
93, 6, 7, 8oppcval 13632 . . . . 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 5545 . . 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 5899 . . . 4  |-  ( C sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  C
) >. )  e.  _V
13 fvex 5555 . . . . . . 7  |-  ( Base `  C )  e.  _V
143, 13eqeltri 2366 . . . . . 6  |-  B  e. 
_V
1514, 14xpex 4817 . . . . 5  |-  ( B  X.  B )  e. 
_V
1615, 14mpt2ex 6214 . . . 4  |-  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <.
z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) )  e.  _V
17 df-cco 13249 . . . . . 6  |- comp  = Slot ; 1 5
18 1nn0 9997 . . . . . . 7  |-  1  e.  NN0
19 5nn 9896 . . . . . . 7  |-  5  e.  NN
2018, 19decnncl 10153 . . . . . 6  |- ; 1 5  e.  NN
2117, 20ndxid 13185 . . . . 5  |- comp  = Slot  (comp ` 
ndx )
2221setsid 13203 . . . 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 2346 . 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 5545 . . . . . 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 6149 . . . . . . 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 2328 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  u )  =  Y )
3425, 33opeq12d 3820 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  <. z ,  ( 2nd `  u
) >.  =  <. Z ,  Y >. )
3526fveq2d 5545 . . . . 5  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  ( 1st `  <. X ,  Y >. )
)
36 op1stg 6148 . . . . . 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 2328 . . . 4  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  u )  =  X )
3934, 38oveq12d 5892 . . 3  |-  ( (
ph  /\  ( u  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( <. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) )  =  ( <. Z ,  Y >.  .x.  X )
)
40 tposeq 6252 . . 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 4737 . . 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 5899 . . . 4  |-  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4645tposex 6284 . . 3  |- tpos  ( <. Z ,  Y >.  .x. 
X )  e.  _V
4746a1i 10 . 2  |-  ( ph  -> tpos  ( <. Z ,  Y >.  .x.  X )  e. 
_V )
4824, 41, 43, 44, 47ovmpt2d 5991 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137  tpos ctpos 6249   1c1 8754   5c5 9814  ;cdc 10140   ndxcnx 13161   sSet csts 13162   Basecbs 13164    Hom chom 13235  compcco 13236  oppCatcoppc 13630
This theorem is referenced by:  oppcco  13636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-dec 10141  df-ndx 13167  df-slot 13168  df-sets 13170  df-cco 13249  df-oppc 13631
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