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Theorem xpcco1st 14990
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpcco1st.t  |-  T  =  ( C  X.c  D )
xpcco1st.b  |-  B  =  ( Base `  T
)
xpcco1st.k  |-  K  =  ( Hom  `  T
)
xpcco1st.o  |-  O  =  (comp `  T )
xpcco1st.x  |-  ( ph  ->  X  e.  B )
xpcco1st.y  |-  ( ph  ->  Y  e.  B )
xpcco1st.z  |-  ( ph  ->  Z  e.  B )
xpcco1st.f  |-  ( ph  ->  F  e.  ( X K Y ) )
xpcco1st.g  |-  ( ph  ->  G  e.  ( Y K Z ) )
xpcco1st.1  |-  .x.  =  (comp `  C )
Assertion
Ref Expression
xpcco1st  |-  ( ph  ->  ( 1st `  ( G ( <. X ,  Y >. O Z ) F ) )  =  ( ( 1st `  G
) ( <. ( 1st `  X ) ,  ( 1st `  Y
) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) )

Proof of Theorem xpcco1st
StepHypRef Expression
1 xpcco1st.t . . 3  |-  T  =  ( C  X.c  D )
2 xpcco1st.b . . 3  |-  B  =  ( Base `  T
)
3 xpcco1st.k . . 3  |-  K  =  ( Hom  `  T
)
4 xpcco1st.1 . . 3  |-  .x.  =  (comp `  C )
5 eqid 2441 . . 3  |-  (comp `  D )  =  (comp `  D )
6 xpcco1st.o . . 3  |-  O  =  (comp `  T )
7 xpcco1st.x . . 3  |-  ( ph  ->  X  e.  B )
8 xpcco1st.y . . 3  |-  ( ph  ->  Y  e.  B )
9 xpcco1st.z . . 3  |-  ( ph  ->  Z  e.  B )
10 xpcco1st.f . . 3  |-  ( ph  ->  F  e.  ( X K Y ) )
11 xpcco1st.g . . 3  |-  ( ph  ->  G  e.  ( Y K Z ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11xpcco 14989 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  <. ( ( 1st `  G ) ( <.
( 1st `  X
) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F
) ) ,  ( ( 2nd `  G
) ( <. ( 2nd `  X ) ,  ( 2nd `  Y
) >. (comp `  D
) ( 2nd `  Z
) ) ( 2nd `  F ) ) >.
)
13 ovex 6115 . . 3  |-  ( ( 1st `  G ) ( <. ( 1st `  X
) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F
) )  e.  _V
14 ovex 6115 . . 3  |-  ( ( 2nd `  G ) ( <. ( 2nd `  X
) ,  ( 2nd `  Y ) >. (comp `  D ) ( 2nd `  Z ) ) ( 2nd `  F ) )  e.  _V
1513, 14op1std 6586 . 2  |-  ( ( G ( <. X ,  Y >. O Z ) F )  =  <. ( ( 1st `  G
) ( <. ( 1st `  X ) ,  ( 1st `  Y
) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) ,  ( ( 2nd `  G ) ( <. ( 2nd `  X
) ,  ( 2nd `  Y ) >. (comp `  D ) ( 2nd `  Z ) ) ( 2nd `  F ) ) >.  ->  ( 1st `  ( G ( <. X ,  Y >. O Z ) F ) )  =  ( ( 1st `  G ) ( <. ( 1st `  X
) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F
) ) )
1612, 15syl 16 1  |-  ( ph  ->  ( 1st `  ( G ( <. X ,  Y >. O Z ) F ) )  =  ( ( 1st `  G
) ( <. ( 1st `  X ) ,  ( 1st `  Y
) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   <.cop 3880   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   Hom chom 14245  compcco 14246    X.c cxpc 14974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-hom 14258  df-cco 14259  df-xpc 14978
This theorem is referenced by:  1stfcl  15003
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