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Theorem xpcco2 15096
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpcco2.t  |-  T  =  ( C  X.c  D )
xpcco2.x  |-  X  =  ( Base `  C
)
xpcco2.y  |-  Y  =  ( Base `  D
)
xpcco2.h  |-  H  =  ( Hom  `  C
)
xpcco2.j  |-  J  =  ( Hom  `  D
)
xpcco2.m  |-  ( ph  ->  M  e.  X )
xpcco2.n  |-  ( ph  ->  N  e.  Y )
xpcco2.p  |-  ( ph  ->  P  e.  X )
xpcco2.q  |-  ( ph  ->  Q  e.  Y )
xpcco2.o1  |-  .x.  =  (comp `  C )
xpcco2.o2  |-  .xb  =  (comp `  D )
xpcco2.o  |-  O  =  (comp `  T )
xpcco2.r  |-  ( ph  ->  R  e.  X )
xpcco2.s  |-  ( ph  ->  S  e.  Y )
xpcco2.f  |-  ( ph  ->  F  e.  ( M H P ) )
xpcco2.g  |-  ( ph  ->  G  e.  ( N J Q ) )
xpcco2.k  |-  ( ph  ->  K  e.  ( P H R ) )
xpcco2.l  |-  ( ph  ->  L  e.  ( Q J S ) )
Assertion
Ref Expression
xpcco2  |-  ( ph  ->  ( <. K ,  L >. ( <. <. M ,  N >. ,  <. P ,  Q >. >. O <. R ,  S >. ) <. F ,  G >. )  =  <. ( K ( <. M ,  P >.  .x.  R ) F ) ,  ( L ( <. N ,  Q >.  .xb  S ) G ) >. )

Proof of Theorem xpcco2
StepHypRef Expression
1 xpcco2.t . . 3  |-  T  =  ( C  X.c  D )
2 xpcco2.x . . . 4  |-  X  =  ( Base `  C
)
3 xpcco2.y . . . 4  |-  Y  =  ( Base `  D
)
41, 2, 3xpcbas 15087 . . 3  |-  ( X  X.  Y )  =  ( Base `  T
)
5 eqid 2451 . . 3  |-  ( Hom  `  T )  =  ( Hom  `  T )
6 xpcco2.o1 . . 3  |-  .x.  =  (comp `  C )
7 xpcco2.o2 . . 3  |-  .xb  =  (comp `  D )
8 xpcco2.o . . 3  |-  O  =  (comp `  T )
9 xpcco2.m . . . 4  |-  ( ph  ->  M  e.  X )
10 xpcco2.n . . . 4  |-  ( ph  ->  N  e.  Y )
11 opelxpi 4966 . . . 4  |-  ( ( M  e.  X  /\  N  e.  Y )  -> 
<. M ,  N >.  e.  ( X  X.  Y
) )
129, 10, 11syl2anc 661 . . 3  |-  ( ph  -> 
<. M ,  N >.  e.  ( X  X.  Y
) )
13 xpcco2.p . . . 4  |-  ( ph  ->  P  e.  X )
14 xpcco2.q . . . 4  |-  ( ph  ->  Q  e.  Y )
15 opelxpi 4966 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  Y )  -> 
<. P ,  Q >.  e.  ( X  X.  Y
) )
1613, 14, 15syl2anc 661 . . 3  |-  ( ph  -> 
<. P ,  Q >.  e.  ( X  X.  Y
) )
17 xpcco2.r . . . 4  |-  ( ph  ->  R  e.  X )
18 xpcco2.s . . . 4  |-  ( ph  ->  S  e.  Y )
19 opelxpi 4966 . . . 4  |-  ( ( R  e.  X  /\  S  e.  Y )  -> 
<. R ,  S >.  e.  ( X  X.  Y
) )
2017, 18, 19syl2anc 661 . . 3  |-  ( ph  -> 
<. R ,  S >.  e.  ( X  X.  Y
) )
21 xpcco2.f . . . . 5  |-  ( ph  ->  F  e.  ( M H P ) )
22 xpcco2.g . . . . 5  |-  ( ph  ->  G  e.  ( N J Q ) )
23 opelxpi 4966 . . . . 5  |-  ( ( F  e.  ( M H P )  /\  G  e.  ( N J Q ) )  ->  <. F ,  G >.  e.  ( ( M H P )  X.  ( N J Q ) ) )
2421, 22, 23syl2anc 661 . . . 4  |-  ( ph  -> 
<. F ,  G >.  e.  ( ( M H P )  X.  ( N J Q ) ) )
25 xpcco2.h . . . . 5  |-  H  =  ( Hom  `  C
)
26 xpcco2.j . . . . 5  |-  J  =  ( Hom  `  D
)
271, 2, 3, 25, 26, 9, 10, 13, 14, 5xpchom2 15095 . . . 4  |-  ( ph  ->  ( <. M ,  N >. ( Hom  `  T
) <. P ,  Q >. )  =  ( ( M H P )  X.  ( N J Q ) ) )
2824, 27eleqtrrd 2540 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( <. M ,  N >. ( Hom  `  T
) <. P ,  Q >. ) )
29 xpcco2.k . . . . 5  |-  ( ph  ->  K  e.  ( P H R ) )
30 xpcco2.l . . . . 5  |-  ( ph  ->  L  e.  ( Q J S ) )
31 opelxpi 4966 . . . . 5  |-  ( ( K  e.  ( P H R )  /\  L  e.  ( Q J S ) )  ->  <. K ,  L >.  e.  ( ( P H R )  X.  ( Q J S ) ) )
3229, 30, 31syl2anc 661 . . . 4  |-  ( ph  -> 
<. K ,  L >.  e.  ( ( P H R )  X.  ( Q J S ) ) )
331, 2, 3, 25, 26, 13, 14, 17, 18, 5xpchom2 15095 . . . 4  |-  ( ph  ->  ( <. P ,  Q >. ( Hom  `  T
) <. R ,  S >. )  =  ( ( P H R )  X.  ( Q J S ) ) )
3432, 33eleqtrrd 2540 . . 3  |-  ( ph  -> 
<. K ,  L >.  e.  ( <. P ,  Q >. ( Hom  `  T
) <. R ,  S >. ) )
351, 4, 5, 6, 7, 8, 12, 16, 20, 28, 34xpcco 15092 . 2  |-  ( ph  ->  ( <. K ,  L >. ( <. <. M ,  N >. ,  <. P ,  Q >. >. O <. R ,  S >. ) <. F ,  G >. )  =  <. ( ( 1st `  <. K ,  L >. )
( <. ( 1st `  <. M ,  N >. ) ,  ( 1st `  <. P ,  Q >. ) >.  .x.  ( 1st `  <. R ,  S >. )
) ( 1st `  <. F ,  G >. )
) ,  ( ( 2nd `  <. K ,  L >. ) ( <.
( 2nd `  <. M ,  N >. ) ,  ( 2nd `  <. P ,  Q >. ) >. 
.xb  ( 2nd `  <. R ,  S >. )
) ( 2nd `  <. F ,  G >. )
) >. )
36 op1stg 6686 . . . . . . 7  |-  ( ( M  e.  X  /\  N  e.  Y )  ->  ( 1st `  <. M ,  N >. )  =  M )
379, 10, 36syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 1st `  <. M ,  N >. )  =  M )
38 op1stg 6686 . . . . . . 7  |-  ( ( P  e.  X  /\  Q  e.  Y )  ->  ( 1st `  <. P ,  Q >. )  =  P )
3913, 14, 38syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 1st `  <. P ,  Q >. )  =  P )
4037, 39opeq12d 4162 . . . . 5  |-  ( ph  -> 
<. ( 1st `  <. M ,  N >. ) ,  ( 1st `  <. P ,  Q >. ) >.  =  <. M ,  P >. )
41 op1stg 6686 . . . . . 6  |-  ( ( R  e.  X  /\  S  e.  Y )  ->  ( 1st `  <. R ,  S >. )  =  R )
4217, 18, 41syl2anc 661 . . . . 5  |-  ( ph  ->  ( 1st `  <. R ,  S >. )  =  R )
4340, 42oveq12d 6205 . . . 4  |-  ( ph  ->  ( <. ( 1st `  <. M ,  N >. ) ,  ( 1st `  <. P ,  Q >. ) >.  .x.  ( 1st `  <. R ,  S >. )
)  =  ( <. M ,  P >.  .x. 
R ) )
44 op1stg 6686 . . . . 5  |-  ( ( K  e.  ( P H R )  /\  L  e.  ( Q J S ) )  -> 
( 1st `  <. K ,  L >. )  =  K )
4529, 30, 44syl2anc 661 . . . 4  |-  ( ph  ->  ( 1st `  <. K ,  L >. )  =  K )
46 op1stg 6686 . . . . 5  |-  ( ( F  e.  ( M H P )  /\  G  e.  ( N J Q ) )  -> 
( 1st `  <. F ,  G >. )  =  F )
4721, 22, 46syl2anc 661 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
4843, 45, 47oveq123d 6208 . . 3  |-  ( ph  ->  ( ( 1st `  <. K ,  L >. )
( <. ( 1st `  <. M ,  N >. ) ,  ( 1st `  <. P ,  Q >. ) >.  .x.  ( 1st `  <. R ,  S >. )
) ( 1st `  <. F ,  G >. )
)  =  ( K ( <. M ,  P >.  .x.  R ) F ) )
49 op2ndg 6687 . . . . . . 7  |-  ( ( M  e.  X  /\  N  e.  Y )  ->  ( 2nd `  <. M ,  N >. )  =  N )
509, 10, 49syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. M ,  N >. )  =  N )
51 op2ndg 6687 . . . . . . 7  |-  ( ( P  e.  X  /\  Q  e.  Y )  ->  ( 2nd `  <. P ,  Q >. )  =  Q )
5213, 14, 51syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. P ,  Q >. )  =  Q )
5350, 52opeq12d 4162 . . . . 5  |-  ( ph  -> 
<. ( 2nd `  <. M ,  N >. ) ,  ( 2nd `  <. P ,  Q >. ) >.  =  <. N ,  Q >. )
54 op2ndg 6687 . . . . . 6  |-  ( ( R  e.  X  /\  S  e.  Y )  ->  ( 2nd `  <. R ,  S >. )  =  S )
5517, 18, 54syl2anc 661 . . . . 5  |-  ( ph  ->  ( 2nd `  <. R ,  S >. )  =  S )
5653, 55oveq12d 6205 . . . 4  |-  ( ph  ->  ( <. ( 2nd `  <. M ,  N >. ) ,  ( 2nd `  <. P ,  Q >. ) >. 
.xb  ( 2nd `  <. R ,  S >. )
)  =  ( <. N ,  Q >.  .xb 
S ) )
57 op2ndg 6687 . . . . 5  |-  ( ( K  e.  ( P H R )  /\  L  e.  ( Q J S ) )  -> 
( 2nd `  <. K ,  L >. )  =  L )
5829, 30, 57syl2anc 661 . . . 4  |-  ( ph  ->  ( 2nd `  <. K ,  L >. )  =  L )
59 op2ndg 6687 . . . . 5  |-  ( ( F  e.  ( M H P )  /\  G  e.  ( N J Q ) )  -> 
( 2nd `  <. F ,  G >. )  =  G )
6021, 22, 59syl2anc 661 . . . 4  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
6156, 58, 60oveq123d 6208 . . 3  |-  ( ph  ->  ( ( 2nd `  <. K ,  L >. )
( <. ( 2nd `  <. M ,  N >. ) ,  ( 2nd `  <. P ,  Q >. ) >. 
.xb  ( 2nd `  <. R ,  S >. )
) ( 2nd `  <. F ,  G >. )
)  =  ( L ( <. N ,  Q >. 
.xb  S ) G ) )
6248, 61opeq12d 4162 . 2  |-  ( ph  -> 
<. ( ( 1st `  <. K ,  L >. )
( <. ( 1st `  <. M ,  N >. ) ,  ( 1st `  <. P ,  Q >. ) >.  .x.  ( 1st `  <. R ,  S >. )
) ( 1st `  <. F ,  G >. )
) ,  ( ( 2nd `  <. K ,  L >. ) ( <.
( 2nd `  <. M ,  N >. ) ,  ( 2nd `  <. P ,  Q >. ) >. 
.xb  ( 2nd `  <. R ,  S >. )
) ( 2nd `  <. F ,  G >. )
) >.  =  <. ( K ( <. M ,  P >.  .x.  R ) F ) ,  ( L ( <. N ,  Q >.  .xb  S ) G ) >. )
6335, 62eqtrd 2491 1  |-  ( ph  ->  ( <. K ,  L >. ( <. <. M ,  N >. ,  <. P ,  Q >. >. O <. R ,  S >. ) <. F ,  G >. )  =  <. ( K ( <. M ,  P >.  .x.  R ) F ) ,  ( L ( <. N ,  Q >.  .xb  S ) G ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   <.cop 3978    X. cxp 4933   ` cfv 5513  (class class class)co 6187   1stc1st 6672   2ndc2nd 6673   Basecbs 14273   Hom chom 14348  compcco 14349    X.c cxpc 15077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-fz 11536  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-hom 14361  df-cco 14362  df-xpc 15081
This theorem is referenced by:  prfcl  15112  evlfcllem  15130  curf1cl  15137  curf2cl  15140  curfcl  15141  uncfcurf  15148  hofcl  15168
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