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Theorem xpcco2 15658
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 15649 . . 3  |-  ( X  X.  Y )  =  ( Base `  T
)
5 eqid 2454 . . 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 5020 . . . 4  |-  ( ( M  e.  X  /\  N  e.  Y )  -> 
<. M ,  N >.  e.  ( X  X.  Y
) )
129, 10, 11syl2anc 659 . . 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 5020 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  Y )  -> 
<. P ,  Q >.  e.  ( X  X.  Y
) )
1613, 14, 15syl2anc 659 . . 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 5020 . . . 4  |-  ( ( R  e.  X  /\  S  e.  Y )  -> 
<. R ,  S >.  e.  ( X  X.  Y
) )
2017, 18, 19syl2anc 659 . . 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 5020 . . . . 5  |-  ( ( F  e.  ( M H P )  /\  G  e.  ( N J Q ) )  ->  <. F ,  G >.  e.  ( ( M H P )  X.  ( N J Q ) ) )
2421, 22, 23syl2anc 659 . . . 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 15657 . . . 4  |-  ( ph  ->  ( <. M ,  N >. ( Hom  `  T
) <. P ,  Q >. )  =  ( ( M H P )  X.  ( N J Q ) ) )
2824, 27eleqtrrd 2545 . . 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 5020 . . . . 5  |-  ( ( K  e.  ( P H R )  /\  L  e.  ( Q J S ) )  ->  <. K ,  L >.  e.  ( ( P H R )  X.  ( Q J S ) ) )
3229, 30, 31syl2anc 659 . . . 4  |-  ( ph  -> 
<. K ,  L >.  e.  ( ( P H R )  X.  ( Q J S ) ) )
331, 2, 3, 25, 26, 13, 14, 17, 18, 5xpchom2 15657 . . . 4  |-  ( ph  ->  ( <. P ,  Q >. ( Hom  `  T
) <. R ,  S >. )  =  ( ( P H R )  X.  ( Q J S ) ) )
3432, 33eleqtrrd 2545 . . 3  |-  ( ph  -> 
<. K ,  L >.  e.  ( <. P ,  Q >. ( Hom  `  T
) <. R ,  S >. ) )
351, 4, 5, 6, 7, 8, 12, 16, 20, 28, 34xpcco 15654 . 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 6785 . . . . . . 7  |-  ( ( M  e.  X  /\  N  e.  Y )  ->  ( 1st `  <. M ,  N >. )  =  M )
379, 10, 36syl2anc 659 . . . . . 6  |-  ( ph  ->  ( 1st `  <. M ,  N >. )  =  M )
38 op1stg 6785 . . . . . . 7  |-  ( ( P  e.  X  /\  Q  e.  Y )  ->  ( 1st `  <. P ,  Q >. )  =  P )
3913, 14, 38syl2anc 659 . . . . . 6  |-  ( ph  ->  ( 1st `  <. P ,  Q >. )  =  P )
4037, 39opeq12d 4211 . . . . 5  |-  ( ph  -> 
<. ( 1st `  <. M ,  N >. ) ,  ( 1st `  <. P ,  Q >. ) >.  =  <. M ,  P >. )
41 op1stg 6785 . . . . . 6  |-  ( ( R  e.  X  /\  S  e.  Y )  ->  ( 1st `  <. R ,  S >. )  =  R )
4217, 18, 41syl2anc 659 . . . . 5  |-  ( ph  ->  ( 1st `  <. R ,  S >. )  =  R )
4340, 42oveq12d 6288 . . . 4  |-  ( ph  ->  ( <. ( 1st `  <. M ,  N >. ) ,  ( 1st `  <. P ,  Q >. ) >.  .x.  ( 1st `  <. R ,  S >. )
)  =  ( <. M ,  P >.  .x. 
R ) )
44 op1stg 6785 . . . . 5  |-  ( ( K  e.  ( P H R )  /\  L  e.  ( Q J S ) )  -> 
( 1st `  <. K ,  L >. )  =  K )
4529, 30, 44syl2anc 659 . . . 4  |-  ( ph  ->  ( 1st `  <. K ,  L >. )  =  K )
46 op1stg 6785 . . . . 5  |-  ( ( F  e.  ( M H P )  /\  G  e.  ( N J Q ) )  -> 
( 1st `  <. F ,  G >. )  =  F )
4721, 22, 46syl2anc 659 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
4843, 45, 47oveq123d 6291 . . 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 6786 . . . . . . 7  |-  ( ( M  e.  X  /\  N  e.  Y )  ->  ( 2nd `  <. M ,  N >. )  =  N )
509, 10, 49syl2anc 659 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. M ,  N >. )  =  N )
51 op2ndg 6786 . . . . . . 7  |-  ( ( P  e.  X  /\  Q  e.  Y )  ->  ( 2nd `  <. P ,  Q >. )  =  Q )
5213, 14, 51syl2anc 659 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. P ,  Q >. )  =  Q )
5350, 52opeq12d 4211 . . . . 5  |-  ( ph  -> 
<. ( 2nd `  <. M ,  N >. ) ,  ( 2nd `  <. P ,  Q >. ) >.  =  <. N ,  Q >. )
54 op2ndg 6786 . . . . . 6  |-  ( ( R  e.  X  /\  S  e.  Y )  ->  ( 2nd `  <. R ,  S >. )  =  S )
5517, 18, 54syl2anc 659 . . . . 5  |-  ( ph  ->  ( 2nd `  <. R ,  S >. )  =  S )
5653, 55oveq12d 6288 . . . 4  |-  ( ph  ->  ( <. ( 2nd `  <. M ,  N >. ) ,  ( 2nd `  <. P ,  Q >. ) >. 
.xb  ( 2nd `  <. R ,  S >. )
)  =  ( <. N ,  Q >.  .xb 
S ) )
57 op2ndg 6786 . . . . 5  |-  ( ( K  e.  ( P H R )  /\  L  e.  ( Q J S ) )  -> 
( 2nd `  <. K ,  L >. )  =  L )
5829, 30, 57syl2anc 659 . . . 4  |-  ( ph  ->  ( 2nd `  <. K ,  L >. )  =  L )
59 op2ndg 6786 . . . . 5  |-  ( ( F  e.  ( M H P )  /\  G  e.  ( N J Q ) )  -> 
( 2nd `  <. F ,  G >. )  =  G )
6021, 22, 59syl2anc 659 . . . 4  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
6156, 58, 60oveq123d 6291 . . 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 4211 . 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 2495 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 1398    e. wcel 1823   <.cop 4022    X. cxp 4986   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   Basecbs 14719   Hom chom 14798  compcco 14799    X.c cxpc 15639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-hom 14811  df-cco 14812  df-xpc 15643
This theorem is referenced by:  prfcl  15674  evlfcllem  15692  curf1cl  15699  curf2cl  15702  curfcl  15703  uncfcurf  15710  hofcl  15730
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