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Theorem sectco 13945
Description: Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
sectco.b  |-  B  =  ( Base `  C
)
sectco.o  |-  .x.  =  (comp `  C )
sectco.s  |-  S  =  (Sect `  C )
sectco.c  |-  ( ph  ->  C  e.  Cat )
sectco.x  |-  ( ph  ->  X  e.  B )
sectco.y  |-  ( ph  ->  Y  e.  B )
sectco.z  |-  ( ph  ->  Z  e.  B )
sectco.1  |-  ( ph  ->  F ( X S Y ) G )
sectco.2  |-  ( ph  ->  H ( Y S Z ) K )
Assertion
Ref Expression
sectco  |-  ( ph  ->  ( H ( <. X ,  Y >.  .x. 
Z ) F ) ( X S Z ) ( G (
<. Z ,  Y >.  .x. 
X ) K ) )

Proof of Theorem sectco
StepHypRef Expression
1 sectco.b . . . 4  |-  B  =  ( Base `  C
)
2 eqid 2412 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 sectco.o . . . 4  |-  .x.  =  (comp `  C )
4 sectco.c . . . 4  |-  ( ph  ->  C  e.  Cat )
5 sectco.x . . . 4  |-  ( ph  ->  X  e.  B )
6 sectco.z . . . 4  |-  ( ph  ->  Z  e.  B )
7 sectco.y . . . 4  |-  ( ph  ->  Y  e.  B )
8 sectco.1 . . . . . . 7  |-  ( ph  ->  F ( X S Y ) G )
9 eqid 2412 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
10 sectco.s . . . . . . . 8  |-  S  =  (Sect `  C )
111, 2, 3, 9, 10, 4, 5, 7issect 13942 . . . . . . 7  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( F  e.  ( X (  Hom  `  C
) Y )  /\  G  e.  ( Y
(  Hom  `  C ) X )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  ( ( Id `  C
) `  X )
) ) )
128, 11mpbid 202 . . . . . 6  |-  ( ph  ->  ( F  e.  ( X (  Hom  `  C
) Y )  /\  G  e.  ( Y
(  Hom  `  C ) X )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  ( ( Id `  C
) `  X )
) )
1312simp1d 969 . . . . 5  |-  ( ph  ->  F  e.  ( X (  Hom  `  C
) Y ) )
14 sectco.2 . . . . . . 7  |-  ( ph  ->  H ( Y S Z ) K )
151, 2, 3, 9, 10, 4, 7, 6issect 13942 . . . . . . 7  |-  ( ph  ->  ( H ( Y S Z ) K  <-> 
( H  e.  ( Y (  Hom  `  C
) Z )  /\  K  e.  ( Z
(  Hom  `  C ) Y )  /\  ( K ( <. Y ,  Z >.  .x.  Y ) H )  =  ( ( Id `  C
) `  Y )
) ) )
1614, 15mpbid 202 . . . . . 6  |-  ( ph  ->  ( H  e.  ( Y (  Hom  `  C
) Z )  /\  K  e.  ( Z
(  Hom  `  C ) Y )  /\  ( K ( <. Y ,  Z >.  .x.  Y ) H )  =  ( ( Id `  C
) `  Y )
) )
1716simp1d 969 . . . . 5  |-  ( ph  ->  H  e.  ( Y (  Hom  `  C
) Z ) )
181, 2, 3, 4, 5, 7, 6, 13, 17catcocl 13873 . . . 4  |-  ( ph  ->  ( H ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X (  Hom  `  C ) Z ) )
1916simp2d 970 . . . 4  |-  ( ph  ->  K  e.  ( Z (  Hom  `  C
) Y ) )
2012simp2d 970 . . . 4  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) X ) )
211, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20catass 13874 . . 3  |-  ( ph  ->  ( ( G (
<. Z ,  Y >.  .x. 
X ) K ) ( <. X ,  Z >.  .x.  X ) ( H ( <. X ,  Y >.  .x.  Z ) F ) )  =  ( G ( <. X ,  Y >.  .x. 
X ) ( K ( <. X ,  Z >.  .x.  Y ) ( H ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
2216simp3d 971 . . . . . 6  |-  ( ph  ->  ( K ( <. Y ,  Z >.  .x. 
Y ) H )  =  ( ( Id
`  C ) `  Y ) )
2322oveq1d 6063 . . . . 5  |-  ( ph  ->  ( ( K (
<. Y ,  Z >.  .x. 
Y ) H ) ( <. X ,  Y >.  .x.  Y ) F )  =  ( ( ( Id `  C
) `  Y )
( <. X ,  Y >.  .x.  Y ) F ) )
241, 2, 3, 4, 5, 7, 6, 13, 17, 7, 19catass 13874 . . . . 5  |-  ( ph  ->  ( ( K (
<. Y ,  Z >.  .x. 
Y ) H ) ( <. X ,  Y >.  .x.  Y ) F )  =  ( K ( <. X ,  Z >.  .x.  Y ) ( H ( <. X ,  Y >.  .x.  Z ) F ) ) )
251, 2, 9, 4, 5, 3, 7, 13catlid 13871 . . . . 5  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >.  .x. 
Y ) F )  =  F )
2623, 24, 253eqtr3d 2452 . . . 4  |-  ( ph  ->  ( K ( <. X ,  Z >.  .x. 
Y ) ( H ( <. X ,  Y >.  .x.  Z ) F ) )  =  F )
2726oveq2d 6064 . . 3  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
X ) ( K ( <. X ,  Z >.  .x.  Y ) ( H ( <. X ,  Y >.  .x.  Z ) F ) ) )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
2812simp3d 971 . . 3  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
X ) F )  =  ( ( Id
`  C ) `  X ) )
2921, 27, 283eqtrd 2448 . 2  |-  ( ph  ->  ( ( G (
<. Z ,  Y >.  .x. 
X ) K ) ( <. X ,  Z >.  .x.  X ) ( H ( <. X ,  Y >.  .x.  Z ) F ) )  =  ( ( Id `  C ) `  X
) )
301, 2, 3, 4, 6, 7, 5, 19, 20catcocl 13873 . . 3  |-  ( ph  ->  ( G ( <. Z ,  Y >.  .x. 
X ) K )  e.  ( Z (  Hom  `  C ) X ) )
311, 2, 3, 9, 10, 4, 5, 6, 18, 30issect2 13943 . 2  |-  ( ph  ->  ( ( H (
<. X ,  Y >.  .x. 
Z ) F ) ( X S Z ) ( G (
<. Z ,  Y >.  .x. 
X ) K )  <-> 
( ( G (
<. Z ,  Y >.  .x. 
X ) K ) ( <. X ,  Z >.  .x.  X ) ( H ( <. X ,  Y >.  .x.  Z ) F ) )  =  ( ( Id `  C ) `  X
) ) )
3229, 31mpbird 224 1  |-  ( ph  ->  ( H ( <. X ,  Y >.  .x. 
Z ) F ) ( X S Z ) ( G (
<. Z ,  Y >.  .x. 
X ) K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3785   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432    Hom chom 13503  compcco 13504   Catccat 13852   Idccid 13853  Sectcsect 13933
This theorem is referenced by:  invco  13959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-cat 13856  df-cid 13857  df-sect 13936
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