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Theorem sectco 13659
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 2283 . . . 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 2283 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
10 sectco.s . . . . . . . 8  |-  S  =  (Sect `  C )
111, 2, 3, 9, 10, 4, 5, 7issect 13656 . . . . . . 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 201 . . . . . 6  |-  ( ph  ->  ( F  e.  ( X (  Hom  `  C
) Y )  /\  G  e.  ( Y
(  Hom  `  C ) X )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  ( ( Id `  C
) `  X )
) )
1312simp1d 967 . . . . 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 13656 . . . . . . 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 201 . . . . . 6  |-  ( ph  ->  ( H  e.  ( Y (  Hom  `  C
) Z )  /\  K  e.  ( Z
(  Hom  `  C ) Y )  /\  ( K ( <. Y ,  Z >.  .x.  Y ) H )  =  ( ( Id `  C
) `  Y )
) )
1716simp1d 967 . . . . 5  |-  ( ph  ->  H  e.  ( Y (  Hom  `  C
) Z ) )
181, 2, 3, 4, 5, 7, 6, 13, 17catcocl 13587 . . . 4  |-  ( ph  ->  ( H ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X (  Hom  `  C ) Z ) )
1916simp2d 968 . . . 4  |-  ( ph  ->  K  e.  ( Z (  Hom  `  C
) Y ) )
2012simp2d 968 . . . 4  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) X ) )
211, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20catass 13588 . . 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 969 . . . . . 6  |-  ( ph  ->  ( K ( <. Y ,  Z >.  .x. 
Y ) H )  =  ( ( Id
`  C ) `  Y ) )
2322oveq1d 5873 . . . . 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 13588 . . . . 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 13585 . . . . 5  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >.  .x. 
Y ) F )  =  F )
2623, 24, 253eqtr3d 2323 . . . 4  |-  ( ph  ->  ( K ( <. X ,  Z >.  .x. 
Y ) ( H ( <. X ,  Y >.  .x.  Z ) F ) )  =  F )
2726oveq2d 5874 . . 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 969 . . 3  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
X ) F )  =  ( ( Id
`  C ) `  X ) )
2921, 27, 283eqtrd 2319 . 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 13587 . . 3  |-  ( ph  ->  ( G ( <. Z ,  Y >.  .x. 
X ) K )  e.  ( Z (  Hom  `  C ) X ) )
311, 2, 3, 9, 10, 4, 5, 6, 18, 30issect2 13657 . 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 223 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 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Sectcsect 13647
This theorem is referenced by:  invco  13673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-cat 13570  df-cid 13571  df-sect 13650
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