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Theorem sectco 15172
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 2457 . . . 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 2457 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
10 sectco.s . . . . . . . 8  |-  S  =  (Sect `  C )
111, 2, 3, 9, 10, 4, 5, 7issect 15169 . . . . . . 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 210 . . . . . 6  |-  ( ph  ->  ( F  e.  ( X ( Hom  `  C
) Y )  /\  G  e.  ( Y
( Hom  `  C ) X )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  ( ( Id `  C
) `  X )
) )
1312simp1d 1008 . . . . 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 15169 . . . . . . 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 210 . . . . . 6  |-  ( ph  ->  ( H  e.  ( Y ( Hom  `  C
) Z )  /\  K  e.  ( Z
( Hom  `  C ) Y )  /\  ( K ( <. Y ,  Z >.  .x.  Y ) H )  =  ( ( Id `  C
) `  Y )
) )
1716simp1d 1008 . . . . 5  |-  ( ph  ->  H  e.  ( Y ( Hom  `  C
) Z ) )
181, 2, 3, 4, 5, 7, 6, 13, 17catcocl 15102 . . . 4  |-  ( ph  ->  ( H ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X ( Hom  `  C ) Z ) )
1916simp2d 1009 . . . 4  |-  ( ph  ->  K  e.  ( Z ( Hom  `  C
) Y ) )
2012simp2d 1009 . . . 4  |-  ( ph  ->  G  e.  ( Y ( Hom  `  C
) X ) )
211, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20catass 15103 . . 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 1010 . . . . . 6  |-  ( ph  ->  ( K ( <. Y ,  Z >.  .x. 
Y ) H )  =  ( ( Id
`  C ) `  Y ) )
2322oveq1d 6311 . . . . 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 15103 . . . . 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 15100 . . . . 5  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >.  .x. 
Y ) F )  =  F )
2623, 24, 253eqtr3d 2506 . . . 4  |-  ( ph  ->  ( K ( <. X ,  Z >.  .x. 
Y ) ( H ( <. X ,  Y >.  .x.  Z ) F ) )  =  F )
2726oveq2d 6312 . . 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 1010 . . 3  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
X ) F )  =  ( ( Id
`  C ) `  X ) )
2921, 27, 283eqtrd 2502 . 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 15102 . . 3  |-  ( ph  ->  ( G ( <. Z ,  Y >.  .x. 
X ) K )  e.  ( Z ( Hom  `  C ) X ) )
311, 2, 3, 9, 10, 4, 5, 6, 18, 30issect2 15170 . 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 232 1  |-  ( ph  ->  ( H ( <. X ,  Y >.  .x. 
Z ) F ) ( X S Z ) ( G (
<. Z ,  Y >.  .x. 
X ) K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   Hom chom 14723  compcco 14724   Catccat 15081   Idccid 15082  Sectcsect 15160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-cat 15085  df-cid 15086  df-sect 15163
This theorem is referenced by:  invco  15187
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