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Theorem invco 13667
Description: The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b  |-  B  =  ( Base `  C
)
invfval.n  |-  N  =  (Inv `  C )
invfval.c  |-  ( ph  ->  C  e.  Cat )
invfval.x  |-  ( ph  ->  X  e.  B )
invfval.y  |-  ( ph  ->  Y  e.  B )
isoval.n  |-  I  =  (  Iso  `  C
)
invinv.f  |-  ( ph  ->  F  e.  ( X I Y ) )
invco.o  |-  .x.  =  (comp `  C )
invco.z  |-  ( ph  ->  Z  e.  B )
invco.f  |-  ( ph  ->  G  e.  ( Y I Z ) )
Assertion
Ref Expression
invco  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F ) ( X N Z ) ( ( ( X N Y ) `
 F ) (
<. Z ,  Y >.  .x. 
X ) ( ( Y N Z ) `
 G ) ) )

Proof of Theorem invco
StepHypRef Expression
1 invfval.b . . 3  |-  B  =  ( Base `  C
)
2 invco.o . . 3  |-  .x.  =  (comp `  C )
3 eqid 2284 . . 3  |-  (Sect `  C )  =  (Sect `  C )
4 invfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
5 invfval.x . . 3  |-  ( ph  ->  X  e.  B )
6 invfval.y . . 3  |-  ( ph  ->  Y  e.  B )
7 invco.z . . 3  |-  ( ph  ->  Z  e.  B )
8 invinv.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X I Y ) )
9 invfval.n . . . . . . . 8  |-  N  =  (Inv `  C )
10 isoval.n . . . . . . . 8  |-  I  =  (  Iso  `  C
)
111, 9, 4, 5, 6, 10isoval 13661 . . . . . . 7  |-  ( ph  ->  ( X I Y )  =  dom  (  X N Y ) )
128, 11eleqtrd 2360 . . . . . 6  |-  ( ph  ->  F  e.  dom  (  X N Y ) )
131, 9, 4, 5, 6invfun 13660 . . . . . . 7  |-  ( ph  ->  Fun  ( X N Y ) )
14 funfvbrb 5599 . . . . . . 7  |-  ( Fun  ( X N Y )  ->  ( F  e.  dom  (  X N Y )  <->  F ( X N Y ) ( ( X N Y ) `  F ) ) )
1513, 14syl 17 . . . . . 6  |-  ( ph  ->  ( F  e.  dom  (  X N Y )  <-> 
F ( X N Y ) ( ( X N Y ) `
 F ) ) )
1612, 15mpbid 203 . . . . 5  |-  ( ph  ->  F ( X N Y ) ( ( X N Y ) `
 F ) )
171, 9, 4, 5, 6, 3isinv 13656 . . . . 5  |-  ( ph  ->  ( F ( X N Y ) ( ( X N Y ) `  F )  <-> 
( F ( X (Sect `  C ) Y ) ( ( X N Y ) `
 F )  /\  ( ( X N Y ) `  F
) ( Y (Sect `  C ) X ) F ) ) )
1816, 17mpbid 203 . . . 4  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) ( ( X N Y ) `
 F )  /\  ( ( X N Y ) `  F
) ( Y (Sect `  C ) X ) F ) )
1918simpld 447 . . 3  |-  ( ph  ->  F ( X (Sect `  C ) Y ) ( ( X N Y ) `  F
) )
20 invco.f . . . . . . 7  |-  ( ph  ->  G  e.  ( Y I Z ) )
211, 9, 4, 6, 7, 10isoval 13661 . . . . . . 7  |-  ( ph  ->  ( Y I Z )  =  dom  (  Y N Z ) )
2220, 21eleqtrd 2360 . . . . . 6  |-  ( ph  ->  G  e.  dom  (  Y N Z ) )
231, 9, 4, 6, 7invfun 13660 . . . . . . 7  |-  ( ph  ->  Fun  ( Y N Z ) )
24 funfvbrb 5599 . . . . . . 7  |-  ( Fun  ( Y N Z )  ->  ( G  e.  dom  (  Y N Z )  <->  G ( Y N Z ) ( ( Y N Z ) `  G ) ) )
2523, 24syl 17 . . . . . 6  |-  ( ph  ->  ( G  e.  dom  (  Y N Z )  <-> 
G ( Y N Z ) ( ( Y N Z ) `
 G ) ) )
2622, 25mpbid 203 . . . . 5  |-  ( ph  ->  G ( Y N Z ) ( ( Y N Z ) `
 G ) )
271, 9, 4, 6, 7, 3isinv 13656 . . . . 5  |-  ( ph  ->  ( G ( Y N Z ) ( ( Y N Z ) `  G )  <-> 
( G ( Y (Sect `  C ) Z ) ( ( Y N Z ) `
 G )  /\  ( ( Y N Z ) `  G
) ( Z (Sect `  C ) Y ) G ) ) )
2826, 27mpbid 203 . . . 4  |-  ( ph  ->  ( G ( Y (Sect `  C ) Z ) ( ( Y N Z ) `
 G )  /\  ( ( Y N Z ) `  G
) ( Z (Sect `  C ) Y ) G ) )
2928simpld 447 . . 3  |-  ( ph  ->  G ( Y (Sect `  C ) Z ) ( ( Y N Z ) `  G
) )
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 13653 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F ) ( X (Sect `  C ) Z ) ( ( ( X N Y ) `  F ) ( <. Z ,  Y >.  .x. 
X ) ( ( Y N Z ) `
 G ) ) )
3128simprd 451 . . 3  |-  ( ph  ->  ( ( Y N Z ) `  G
) ( Z (Sect `  C ) Y ) G )
3218simprd 451 . . 3  |-  ( ph  ->  ( ( X N Y ) `  F
) ( Y (Sect `  C ) X ) F )
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 13653 . 2  |-  ( ph  ->  ( ( ( X N Y ) `  F ) ( <. Z ,  Y >.  .x. 
X ) ( ( Y N Z ) `
 G ) ) ( Z (Sect `  C ) X ) ( G ( <. X ,  Y >.  .x. 
Z ) F ) )
341, 9, 4, 5, 7, 3isinv 13656 . 2  |-  ( ph  ->  ( ( G (
<. X ,  Y >.  .x. 
Z ) F ) ( X N Z ) ( ( ( X N Y ) `
 F ) (
<. Z ,  Y >.  .x. 
X ) ( ( Y N Z ) `
 G ) )  <-> 
( ( G (
<. X ,  Y >.  .x. 
Z ) F ) ( X (Sect `  C ) Z ) ( ( ( X N Y ) `  F ) ( <. Z ,  Y >.  .x. 
X ) ( ( Y N Z ) `
 G ) )  /\  ( ( ( X N Y ) `
 F ) (
<. Z ,  Y >.  .x. 
X ) ( ( Y N Z ) `
 G ) ) ( Z (Sect `  C ) X ) ( G ( <. X ,  Y >.  .x. 
Z ) F ) ) ) )
3530, 33, 34mpbir2and 890 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F ) ( X N Z ) ( ( ( X N Y ) `
 F ) (
<. Z ,  Y >.  .x. 
X ) ( ( Y N Z ) `
 G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   <.cop 3644   class class class wbr 4024   dom cdm 4688   Fun wfun 5215   ` cfv 5221  (class class class)co 5819   Basecbs 13142  compcco 13214   Catccat 13560  Sectcsect 13641  Invcinv 13642    Iso ciso 13643
This theorem is referenced by:  isoco  13669
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-cat 13564  df-cid 13565  df-sect 13644  df-inv 13645  df-iso 13646
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