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Theorem invco 13984
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 2435 . . 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 13978 . . . . . . 7  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
128, 11eleqtrd 2511 . . . . . 6  |-  ( ph  ->  F  e.  dom  ( X N Y ) )
131, 9, 4, 5, 6invfun 13977 . . . . . . 7  |-  ( ph  ->  Fun  ( X N Y ) )
14 funfvbrb 5834 . . . . . . 7  |-  ( Fun  ( X N Y )  ->  ( F  e.  dom  ( X N Y )  <->  F ( X N Y ) ( ( X N Y ) `  F ) ) )
1513, 14syl 16 . . . . . 6  |-  ( ph  ->  ( F  e.  dom  ( X N Y )  <-> 
F ( X N Y ) ( ( X N Y ) `
 F ) ) )
1612, 15mpbid 202 . . . . 5  |-  ( ph  ->  F ( X N Y ) ( ( X N Y ) `
 F ) )
171, 9, 4, 5, 6, 3isinv 13973 . . . . 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 202 . . . 4  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) ( ( X N Y ) `
 F )  /\  ( ( X N Y ) `  F
) ( Y (Sect `  C ) X ) F ) )
1918simpld 446 . . 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 13978 . . . . . . 7  |-  ( ph  ->  ( Y I Z )  =  dom  ( Y N Z ) )
2220, 21eleqtrd 2511 . . . . . 6  |-  ( ph  ->  G  e.  dom  ( Y N Z ) )
231, 9, 4, 6, 7invfun 13977 . . . . . . 7  |-  ( ph  ->  Fun  ( Y N Z ) )
24 funfvbrb 5834 . . . . . . 7  |-  ( Fun  ( Y N Z )  ->  ( G  e.  dom  ( Y N Z )  <->  G ( Y N Z ) ( ( Y N Z ) `  G ) ) )
2523, 24syl 16 . . . . . 6  |-  ( ph  ->  ( G  e.  dom  ( Y N Z )  <-> 
G ( Y N Z ) ( ( Y N Z ) `
 G ) ) )
2622, 25mpbid 202 . . . . 5  |-  ( ph  ->  G ( Y N Z ) ( ( Y N Z ) `
 G ) )
271, 9, 4, 6, 7, 3isinv 13973 . . . . 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 202 . . . 4  |-  ( ph  ->  ( G ( Y (Sect `  C ) Z ) ( ( Y N Z ) `
 G )  /\  ( ( Y N Z ) `  G
) ( Z (Sect `  C ) Y ) G ) )
2928simpld 446 . . 3  |-  ( ph  ->  G ( Y (Sect `  C ) Z ) ( ( Y N Z ) `  G
) )
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 13970 . 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 450 . . 3  |-  ( ph  ->  ( ( Y N Z ) `  G
) ( Z (Sect `  C ) Y ) G )
3218simprd 450 . . 3  |-  ( ph  ->  ( ( X N Y ) `  F
) ( Y (Sect `  C ) X ) F )
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 13970 . 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 13973 . 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 889 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 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204   dom cdm 4869   Fun wfun 5439   ` cfv 5445  (class class class)co 6072   Basecbs 13457  compcco 13529   Catccat 13877  Sectcsect 13958  Invcinv 13959    Iso ciso 13960
This theorem is referenced by:  isoco  13986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-cat 13881  df-cid 13882  df-sect 13961  df-inv 13962  df-iso 13963
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