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Theorem invco 13673
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 2283 . . 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 13667 . . . . . . 7  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
128, 11eleqtrd 2359 . . . . . 6  |-  ( ph  ->  F  e.  dom  ( X N Y ) )
131, 9, 4, 5, 6invfun 13666 . . . . . . 7  |-  ( ph  ->  Fun  ( X N Y ) )
14 funfvbrb 5638 . . . . . . 7  |-  ( Fun  ( X N Y )  ->  ( F  e.  dom  ( X N Y )  <->  F ( X N Y ) ( ( X N Y ) `  F ) ) )
1513, 14syl 15 . . . . . 6  |-  ( ph  ->  ( F  e.  dom  ( X N Y )  <-> 
F ( X N Y ) ( ( X N Y ) `
 F ) ) )
1612, 15mpbid 201 . . . . 5  |-  ( ph  ->  F ( X N Y ) ( ( X N Y ) `
 F ) )
171, 9, 4, 5, 6, 3isinv 13662 . . . . 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 201 . . . 4  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) ( ( X N Y ) `
 F )  /\  ( ( X N Y ) `  F
) ( Y (Sect `  C ) X ) F ) )
1918simpld 445 . . 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 13667 . . . . . . 7  |-  ( ph  ->  ( Y I Z )  =  dom  ( Y N Z ) )
2220, 21eleqtrd 2359 . . . . . 6  |-  ( ph  ->  G  e.  dom  ( Y N Z ) )
231, 9, 4, 6, 7invfun 13666 . . . . . . 7  |-  ( ph  ->  Fun  ( Y N Z ) )
24 funfvbrb 5638 . . . . . . 7  |-  ( Fun  ( Y N Z )  ->  ( G  e.  dom  ( Y N Z )  <->  G ( Y N Z ) ( ( Y N Z ) `  G ) ) )
2523, 24syl 15 . . . . . 6  |-  ( ph  ->  ( G  e.  dom  ( Y N Z )  <-> 
G ( Y N Z ) ( ( Y N Z ) `
 G ) ) )
2622, 25mpbid 201 . . . . 5  |-  ( ph  ->  G ( Y N Z ) ( ( Y N Z ) `
 G ) )
271, 9, 4, 6, 7, 3isinv 13662 . . . . 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 201 . . . 4  |-  ( ph  ->  ( G ( Y (Sect `  C ) Z ) ( ( Y N Z ) `
 G )  /\  ( ( Y N Z ) `  G
) ( Z (Sect `  C ) Y ) G ) )
2928simpld 445 . . 3  |-  ( ph  ->  G ( Y (Sect `  C ) Z ) ( ( Y N Z ) `  G
) )
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 13659 . 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 449 . . 3  |-  ( ph  ->  ( ( Y N Z ) `  G
) ( Z (Sect `  C ) Y ) G )
3218simprd 449 . . 3  |-  ( ph  ->  ( ( X N Y ) `  F
) ( Y (Sect `  C ) X ) F )
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 13659 . 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 13662 . 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 888 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   dom cdm 4689   Fun wfun 5249   ` cfv 5255  (class class class)co 5858   Basecbs 13148  compcco 13220   Catccat 13566  Sectcsect 13647  Invcinv 13648    Iso ciso 13649
This theorem is referenced by:  isoco  13675
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  df-inv 13651  df-iso 13652
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