Theorem invco 16431

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

Step	Hyp	Ref	Expression
1		invfval.b	. . 3 |- B = ( Base ` C )
2		invco.o	. . 3 |- .x. = (comp ` C )
3		eqid 2622	. . 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 )
11	1, 9, 4, 5, 6, 10	isoval 16425	. . . . . . 7 |- ( ph -> ( X I Y ) = dom ( X N Y ) )
12	8, 11	eleqtrd 2703	. . . . . 6 |- ( ph -> F e. dom ( X N Y ) )
13	1, 9, 4, 5, 6	invfun 16424	. . . . . . 7 |- ( ph -> Fun ( X N Y ) )
14		funfvbrb 6330	. . . . . . 7 |- ( Fun ( X N Y ) -> ( F e. dom ( X N Y ) <-> F ( X N Y ) ( ( X N Y ) ` F ) ) )
15	13, 14	syl 17	. . . . . 6 |- ( ph -> ( F e. dom ( X N Y ) <-> F ( X N Y ) ( ( X N Y ) ` F ) ) )
16	12, 15	mpbid 222	. . . . 5 |- ( ph -> F ( X N Y ) ( ( X N Y ) ` F ) )
17	1, 9, 4, 5, 6, 3	isinv 16420	. . . . 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 ) ) )
18	16, 17	mpbid 222	. . . 4 |- ( ph -> ( F ( X (Sect ` C ) Y ) ( ( X N Y ) ` F ) /\ ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F ) )
19	18	simpld 475	. . 3 |- ( ph -> F ( X (Sect ` C ) Y ) ( ( X N Y ) ` F ) )
20		invco.f	. . . . . . 7 |- ( ph -> G e. ( Y I Z ) )
21	1, 9, 4, 6, 7, 10	isoval 16425	. . . . . . 7 |- ( ph -> ( Y I Z ) = dom ( Y N Z ) )
22	20, 21	eleqtrd 2703	. . . . . 6 |- ( ph -> G e. dom ( Y N Z ) )
23	1, 9, 4, 6, 7	invfun 16424	. . . . . . 7 |- ( ph -> Fun ( Y N Z ) )
24		funfvbrb 6330	. . . . . . 7 |- ( Fun ( Y N Z ) -> ( G e. dom ( Y N Z ) <-> G ( Y N Z ) ( ( Y N Z ) ` G ) ) )
25	23, 24	syl 17	. . . . . 6 |- ( ph -> ( G e. dom ( Y N Z ) <-> G ( Y N Z ) ( ( Y N Z ) ` G ) ) )
26	22, 25	mpbid 222	. . . . 5 |- ( ph -> G ( Y N Z ) ( ( Y N Z ) ` G ) )
27	1, 9, 4, 6, 7, 3	isinv 16420	. . . . 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 ) ) )
28	26, 27	mpbid 222	. . . 4 |- ( ph -> ( G ( Y (Sect ` C ) Z ) ( ( Y N Z ) ` G ) /\ ( ( Y N Z ) ` G ) ( Z (Sect ` C ) Y ) G ) )
29	28	simpld 475	. . 3 |- ( ph -> G ( Y (Sect ` C ) Z ) ( ( Y N Z ) ` G ) )
30	1, 2, 3, 4, 5, 6, 7, 19, 29	sectco 16416	. 2 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) ( X (Sect ` C ) Z ) ( ( ( X N Y ) ` F ) ( <. Z , Y >. .x. X ) ( ( Y N Z ) ` G ) ) )
31	28	simprd 479	. . 3 |- ( ph -> ( ( Y N Z ) ` G ) ( Z (Sect ` C ) Y ) G )
32	18	simprd 479	. . 3 |- ( ph -> ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F )
33	1, 2, 3, 4, 7, 6, 5, 31, 32	sectco 16416	. 2 |- ( ph -> ( ( ( X N Y ) ` F ) ( <. Z , Y >. .x. X ) ( ( Y N Z ) ` G ) ) ( Z (Sect ` C ) X ) ( G ( <. X , Y >. .x. Z ) F ) )
34	1, 9, 4, 5, 7, 3	isinv 16420	. 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 ) ) ) )
35	30, 33, 34	mpbir2and 957	1 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) ( X N Z ) ( ( ( X N Y ) ` F ) ( <. Z , Y >. .x. X ) ( ( Y N Z ) ` G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   Basecbs 15857  compcco 15953   Catccat 16325  Sectcsect 16404  Invcinv 16405   Iso ciso 16406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409

This theorem is referenced by:  isoco  16437  invisoinvl  16450