Theorem monsect 16443

Description: If F is a monomorphism and G is a section of F, then G is an inverse of F and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
sectmon.b	|- B = ( Base ` C )
sectmon.m	|- M = (Mono ` C )
sectmon.s	|- S = (Sect ` C )
sectmon.c	|- ( ph -> C e. Cat )
sectmon.x	|- ( ph -> X e. B )
sectmon.y	|- ( ph -> Y e. B )
monsect.n	|- N = (Inv ` C )
monsect.1	|- ( ph -> F e. ( X M Y ) )
monsect.2	|- ( ph -> G ( Y S X ) F )

Assertion

Ref	Expression
monsect	|- ( ph -> F ( X N Y ) G )

Proof of Theorem monsect

Step	Hyp	Ref	Expression
1		monsect.2	. . . . . . . 8 |- ( ph -> G ( Y S X ) F )
2		sectmon.b	. . . . . . . . 9 |- B = ( Base ` C )
3		eqid 2622	. . . . . . . . 9 |- ( Hom ` C ) = ( Hom ` C )
4		eqid 2622	. . . . . . . . 9 |- (comp ` C ) = (comp ` C )
5		eqid 2622	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
6		sectmon.s	. . . . . . . . 9 |- S = (Sect ` C )
7		sectmon.c	. . . . . . . . 9 |- ( ph -> C e. Cat )
8		sectmon.y	. . . . . . . . 9 |- ( ph -> Y e. B )
9		sectmon.x	. . . . . . . . 9 |- ( ph -> X e. B )
10	2, 3, 4, 5, 6, 7, 8, 9	issect 16413	. . . . . . . 8 |- ( ph -> ( G ( Y S X ) F <-> ( G e. ( Y ( Hom ` C ) X ) /\ F e. ( X ( Hom ` C ) Y ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) G ) = ( ( Id ` C ) ` Y ) ) ) )
11	1, 10	mpbid 222	. . . . . . 7 |- ( ph -> ( G e. ( Y ( Hom ` C ) X ) /\ F e. ( X ( Hom ` C ) Y ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) G ) = ( ( Id ` C ) ` Y ) ) )
12	11	simp3d 1075	. . . . . 6 |- ( ph -> ( F ( <. Y , X >. (comp ` C ) Y ) G ) = ( ( Id ` C ) ` Y ) )
13	12	oveq1d 6665	. . . . 5 |- ( ph -> ( ( F ( <. Y , X >. (comp ` C ) Y ) G ) ( <. X , Y >. (comp ` C ) Y ) F ) = ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) F ) )
14	11	simp2d 1074	. . . . . 6 |- ( ph -> F e. ( X ( Hom ` C ) Y ) )
15	11	simp1d 1073	. . . . . 6 |- ( ph -> G e. ( Y ( Hom ` C ) X ) )
16	2, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14	catass 16347	. . . . 5 |- ( ph -> ( ( F ( <. Y , X >. (comp ` C ) Y ) G ) ( <. X , Y >. (comp ` C ) Y ) F ) = ( F ( <. X , X >. (comp ` C ) Y ) ( G ( <. X , Y >. (comp ` C ) X ) F ) ) )
17	2, 3, 5, 7, 9, 4, 8, 14	catlid 16344	. . . . . 6 |- ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) F ) = F )
18	2, 3, 5, 7, 9, 4, 8, 14	catrid 16345	. . . . . 6 |- ( ph -> ( F ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) = F )
19	17, 18	eqtr4d 2659	. . . . 5 |- ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) F ) = ( F ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) )
20	13, 16, 19	3eqtr3d 2664	. . . 4 |- ( ph -> ( F ( <. X , X >. (comp ` C ) Y ) ( G ( <. X , Y >. (comp ` C ) X ) F ) ) = ( F ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) )
21		sectmon.m	. . . . 5 |- M = (Mono ` C )
22		monsect.1	. . . . 5 |- ( ph -> F e. ( X M Y ) )
23	2, 3, 4, 7, 9, 8, 9, 14, 15	catcocl 16346	. . . . 5 |- ( ph -> ( G ( <. X , Y >. (comp ` C ) X ) F ) e. ( X ( Hom ` C ) X ) )
24	2, 3, 5, 7, 9	catidcl 16343	. . . . 5 |- ( ph -> ( ( Id ` C ) ` X ) e. ( X ( Hom ` C ) X ) )
25	2, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24	moni 16396	. . . 4 |- ( ph -> ( ( F ( <. X , X >. (comp ` C ) Y ) ( G ( <. X , Y >. (comp ` C ) X ) F ) ) = ( F ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) <-> ( G ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) )
26	20, 25	mpbid 222	. . 3 |- ( ph -> ( G ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) )
27	2, 3, 4, 5, 6, 7, 9, 8, 14, 15	issect2 16414	. . 3 |- ( ph -> ( F ( X S Y ) G <-> ( G ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) )
28	26, 27	mpbird 247	. 2 |- ( ph -> F ( X S Y ) G )
29		monsect.n	. . 3 |- N = (Inv ` C )
30	2, 29, 7, 9, 8, 6	isinv 16420	. 2 |- ( ph -> ( F ( X N Y ) G <-> ( F ( X S Y ) G /\ G ( Y S X ) F ) ) )
31	28, 1, 30	mpbir2and 957	1 |- ( ph -> F ( X N Y ) G )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Monocmon 16388  Sectcsect 16404  Invcinv 16405

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-mon 16390  df-sect 16407  df-inv 16408

This theorem is referenced by:  episect  16445