Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  monsect Structured version   Unicode version

Theorem monsect 15198
 Description: If is a monomorphism and is a section of , then is an inverse of 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
sectmon.m Mono
sectmon.s Sect
sectmon.c
sectmon.x
sectmon.y
monsect.n Inv
monsect.1
monsect.2
Assertion
Ref Expression
monsect

Proof of Theorem monsect
StepHypRef Expression
1 monsect.2 . . . . . . . 8
2 sectmon.b . . . . . . . . 9
3 eqid 2457 . . . . . . . . 9
4 eqid 2457 . . . . . . . . 9 comp comp
5 eqid 2457 . . . . . . . . 9
6 sectmon.s . . . . . . . . 9 Sect
7 sectmon.c . . . . . . . . 9
8 sectmon.y . . . . . . . . 9
9 sectmon.x . . . . . . . . 9
102, 3, 4, 5, 6, 7, 8, 9issect 15168 . . . . . . . 8 comp
111, 10mpbid 210 . . . . . . 7 comp
1211simp3d 1010 . . . . . 6 comp
1312oveq1d 6311 . . . . 5 comp comp comp
1411simp2d 1009 . . . . . 6
1511simp1d 1008 . . . . . 6
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 15102 . . . . 5 comp comp comp comp
172, 3, 5, 7, 9, 4, 8, 14catlid 15099 . . . . . 6 comp
182, 3, 5, 7, 9, 4, 8, 14catrid 15100 . . . . . 6 comp
1917, 18eqtr4d 2501 . . . . 5 comp comp
2013, 16, 193eqtr3d 2506 . . . 4 comp comp comp
21 sectmon.m . . . . 5 Mono
22 monsect.1 . . . . 5
232, 3, 4, 7, 9, 8, 9, 14, 15catcocl 15101 . . . . 5 comp
242, 3, 5, 7, 9catidcl 15098 . . . . 5
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 15151 . . . 4 comp comp comp comp
2620, 25mpbid 210 . . 3 comp
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 15169 . . 3 comp
2826, 27mpbird 232 . 2
29 monsect.n . . 3 Inv
302, 29, 7, 9, 8, 6isinv 15175 . 2
3128, 1, 30mpbir2and 922 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3a 973   wceq 1395   wcel 1819  cop 4038   class class class wbr 4456  cfv 5594  (class class class)co 6296  cbs 14643   chom 14722  compcco 14723  ccat 15080  ccid 15081  Monocmon 15143  Sectcsect 15159  Invcinv 15160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-cat 15084  df-cid 15085  df-mon 15145  df-sect 15162  df-inv 15163 This theorem is referenced by:  episect  15200
 Copyright terms: Public domain W3C validator