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Theorem monsect 17049
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 (𝜑𝐶 ∈ Cat)
sectmon.x (𝜑𝑋𝐵)
sectmon.y (𝜑𝑌𝐵)
monsect.n 𝑁 = (Inv‘𝐶)
monsect.1 (𝜑𝐹 ∈ (𝑋𝑀𝑌))
monsect.2 (𝜑𝐺(𝑌𝑆𝑋)𝐹)
Assertion
Ref Expression
monsect (𝜑𝐹(𝑋𝑁𝑌)𝐺)

Proof of Theorem monsect
StepHypRef Expression
1 monsect.2 . . . . . . . 8 (𝜑𝐺(𝑌𝑆𝑋)𝐹)
2 sectmon.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
3 eqid 2801 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2801 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2801 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
6 sectmon.s . . . . . . . . 9 𝑆 = (Sect‘𝐶)
7 sectmon.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
8 sectmon.y . . . . . . . . 9 (𝜑𝑌𝐵)
9 sectmon.x . . . . . . . . 9 (𝜑𝑋𝐵)
102, 3, 4, 5, 6, 7, 8, 9issect 17019 . . . . . . . 8 (𝜑 → (𝐺(𝑌𝑆𝑋)𝐹 ↔ (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))))
111, 10mpbid 235 . . . . . . 7 (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌)))
1211simp3d 1141 . . . . . 6 (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
1312oveq1d 7154 . . . . 5 (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹))
1411simp2d 1140 . . . . . 6 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
1511simp1d 1139 . . . . . 6 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 16953 . . . . 5 (𝜑 → ((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)))
172, 3, 5, 7, 9, 4, 8, 14catlid 16950 . . . . . 6 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
182, 3, 5, 7, 9, 4, 8, 14catrid 16951 . . . . . 6 (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
1917, 18eqtr4d 2839 . . . . 5 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
2013, 16, 193eqtr3d 2844 . . . 4 (𝜑 → (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 (𝜑𝐹 ∈ (𝑋𝑀𝑌))
232, 3, 4, 7, 9, 8, 9, 14, 15catcocl 16952 . . . . 5 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
242, 3, 5, 7, 9catidcl 16949 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 17002 . . . 4 (𝜑 → ((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)))
2620, 25mpbid 235 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 17020 . . 3 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
2826, 27mpbird 260 . 2 (𝜑𝐹(𝑋𝑆𝑌)𝐺)
29 monsect.n . . 3 𝑁 = (Inv‘𝐶)
302, 29, 7, 9, 8, 6isinv 17026 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺𝐺(𝑌𝑆𝑋)𝐹)))
3128, 1, 30mpbir2and 712 1 (𝜑𝐹(𝑋𝑁𝑌)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  cop 4534   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16479  Hom chom 16572  compcco 16573  Catccat 16931  Idccid 16932  Monocmon 16994  Sectcsect 17010  Invcinv 17011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-cat 16935  df-cid 16936  df-mon 16996  df-sect 17013  df-inv 17014
This theorem is referenced by:  episect  17051
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