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Theorem monsect 16383
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 2621 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2621 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2621 . . . . . . . . 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 16353 . . . . . . . 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 222 . . . . . . 7 (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌)))
1211simp3d 1073 . . . . . 6 (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
1312oveq1d 6630 . . . . 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 1072 . . . . . 6 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
1511simp1d 1071 . . . . . 6 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 16287 . . . . 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 16284 . . . . . 6 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
182, 3, 5, 7, 9, 4, 8, 14catrid 16285 . . . . . 6 (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
1917, 18eqtr4d 2658 . . . . 5 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
2013, 16, 193eqtr3d 2663 . . . 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 16286 . . . . 5 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
242, 3, 5, 7, 9catidcl 16283 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 16336 . . . 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 222 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 16354 . . 3 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
2826, 27mpbird 247 . 2 (𝜑𝐹(𝑋𝑆𝑌)𝐺)
29 monsect.n . . 3 𝑁 = (Inv‘𝐶)
302, 29, 7, 9, 8, 6isinv 16360 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺𝐺(𝑌𝑆𝑋)𝐹)))
3128, 1, 30mpbir2and 956 1 (𝜑𝐹(𝑋𝑁𝑌)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  cop 4161   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  Hom chom 15892  compcco 15893  Catccat 16265  Idccid 16266  Monocmon 16328  Sectcsect 16344  Invcinv 16345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-cat 16269  df-cid 16270  df-mon 16330  df-sect 16347  df-inv 16348
This theorem is referenced by:  episect  16385
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