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Theorem monsect 16649
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 2770 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2770 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2770 . . . . . . . . 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 16619 . . . . . . . 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 1137 . . . . . 6 (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
1312oveq1d 6807 . . . . 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 1136 . . . . . 6 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
1511simp1d 1135 . . . . . 6 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 16553 . . . . 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 16550 . . . . . 6 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
182, 3, 5, 7, 9, 4, 8, 14catrid 16551 . . . . . 6 (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
1917, 18eqtr4d 2807 . . . . 5 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
2013, 16, 193eqtr3d 2812 . . . 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 16552 . . . . 5 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
242, 3, 5, 7, 9catidcl 16549 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 16602 . . . 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 16620 . . 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 16626 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺𝐺(𝑌𝑆𝑋)𝐹)))
3128, 1, 30mpbir2and 684 1 (𝜑𝐹(𝑋𝑁𝑌)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1070   = wceq 1630  wcel 2144  cop 4320   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  Hom chom 16159  compcco 16160  Catccat 16531  Idccid 16532  Monocmon 16594  Sectcsect 16610  Invcinv 16611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-cat 16535  df-cid 16536  df-mon 16596  df-sect 16613  df-inv 16614
This theorem is referenced by:  episect  16651
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