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Theorem monsect 16643
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 2806 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2806 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2806 . . . . . . . . 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 16613 . . . . . . . 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 223 . . . . . . 7 (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌)))
1211simp3d 1167 . . . . . 6 (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
1312oveq1d 6885 . . . . 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 1166 . . . . . 6 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
1511simp1d 1165 . . . . . 6 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 16547 . . . . 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 16544 . . . . . 6 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
182, 3, 5, 7, 9, 4, 8, 14catrid 16545 . . . . . 6 (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
1917, 18eqtr4d 2843 . . . . 5 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
2013, 16, 193eqtr3d 2848 . . . 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 16546 . . . . 5 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
242, 3, 5, 7, 9catidcl 16543 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 16596 . . . 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 223 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 16614 . . 3 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
2826, 27mpbird 248 . 2 (𝜑𝐹(𝑋𝑆𝑌)𝐺)
29 monsect.n . . 3 𝑁 = (Inv‘𝐶)
302, 29, 7, 9, 8, 6isinv 16620 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺𝐺(𝑌𝑆𝑋)𝐹)))
3128, 1, 30mpbir2and 695 1 (𝜑𝐹(𝑋𝑁𝑌)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1100   = wceq 1637  wcel 2156  cop 4376   class class class wbr 4844  cfv 6097  (class class class)co 6870  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16525  Idccid 16526  Monocmon 16588  Sectcsect 16604  Invcinv 16605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-1st 7394  df-2nd 7395  df-cat 16529  df-cid 16530  df-mon 16590  df-sect 16607  df-inv 16608
This theorem is referenced by:  episect  16645
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