Theorem monsect 17680

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	⊢ (𝜑 → 𝐶 ∈ Cat)
sectmon.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
sectmon.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
monsect.n	⊢ 𝑁 = (Inv‘𝐶)
monsect.1	⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌))
monsect.2	⊢ (𝜑 → 𝐺(𝑌𝑆𝑋)𝐹)

Assertion

Ref	Expression
monsect	⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)

Proof of Theorem monsect

Step	Hyp	Ref	Expression
1		monsect.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺(𝑌𝑆𝑋)𝐹)
2		sectmon.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2731	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		eqid 2731	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		sectmon.s	. . . . . . . . 9 ⊢ 𝑆 = (Sect‘𝐶)
7		sectmon.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		sectmon.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		sectmon.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10	2, 3, 4, 5, 6, 7, 8, 9	issect 17650	. . . . . . . 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))))
11	1, 10	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌)))
12	11	simp3d 1144	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
13	12	oveq1d 7377	. . . . 5 ⊢ (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹))
14	11	simp2d 1143	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
15	11	simp1d 1142	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
16	2, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14	catass 17580	. . . . 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)))
17	2, 3, 5, 7, 9, 4, 8, 14	catlid 17577	. . . . . 6 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
18	2, 3, 5, 7, 9, 4, 8, 14	catrid 17578	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
19	17, 18	eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
20	13, 16, 19	3eqtr3d 2779	. . . 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 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌))
23	2, 3, 4, 7, 9, 8, 9, 14, 15	catcocl 17579	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
24	2, 3, 5, 7, 9	catidcl 17576	. . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
25	2, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24	moni 17633	. . . 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)))
26	20, 25	mpbid 231	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
27	2, 3, 4, 5, 6, 7, 9, 8, 14, 15	issect2 17651	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
28	26, 27	mpbird 256	. 2 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
29		monsect.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
30	2, 29, 7, 9, 8, 6	isinv 17657	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)))
31	28, 1, 30	mpbir2and 711	1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ⟨cop 4597   class class class wbr 5110  ‘cfv 6501  (class class class)co 7362  Basecbs 17094  Hom chom 17158  compcco 17159  Catccat 17558  Idccid 17559  Monocmon 17625  Sectcsect 17641  Invcinv 17642

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-cat 17562  df-cid 17563  df-mon 17627  df-sect 17644  df-inv 17645

This theorem is referenced by:  episect  17682