Theorem monsect 17839

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 2761	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2761	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		eqid 2761	. . . . . . . . 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 17809	. . . . . . . 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 235	. . . . . . 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 1160	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
13	12	oveq1d 7425	. . . . 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 1159	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
15	11	simp1d 1158	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
16	2, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14	catass 17741	. . . . 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 17738	. . . . . 6 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
18	2, 3, 5, 7, 9, 4, 8, 14	catrid 17739	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
19	17, 18	eqtr4d 2799	. . . . 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 2804	. . . 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 17740	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
24	2, 3, 5, 7, 9	catidcl 17737	. . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
25	2, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24	moni 17792	. . . 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 235	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
27	2, 3, 4, 5, 6, 7, 9, 8, 14, 15	issect2 17810	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
28	26, 27	mpbird 260	. 2 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
29		monsect.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
30	2, 29, 7, 9, 8, 6	isinv 17816	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)))
31	28, 1, 30	mpbir2and 725	1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ⟨cop 4594   class class class wbr 5108  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  Hom chom 17320  compcco 17321  Catccat 17719  Idccid 17720  Monocmon 17784  Sectcsect 17800  Invcinv 17801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-cat 17723  df-cid 17724  df-mon 17786  df-sect 17803  df-inv 17804

This theorem is referenced by:  episect  17841