Theorem monsect 17739

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 2735	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2735	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		eqid 2735	. . . . . . . . 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 17709	. . . . . . . 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 232	. . . . . . 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 1145	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
13	12	oveq1d 7371	. . . . 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 1144	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
15	11	simp1d 1143	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
16	2, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14	catass 17641	. . . . 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 17638	. . . . . 6 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
18	2, 3, 5, 7, 9, 4, 8, 14	catrid 17639	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
19	17, 18	eqtr4d 2773	. . . . 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 2778	. . . 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 17640	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
24	2, 3, 5, 7, 9	catidcl 17637	. . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
25	2, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24	moni 17692	. . . 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 232	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
27	2, 3, 4, 5, 6, 7, 9, 8, 14, 15	issect2 17710	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
28	26, 27	mpbird 257	. 2 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
29		monsect.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
30	2, 29, 7, 9, 8, 6	isinv 17716	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)))
31	28, 1, 30	mpbir2and 714	1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4563   class class class wbr 5074  ‘cfv 6487  (class class class)co 7356  Basecbs 17168  Hom chom 17220  compcco 17221  Catccat 17619  Idccid 17620  Monocmon 17684  Sectcsect 17700  Invcinv 17701

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-cat 17623  df-cid 17624  df-mon 17686  df-sect 17703  df-inv 17704

This theorem is referenced by:  episect  17741