Theorem monsect 17958

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 2760	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2760	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		eqid 2760	. . . . . . . . 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 17928	. . . . . . . 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 1162	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐺) = ((Id‘𝐶)‘𝑌))
13	12	oveq1d 7435	. . . . 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 1161	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
15	11	simp1d 1160	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
16	2, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14	catass 17860	. . . . 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 17857	. . . . . 6 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
18	2, 3, 5, 7, 9, 4, 8, 14	catrid 17858	. . . . . 6 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐹)
19	17, 18	eqtr4d 2798	. . . . 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 2803	. . . 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 17859	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑋))
24	2, 3, 5, 7, 9	catidcl 17856	. . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
25	2, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24	moni 17911	. . . 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 17929	. . 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 17935	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)))
31	28, 1, 30	mpbir2and 726	1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ⟨cop 4589   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom chom 17439  compcco 17440  Catccat 17838  Idccid 17839  Monocmon 17903  Sectcsect 17919  Invcinv 17920

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-cat 17842  df-cid 17843  df-mon 17905  df-sect 17922  df-inv 17923

This theorem is used by:  episect  17960