Theorem sectco 16239

Description: Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
sectco.b	⊢ 𝐵 = (Base‘𝐶)
sectco.o	⊢ · = (comp‘𝐶)
sectco.s	⊢ 𝑆 = (Sect‘𝐶)
sectco.c	⊢ (𝜑 → 𝐶 ∈ Cat)
sectco.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
sectco.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
sectco.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
sectco.1	⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
sectco.2	⊢ (𝜑 → 𝐻(𝑌𝑆𝑍)𝐾)

Assertion

Ref	Expression
sectco	⊢ (𝜑 → (𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾))

Proof of Theorem sectco

Step	Hyp	Ref	Expression
1		sectco.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2610	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		sectco.o	. . . 4 ⊢ · = (comp‘𝐶)
4		sectco.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		sectco.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		sectco.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
7		sectco.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		sectco.1	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
9		eqid 2610	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		sectco.s	. . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶)
11	1, 2, 3, 9, 10, 4, 5, 7	issect 16236	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
12	8, 11	mpbid 221	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
13	12	simp1d 1066	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
14		sectco.2	. . . . . . 7 ⊢ (𝜑 → 𝐻(𝑌𝑆𝑍)𝐾)
15	1, 2, 3, 9, 10, 4, 7, 6	issect 16236	. . . . . . 7 ⊢ (𝜑 → (𝐻(𝑌𝑆𝑍)𝐾 ↔ (𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍) ∧ 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌) ∧ (𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))))
16	14, 15	mpbid 221	. . . . . 6 ⊢ (𝜑 → (𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍) ∧ 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌) ∧ (𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻) = ((Id‘𝐶)‘𝑌)))
17	16	simp1d 1066	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
18	1, 2, 3, 4, 5, 7, 6, 13, 17	catcocl 16169	. . . 4 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑍))
19	16	simp2d 1067	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌))
20	12	simp2d 1067	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
21	1, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20	catass 16170	. . 3 ⊢ (𝜑 → ((𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾)(⟨𝑋, 𝑍⟩ · 𝑋)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)(𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))))
22	16	simp3d 1068	. . . . . 6 ⊢ (𝜑 → (𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))
23	22	oveq1d 6564	. . . . 5 ⊢ (𝜑 → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹))
24	1, 2, 3, 4, 5, 7, 6, 13, 17, 7, 19	catass 16170	. . . . 5 ⊢ (𝜑 → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)))
25	1, 2, 9, 4, 5, 3, 7, 13	catlid 16167	. . . . 5 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)
26	23, 24, 25	3eqtr3d 2652	. . . 4 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) = 𝐹)
27	26	oveq2d 6565	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)(𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
28	12	simp3d 1068	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
29	21, 27, 28	3eqtrd 2648	. 2 ⊢ (𝜑 → ((𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾)(⟨𝑋, 𝑍⟩ · 𝑋)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) = ((Id‘𝐶)‘𝑋))
30	1, 2, 3, 4, 6, 7, 5, 19, 20	catcocl 16169	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾) ∈ (𝑍(Hom ‘𝐶)𝑋))
31	1, 2, 3, 9, 10, 4, 5, 6, 18, 30	issect2 16237	. 2 ⊢ (𝜑 → ((𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾) ↔ ((𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾)(⟨𝑋, 𝑍⟩ · 𝑋)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) = ((Id‘𝐶)‘𝑋)))
32	29, 31	mpbird 246	1 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾))

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Sectcsect 16227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-cid 16153  df-sect 16230

This theorem is referenced by:  invco  16254