Theorem sectco 17804

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 2735	. . . 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 2735	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		sectco.s	. . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶)
11	1, 2, 3, 9, 10, 4, 5, 7	issect 17801	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
13	12	simp1d 1141	. . . . 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 17801	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → (𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍) ∧ 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌) ∧ (𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻) = ((Id‘𝐶)‘𝑌)))
17	16	simp1d 1141	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
18	1, 2, 3, 4, 5, 7, 6, 13, 17	catcocl 17730	. . . 4 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑍))
19	16	simp2d 1142	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌))
20	12	simp2d 1142	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
21	1, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20	catass 17731	. . 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 1143	. . . . . 6 ⊢ (𝜑 → (𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))
23	22	oveq1d 7446	. . . . 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 17731	. . . . 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 17728	. . . . 5 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)
26	23, 24, 25	3eqtr3d 2783	. . . 4 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) = 𝐹)
27	26	oveq2d 7447	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)(𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
28	12	simp3d 1143	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
29	21, 27, 28	3eqtrd 2779	. 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 17730	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾) ∈ (𝑍(Hom ‘𝐶)𝑋))
31	1, 2, 3, 9, 10, 4, 5, 6, 18, 30	issect2 17802	. 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 257	1 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ⟨cop 4637   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710  Sectcsect 17792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-cat 17713  df-cid 17714  df-sect 17795

This theorem is referenced by:  invco  17819