Theorem sectco 17660

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 2731	. . . 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 2731	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		sectco.s	. . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶)
11	1, 2, 3, 9, 10, 4, 5, 7	issect 17657	. . . . . . 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 1142	. . . . 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 17657	. . . . . . 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 1142	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
18	1, 2, 3, 4, 5, 7, 6, 13, 17	catcocl 17588	. . . 4 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑍))
19	16	simp2d 1143	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌))
20	12	simp2d 1143	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
21	1, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20	catass 17589	. . 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 1144	. . . . . 6 ⊢ (𝜑 → (𝐾(⟨𝑌, 𝑍⟩ · 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))
23	22	oveq1d 7361	. . . . 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 17589	. . . . 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 17586	. . . . 5 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)
26	23, 24, 25	3eqtr3d 2774	. . . 4 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) = 𝐹)
27	26	oveq2d 7362	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)(𝐾(⟨𝑋, 𝑍⟩ · 𝑌)(𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
28	12	simp3d 1144	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
29	21, 27, 28	3eqtrd 2770	. 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 17588	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾) ∈ (𝑍(Hom ‘𝐶)𝑋))
31	1, 2, 3, 9, 10, 4, 5, 6, 18, 30	issect2 17658	. 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 1541   ∈ wcel 2111  ⟨cop 4582   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  Hom chom 17169  compcco 17170  Catccat 17567  Idccid 17568  Sectcsect 17648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-cat 17571  df-cid 17572  df-sect 17651

This theorem is referenced by:  invco  17675