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Theorem sectco 16416
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 (𝜑𝐶 ∈ Cat)
sectco.x (𝜑𝑋𝐵)
sectco.y (𝜑𝑌𝐵)
sectco.z (𝜑𝑍𝐵)
sectco.1 (𝜑𝐹(𝑋𝑆𝑌)𝐺)
sectco.2 (𝜑𝐻(𝑌𝑆𝑍)𝐾)
Assertion
Ref Expression
sectco (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌· 𝑋)𝐾))

Proof of Theorem sectco
StepHypRef Expression
1 sectco.b . . . 4 𝐵 = (Base‘𝐶)
2 eqid 2622 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 sectco.o . . . 4 · = (comp‘𝐶)
4 sectco.c . . . 4 (𝜑𝐶 ∈ Cat)
5 sectco.x . . . 4 (𝜑𝑋𝐵)
6 sectco.z . . . 4 (𝜑𝑍𝐵)
7 sectco.y . . . 4 (𝜑𝑌𝐵)
8 sectco.1 . . . . . . 7 (𝜑𝐹(𝑋𝑆𝑌)𝐺)
9 eqid 2622 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
10 sectco.s . . . . . . . 8 𝑆 = (Sect‘𝐶)
111, 2, 3, 9, 10, 4, 5, 7issect 16413 . . . . . . 7 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
128, 11mpbid 222 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
1312simp1d 1073 . . . . 5 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
14 sectco.2 . . . . . . 7 (𝜑𝐻(𝑌𝑆𝑍)𝐾)
151, 2, 3, 9, 10, 4, 7, 6issect 16413 . . . . . . 7 (𝜑 → (𝐻(𝑌𝑆𝑍)𝐾 ↔ (𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍) ∧ 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌) ∧ (𝐾(⟨𝑌, 𝑍· 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))))
1614, 15mpbid 222 . . . . . 6 (𝜑 → (𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍) ∧ 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌) ∧ (𝐾(⟨𝑌, 𝑍· 𝑌)𝐻) = ((Id‘𝐶)‘𝑌)))
1716simp1d 1073 . . . . 5 (𝜑𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
181, 2, 3, 4, 5, 7, 6, 13, 17catcocl 16346 . . . 4 (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑍))
1916simp2d 1074 . . . 4 (𝜑𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌))
2012simp2d 1074 . . . 4 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
211, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20catass 16347 . . 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))))
2216simp3d 1075 . . . . . 6 (𝜑 → (𝐾(⟨𝑌, 𝑍· 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))
2322oveq1d 6665 . . . . 5 (𝜑 → ((𝐾(⟨𝑌, 𝑍· 𝑌)𝐻)(⟨𝑋, 𝑌· 𝑌)𝐹) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹))
241, 2, 3, 4, 5, 7, 6, 13, 17, 7, 19catass 16347 . . . . 5 (𝜑 → ((𝐾(⟨𝑌, 𝑍· 𝑌)𝐻)(⟨𝑋, 𝑌· 𝑌)𝐹) = (𝐾(⟨𝑋, 𝑍· 𝑌)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)))
251, 2, 9, 4, 5, 3, 7, 13catlid 16344 . . . . 5 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)
2623, 24, 253eqtr3d 2664 . . . 4 (𝜑 → (𝐾(⟨𝑋, 𝑍· 𝑌)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)) = 𝐹)
2726oveq2d 6666 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑋)(𝐾(⟨𝑋, 𝑍· 𝑌)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹))) = (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹))
2812simp3d 1075 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
2921, 27, 283eqtrd 2660 . 2 (𝜑 → ((𝐺(⟨𝑍, 𝑌· 𝑋)𝐾)(⟨𝑋, 𝑍· 𝑋)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)) = ((Id‘𝐶)‘𝑋))
301, 2, 3, 4, 6, 7, 5, 19, 20catcocl 16346 . . 3 (𝜑 → (𝐺(⟨𝑍, 𝑌· 𝑋)𝐾) ∈ (𝑍(Hom ‘𝐶)𝑋))
311, 2, 3, 9, 10, 4, 5, 6, 18, 30issect2 16414 . 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)))
3229, 31mpbird 247 1 (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌· 𝑋)𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1037   = wceq 1483  wcel 1990  cop 4183   class class class wbr 4653  cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Idccid 16326  Sectcsect 16404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407
This theorem is referenced by:  invco  16431
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