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Theorem sectco 17710
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 2731 . . . 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 2731 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
10 sectco.s . . . . . . . 8 𝑆 = (Sect‘𝐶)
111, 2, 3, 9, 10, 4, 5, 7issect 17707 . . . . . . 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 231 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
1312simp1d 1141 . . . . 5 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
14 sectco.2 . . . . . . 7 (𝜑𝐻(𝑌𝑆𝑍)𝐾)
151, 2, 3, 9, 10, 4, 7, 6issect 17707 . . . . . . 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 231 . . . . . 6 (𝜑 → (𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍) ∧ 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌) ∧ (𝐾(⟨𝑌, 𝑍· 𝑌)𝐻) = ((Id‘𝐶)‘𝑌)))
1716simp1d 1141 . . . . 5 (𝜑𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
181, 2, 3, 4, 5, 7, 6, 13, 17catcocl 17636 . . . 4 (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑍))
1916simp2d 1142 . . . 4 (𝜑𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌))
2012simp2d 1142 . . . 4 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
211, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20catass 17637 . . 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 1143 . . . . . 6 (𝜑 → (𝐾(⟨𝑌, 𝑍· 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))
2322oveq1d 7427 . . . . 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 17637 . . . . 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 17634 . . . . 5 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)
2623, 24, 253eqtr3d 2779 . . . 4 (𝜑 → (𝐾(⟨𝑋, 𝑍· 𝑌)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)) = 𝐹)
2726oveq2d 7428 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑋)(𝐾(⟨𝑋, 𝑍· 𝑌)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹))) = (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹))
2812simp3d 1143 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
2921, 27, 283eqtrd 2775 . 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 17636 . . 3 (𝜑 → (𝐺(⟨𝑍, 𝑌· 𝑋)𝐾) ∈ (𝑍(Hom ‘𝐶)𝑋))
311, 2, 3, 9, 10, 4, 5, 6, 18, 30issect2 17708 . 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 257 1 (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌· 𝑋)𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1540  wcel 2105  cop 4634   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615  Idccid 17616  Sectcsect 17698
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-cat 17619  df-cid 17620  df-sect 17701
This theorem is referenced by:  invco  17725
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