MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sectco Structured version   Visualization version   GIF version

Theorem sectco 17022
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 2824 . . . 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 2824 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
10 sectco.s . . . . . . . 8 𝑆 = (Sect‘𝐶)
111, 2, 3, 9, 10, 4, 5, 7issect 17019 . . . . . . 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 235 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
1312simp1d 1139 . . . . 5 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
14 sectco.2 . . . . . . 7 (𝜑𝐻(𝑌𝑆𝑍)𝐾)
151, 2, 3, 9, 10, 4, 7, 6issect 17019 . . . . . . 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 235 . . . . . 6 (𝜑 → (𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍) ∧ 𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌) ∧ (𝐾(⟨𝑌, 𝑍· 𝑌)𝐻) = ((Id‘𝐶)‘𝑌)))
1716simp1d 1139 . . . . 5 (𝜑𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
181, 2, 3, 4, 5, 7, 6, 13, 17catcocl 16952 . . . 4 (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋(Hom ‘𝐶)𝑍))
1916simp2d 1140 . . . 4 (𝜑𝐾 ∈ (𝑍(Hom ‘𝐶)𝑌))
2012simp2d 1140 . . . 4 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
211, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20catass 16953 . . 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 1141 . . . . . 6 (𝜑 → (𝐾(⟨𝑌, 𝑍· 𝑌)𝐻) = ((Id‘𝐶)‘𝑌))
2322oveq1d 7160 . . . . 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 16953 . . . . 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 16950 . . . . 5 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)
2623, 24, 253eqtr3d 2867 . . . 4 (𝜑 → (𝐾(⟨𝑋, 𝑍· 𝑌)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)) = 𝐹)
2726oveq2d 7161 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑋)(𝐾(⟨𝑋, 𝑍· 𝑌)(𝐻(⟨𝑋, 𝑌· 𝑍)𝐹))) = (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹))
2812simp3d 1141 . . 3 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
2921, 27, 283eqtrd 2863 . 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 16952 . . 3 (𝜑 → (𝐺(⟨𝑍, 𝑌· 𝑋)𝐾) ∈ (𝑍(Hom ‘𝐶)𝑋))
311, 2, 3, 9, 10, 4, 5, 6, 18, 30issect2 17020 . 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 260 1 (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌· 𝑋)𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cop 4555   class class class wbr 5052  cfv 6343  (class class class)co 7145  Basecbs 16479  Hom chom 16572  compcco 16573  Catccat 16931  Idccid 16932  Sectcsect 17010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7679  df-2nd 7680  df-cat 16935  df-cid 16936  df-sect 17013
This theorem is referenced by:  invco  17037
  Copyright terms: Public domain W3C validator