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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremyonval 17501 Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝑂 = (oppCat‘𝐶)    &   𝑀 = (HomF𝑂)       (𝜑𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))
 
Theoremyoncl 17502 The Yoneda embedding is a functor from the category to the category 𝑄 of presheaves on 𝐶. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑌 ∈ (𝐶 Func 𝑄))
 
Theoremyon1cl 17503 The Yoneda embedding at an object of 𝐶 is a presheaf on 𝐶, also known as the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
 
Theoremyon11 17504 Value of the Yoneda embedding at an object. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑍𝐵)       (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))
 
Theoremyon12 17505 Value of the Yoneda embedding at a morphism. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑍𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑊𝐵)    &   (𝜑𝐹 ∈ (𝑊𝐻𝑍))    &   (𝜑𝐺 ∈ (𝑍𝐻𝑋))       (𝜑 → (((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))
 
Theoremyon2 17506 Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑍𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑊𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑍))    &   (𝜑𝐺 ∈ (𝑊𝐻𝑋))       (𝜑 → ((((𝑋(2nd𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋· 𝑍)𝐺))
 
Theoremhofpropd 17507 If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (HomF𝐶) = (HomF𝐷))
 
Theoremyonpropd 17508 If two categories have the same set of objects, morphisms, and compositions, then they have the same Yoneda functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (Yon‘𝐶) = (Yon‘𝐷))
 
Theoremoppcyon 17509 Value of the opposite Yoneda embedding. (Contributed by Mario Carneiro, 26-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑌 = (Yon‘𝑂)    &   𝑀 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑𝑌 = (⟨𝑂, 𝐶⟩ curryF 𝑀))
 
Theoremoyoncl 17510 The opposite Yoneda embedding is a functor from oppCat‘𝐶 to the functor category 𝐶 → SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑌 = (Yon‘𝑂)    &   (𝜑𝐶 ∈ Cat)    &   𝑆 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   𝑄 = (𝐶 FuncCat 𝑆)       (𝜑𝑌 ∈ (𝑂 Func 𝑄))
 
Theoremoyon1cl 17511 The opposite Yoneda embedding at an object of 𝐶 is a functor from 𝐶 to Set, also known as the covariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑌 = (Yon‘𝑂)    &   (𝜑𝐶 ∈ Cat)    &   𝑆 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝐶 Func 𝑆))
 
Theoremyonedalem1 17512 Lemma for yoneda 17523. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)       (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
 
Theoremyonedalem21 17513 Lemma for yoneda 17523. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
 
Theoremyonedalem3a 17514* Lemma for yoneda 17523. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))       (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
 
Theoremyonedalem4a 17515* Lemma for yoneda 17523. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))    &   (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))       (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
 
Theoremyonedalem4b 17516* Lemma for yoneda 17523. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))    &   (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))    &   (𝜑𝑃𝐵)    &   (𝜑𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))       (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴))
 
Theoremyonedalem4c 17517* Lemma for yoneda 17523. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))    &   (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))       (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
 
Theoremyonedalem22 17518 Lemma for yoneda 17523. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)    &   (𝜑𝐺 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑃𝐵)    &   (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))    &   (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))       (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
 
Theoremyonedalem3b 17519* Lemma for yoneda 17523. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)    &   (𝜑𝐺 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑃𝐵)    &   (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))    &   (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))       (𝜑 → ((𝐺𝑀𝑃)(⟨(𝐹(1st𝑍)𝑋), (𝐺(1st𝑍)𝑃)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾)(⟨(𝐹(1st𝑍)𝑋), (𝐹(1st𝐸)𝑋)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐹𝑀𝑋)))
 
Theoremyonedalem3 17520* Lemma for yoneda 17523. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))       (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
 
Theoremyonedainv 17521* The Yoneda Lemma with explicit inverse. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))    &   𝐼 = (Inv‘𝑅)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))       (𝜑𝑀(𝑍𝐼𝐸)𝑁)
 
Theoremyonffthlem 17522* Lemma for yonffth 17524. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))    &   𝐼 = (Inv‘𝑅)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))       (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
 
Theoremyoneda 17523* The Yoneda Lemma. There is a natural isomorphism between the functors 𝑍 and 𝐸, where 𝑍(𝐹, 𝑋) is the natural transformations from Yon(𝑋) = Hom ( − , 𝑋) to 𝐹, and 𝐸(𝐹, 𝑋) = 𝐹(𝑋) is the evaluation functor. Here we need two universes to state the claim: the smaller universe 𝑈 is used for forming the functor category 𝑄 = 𝐶 op → SetCat(𝑈), which itself does not (necessarily) live in 𝑈 but instead is an element of the larger universe 𝑉. (If 𝑈 is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set 𝑈 = 𝑉 in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))    &   𝐼 = (Iso‘𝑅)       (𝜑𝑀 ∈ (𝑍𝐼𝐸))
 
Theoremyonffth 17524 The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category 𝐶 as a full subcategory of the category 𝑄 of presheaves on 𝐶. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
 
Theoremyoniso 17525* If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from 𝐶 into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝐷 = (CatCat‘𝑉)    &   𝐵 = (Base‘𝐷)    &   𝐼 = (Iso‘𝐷)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐸 = (𝑄s ran (1st𝑌))    &   (𝜑𝑉𝑋)    &   (𝜑𝐶𝐵)    &   (𝜑𝑈𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑𝐸𝐵)    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)       (𝜑𝑌 ∈ (𝐶𝐼𝐸))
 
PART 9  BASIC ORDER THEORY
 
9.1  Preordered sets and directed sets using extensible structures
 
Syntaxcproset 17526 Extend class notation with the class of all prosets.
class Proset
 
Syntaxcdrs 17527 Extend class notation with the class of all directed sets.
class Dirset
 
Definitiondf-proset 17528* Define the class of preordered sets, or prosets. A proset is a set equipped with a preorder, that is, a transitive and reflexive relation.

Preorders are a natural generalization of partial orders which need not be antisymmetric: there may be pairs of elements such that each is "less than or equal to" the other, so that both elements have the same order-theoretic properties (in some sense, there is a "tie" among them).

If a preorder is required to be antisymmetric, that is, there is no such "tie", then one obtains a partial order. If a preorder is required to be symmetric, that is, all comparable elements are tied, then one obtains an equivalence relation.

Every preorder naturally factors into these two notions: the "tie" relation on a proset is an equivalence relation, and the quotient under that equivalence relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

Proset = {𝑓[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
 
Definitiondf-drs 17529* Define the class of directed sets. A directed set is a nonempty preordered set where every pair of elements have some upper bound. Note that it is not required that there exist a least upper bound.

There is no consensus in the literature over whether directed sets are allowed to be empty. It is slightly more convenient for us if they are not. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
 
Theoremisprs 17530* Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
 
Theoremprslem 17531 Lemma for prsref 17532 and prstr 17533. (Contributed by Mario Carneiro, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
 
Theoremprsref 17532 "Less than or equal to" is reflexive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Proset ∧ 𝑋𝐵) → 𝑋 𝑋)
 
Theoremprstr 17533 "Less than or equal to" is transitive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)
 
Theoremisdrs 17534* Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑧𝑦 𝑧)))
 
Theoremdrsdir 17535* Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Dirset ∧ 𝑋𝐵𝑌𝐵) → ∃𝑧𝐵 (𝑋 𝑧𝑌 𝑧))
 
Theoremdrsprs 17536 A directed set is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Dirset → 𝐾 ∈ Proset )
 
Theoremdrsbn0 17537 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)       (𝐾 ∈ Dirset → 𝐵 ≠ ∅)
 
Theoremdrsdirfi 17538* Any finite number of elements in a directed set have a common upper bound. Here is where the nonemptiness constraint in df-drs 17529 first comes into play; without it we would need an additional constraint that 𝑋 not be empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Dirset ∧ 𝑋𝐵𝑋 ∈ Fin) → ∃𝑦𝐵𝑧𝑋 𝑧 𝑦)
 
Theoremisdrs2 17539* Directed sets may be defined in terms of finite subsets. Again, without nonemptiness we would need to restrict to nonempty subsets here. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (𝒫 𝐵 ∩ Fin)∃𝑦𝐵𝑧𝑥 𝑧 𝑦))
 
9.2  Posets and lattices using extensible structures
 
9.2.1  Posets
 
Syntaxcpo 17540 Extend class notation with the class of posets.
class Poset
 
Syntaxcplt 17541 Extend class notation with less-than for posets.
class lt
 
Syntaxclub 17542 Extend class notation with poset least upper bound.
class lub
 
Syntaxcglb 17543 Extend class notation with poset greatest lower bound.
class glb
 
Syntaxcjn 17544 Extend class notation with poset join.
class join
 
Syntaxcmee 17545 Extend class notation with poset meet.
class meet
 
Definitiondf-poset 17546* Define the class of partially ordered sets (posets). A poset is a set equipped with a partial order, that is, a binary relation which is reflexive, antisymmetric, and transitive. Unlike a total order, in a partial order there may be pairs of elements where neither precedes the other. Definition of poset in [Crawley] p. 1. Note that Crawley-Dilworth require that a poset base set be nonempty, but we follow the convention of most authors who don't make this a requirement.

In our formalism of extensible structures, the base set of a poset 𝑓 is denoted by (Base‘𝑓) and its partial order by (le‘𝑓) (for "less than or equal to"). The quantifiers 𝑏𝑟 provide a notational shorthand to allow us to refer to the base and ordering relation as 𝑏 and 𝑟 in the definition rather than having to repeat (Base‘𝑓) and (le‘𝑓) throughout. These quantifiers can be eliminated with ceqsex2v 3545 and related theorems. (Contributed by NM, 18-Oct-2012.)

Poset = {𝑓 ∣ ∃𝑏𝑟(𝑏 = (Base‘𝑓) ∧ 𝑟 = (le‘𝑓) ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))}
 
Theoremispos 17547* The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
 
Theoremispos2 17548* A poset is an antisymmetric proset.

EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
 
Theoremposprs 17549 A poset is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Poset → 𝐾 ∈ Proset )
 
Theoremposi 17550 Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
 
Theoremposref 17551 A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
 
Theoremposasymb 17552 A poset ordering is asymmetric. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
 
Theorempostr 17553 A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
 
Theorem0pos 17554 Technical lemma to simplify the statement of ipopos 17760. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16525) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
∅ ∈ Poset
 
Theoremisposd 17555* Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.)
(𝜑𝐾 ∈ V)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑 = (le‘𝐾))    &   ((𝜑𝑥𝐵) → 𝑥 𝑥)    &   ((𝜑𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))       (𝜑𝐾 ∈ Poset)
 
Theoremisposi 17556* Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)
𝐾 ∈ V    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝑥𝐵𝑥 𝑥)    &   ((𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))       𝐾 ∈ Poset
 
Theoremisposix 17557* Properties that determine a poset (explicit structure version). Note that the numeric indices of the structure components are not mentioned explicitly in either the theorem or its proof. (Contributed by NM, 9-Nov-2012.)
𝐵 ∈ V    &    ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ⟩}    &   (𝑥𝐵𝑥 𝑥)    &   ((𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))       𝐾 ∈ Poset
 
Definitiondf-plt 17558 Define less-than ordering for posets and related structures. Unlike df-base 16479 and df-ple 16575, this is a derived component extractor and not an extensible structure component extractor that defines the poset. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
 
Theorempltfval 17559 Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
= (le‘𝐾)    &    < = (lt‘𝐾)       (𝐾𝐴< = ( ∖ I ))
 
Theorempltval 17560 Less-than relation. (df-pss 3953 analog.) (Contributed by NM, 12-Oct-2011.)
= (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
 
Theorempltle 17561 "Less than" implies "less than or equal to". (pssss 4071 analog.) (Contributed by NM, 4-Dec-2011.)
= (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋 𝑌))
 
Theorempltne 17562 The "less than" relation is not reflexive. (df-pss 3953 analog.) (Contributed by NM, 2-Dec-2011.)
< = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋𝑌))
 
Theorempltirr 17563 The "less than" relation is not reflexive. (pssirr 4076 analog.) (Contributed by NM, 7-Feb-2012.)
< = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵) → ¬ 𝑋 < 𝑋)
 
Theorempleval2i 17564 One direction of pleval2 17565. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
 
Theorempleval2 17565 "Less than or equal to" in terms of "less than". (sspss 4075 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
 
Theorempltnle 17566 "Less than" implies not converse "less than or equal to". (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 𝑋)
 
Theorempltval3 17567 Alternate expression for the "less than" relation. (dfpss3 4062 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
 
Theorempltnlt 17568 The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 < 𝑋)
 
Theorempltn2lp 17569 The less-than relation has no 2-cycle loops. (pssn2lp 4077 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
 
Theoremplttr 17570 The less-than relation is transitive. (psstr 4080 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
 
Theorempltletr 17571 Transitive law for chained "less than" and "less than or equal to". (psssstr 4082 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
 
Theoremplelttr 17572 Transitive law for chained "less than or equal to" and "less than". (sspsstr 4081 analog.) (Contributed by NM, 2-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
 
Theorempospo 17573 Write a poset structure in terms of the proper-class poset predicate (strict less than version). (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
 
Definitiondf-lub 17574* Define the least upper bound (LUB) of a set of (poset) elements. The domain is restricted to exclude sets 𝑠 for which the LUB doesn't exist uniquely. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
lub = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑧𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑧𝑥(le‘𝑝)𝑧))}))
 
Definitiondf-glb 17575* Define the greatest lower bound (GLB) of a set of (poset) elements. The domain is restricted to exclude sets 𝑠 for which the GLB doesn't exist uniquely. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑧(le‘𝑝)𝑦𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑧(le‘𝑝)𝑦𝑧(le‘𝑝)𝑥))}))
 
Definitiondf-join 17576* Define poset join. (Contributed by NM, 12-Sep-2011.) (Revised by Mario Carneiro, 3-Nov-2015.)
join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
 
Definitiondf-meet 17577* Define poset join. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 8-Sep-2018.)
meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
 
Theoremlubfval 17578* Value of the least upper bound function of a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑠 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑦 𝑧𝑥 𝑧)))    &   (𝜑𝐾𝑉)       (𝜑𝑈 = ((𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥𝐵 𝜓}))
 
Theoremlubdm 17579* Domain of the least upper bound function of a poset. (Contributed by NM, 6-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑠 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑦 𝑧𝑥 𝑧)))    &   (𝜑𝐾𝑉)       (𝜑 → dom 𝑈 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥𝐵 𝜓})
 
Theoremlubfun 17580 The LUB is a function. (Contributed by NM, 9-Sep-2018.)
𝑈 = (lub‘𝐾)       Fun 𝑈
 
Theoremlubeldm 17581* Member of the domain of the least upper bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))    &   (𝜑𝐾𝑉)       (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆𝐵 ∧ ∃!𝑥𝐵 𝜓)))
 
Theoremlubelss 17582 A member of the domain of the least upper bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)       (𝜑𝑆𝐵)
 
Theoremlubeu 17583* Unique existence proper of a member of the domain of the least upper bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)       (𝜑 → ∃!𝑥𝐵 𝜓)
 
Theoremlubval 17584* Value of the least upper bound function of a poset. Out-of-domain arguments (those not satisfying 𝑆 ∈ dom 𝑈) are allowed for convenience, evaluating to the empty set. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))    &   (𝜑𝐾𝑉)    &   (𝜑𝑆𝐵)       (𝜑 → (𝑈𝑆) = (𝑥𝐵 𝜓))
 
Theoremlubcl 17585 The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)       (𝜑 → (𝑈𝑆) ∈ 𝐵)
 
Theoremlubprop 17586* Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)       (𝜑 → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
 
Theoremluble 17587 The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)    &   (𝜑𝑋𝑆)       (𝜑𝑋 (𝑈𝑆))
 
Theoremlublecllem 17588* Lemma for lublecl 17589 and lubid 17590. (Contributed by NM, 8-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)       ((𝜑𝑥𝐵) → ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ 𝑥 = 𝑋))
 
Theoremlublecl 17589* The set of all elements less than a given element has an LUB. (Contributed by NM, 8-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)       (𝜑 → {𝑦𝐵𝑦 𝑋} ∈ dom 𝑈)
 
Theoremlubid 17590* The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑈‘{𝑦𝐵𝑦 𝑋}) = 𝑋)
 
Theoremglbfval 17591* Value of the greatest lower function of a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑠 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)       (𝜑𝐺 = ((𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥𝐵 𝜓}))
 
Theoremglbdm 17592* Domain of the greatest lower bound function of a poset. (Contributed by NM, 6-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑠 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)       (𝜑 → dom 𝐺 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥𝐵 𝜓})
 
Theoremglbfun 17593 The GLB is a function. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glb‘𝐾)       Fun 𝐺
 
Theoremglbeldm 17594* Member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)       (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆𝐵 ∧ ∃!𝑥𝐵 𝜓)))
 
Theoremglbelss 17595 A member of the domain of the greatest lower bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝐺)       (𝜑𝑆𝐵)
 
Theoremglbeu 17596* Unique existence proper of a member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝐺)       (𝜑 → ∃!𝑥𝐵 𝜓)
 
Theoremglbval 17597* Value of the greatest lower bound function of a poset. Out-of-domain arguments (those not satisfying 𝑆 ∈ dom 𝑈) are allowed for convenience, evaluating to the empty set on both sides of the equality. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐺𝑆) = (𝑥𝐵 𝜓))
 
Theoremglbcl 17598 The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝐺)       (𝜑 → (𝐺𝑆) ∈ 𝐵)
 
Theoremglbprop 17599* Properties of greatest lower bound of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)       (𝜑 → (∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))))
 
Theoremglble 17600 The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)    &   (𝜑𝑋𝑆)       (𝜑 → (𝑈𝑆) 𝑋)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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