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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhofcl 17501 Closure of the Hom functor. Note that the codomain is the category SetCat‘𝑈 for any universe 𝑈 which contains each Hom-set. This corresponds to the assertion that 𝐶 be locally small (with respect to 𝑈). (Contributed by Mario Carneiro, 15-Jan-2017.)
𝑀 = (HomF𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝐷 = (SetCat‘𝑈)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))

Theoremoppchofcl 17502 Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑀 = (HomF𝑂)    &   𝐷 = (SetCat‘𝑈)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑀 ∈ ((𝐶 ×c 𝑂) Func 𝐷))

Theoremyonval 17503 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 17504 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 17505 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 17506 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 17507 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 17508 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 17509 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 17510 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 17511 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 17512 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 17513 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 17514 Lemma for yoneda 17525. (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 17515 Lemma for yoneda 17525. (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 17516* Lemma for yoneda 17525. (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 17517* Lemma for yoneda 17525. (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 17518* Lemma for yoneda 17525. (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 17519* Lemma for yoneda 17525. (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 17520 Lemma for yoneda 17525. (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 17521* Lemma for yoneda 17525. (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 17522* Lemma for yoneda 17525. (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 17523* 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 17524* Lemma for yonffth 17526. (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 17525* 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 17526 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 17527* 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 17528 Extend class notation with the class of all prosets.
class Proset

Syntaxcdrs 17529 Extend class notation with the class of all directed sets.
class Dirset

Definitiondf-proset 17530* 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 17531* 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 17532* 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 17533 Lemma for prsref 17534 and prstr 17535. (Contributed by Mario Carneiro, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))

Theoremprsref 17534 "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 17535 "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 17536* 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 17537* 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 17538 A directed set is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Dirset → 𝐾 ∈ Proset )

Theoremdrsbn0 17539 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)       (𝐾 ∈ Dirset → 𝐵 ≠ ∅)

Theoremdrsdirfi 17540* Any finite number of elements in a directed set have a common upper bound. Here is where the nonemptiness constraint in df-drs 17531 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 17541* 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 17542 Extend class notation with the class of posets.
class Poset

Syntaxcplt 17543 Extend class notation with less-than for posets.
class lt

Syntaxclub 17544 Extend class notation with poset least upper bound.
class lub

Syntaxcglb 17545 Extend class notation with poset greatest lower bound.
class glb

Syntaxcjn 17546 Extend class notation with poset join.
class join

Syntaxcmee 17547 Extend class notation with poset meet.
class meet

Definitiondf-poset 17548* 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 3492 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 17549* 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 17550* 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 17551 A poset is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Poset → 𝐾 ∈ Proset )

Theoremposi 17552 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 17553 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 17554 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 17555 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 17556 Technical lemma to simplify the statement of ipopos 17762. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16527) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
∅ ∈ Poset

Theoremisposd 17557* Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by AV, 26-Apr-2024.)
(𝜑𝐾𝑉)    &   (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 17558* 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 17559* 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 17560 Define less-than ordering for posets and related structures. Unlike df-base 16481 and df-ple 16577, 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 17561 Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
= (le‘𝐾)    &    < = (lt‘𝐾)       (𝐾𝐴< = ( ∖ I ))

Theorempltval 17562 Less-than relation. (df-pss 3900 analog.) (Contributed by NM, 12-Oct-2011.)
= (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))

Theorempltle 17563 "Less than" implies "less than or equal to". (pssss 4023 analog.) (Contributed by NM, 4-Dec-2011.)
= (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋 𝑌))

Theorempltne 17564 The "less than" relation is not reflexive. (df-pss 3900 analog.) (Contributed by NM, 2-Dec-2011.)
< = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋𝑌))

Theorempltirr 17565 The "less than" relation is not reflexive. (pssirr 4028 analog.) (Contributed by NM, 7-Feb-2012.)
< = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵) → ¬ 𝑋 < 𝑋)

Theorempleval2i 17566 One direction of pleval2 17567. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))

Theorempleval2 17567 "Less than or equal to" in terms of "less than". (sspss 4027 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 17568 "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 17569 Alternate expression for the "less than" relation. (dfpss3 4014 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))

Theorempltnlt 17570 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 17571 The less-than relation has no 2-cycle loops. (pssn2lp 4029 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))

Theoremplttr 17572 The less-than relation is transitive. (psstr 4032 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Theorempltletr 17573 Transitive law for chained "less than" and "less than or equal to". (psssstr 4034 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))

Theoremplelttr 17574 Transitive law for chained "less than or equal to" and "less than". (sspsstr 4033 analog.) (Contributed by NM, 2-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Theorempospo 17575 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 17576* 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 17577* 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 17578* 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 17579* Define poset meet. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 8-Sep-2018.)
meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})

Theoremlubfval 17580* 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 17581* 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 17582 The LUB is a function. (Contributed by NM, 9-Sep-2018.)
𝑈 = (lub‘𝐾)       Fun 𝑈

Theoremlubeldm 17583* 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 17584 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 17585* 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 17586* 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 17587 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 17588* 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 17589 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 17590* Lemma for lublecl 17591 and lubid 17592. (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 17591* 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 17592* 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 17593* 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 17594* 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 17595 The GLB is a function. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glb‘𝐾)       Fun 𝐺

Theoremglbeldm 17596* 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 17597 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 17598* 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 17599* 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 17600 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)

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