P. 177 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	quss 17601	The scalar field of a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ 𝐾 = (Scalar‘𝑅)    ⇒   ⊢ (𝜑 → 𝐾 = (Scalar‘𝑈))
Theorem	divsfval 17602*	Value of the function in qusval 17597. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
Theorem	ercpbllem 17603*	Lemma for ercpbl 17604. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))
Theorem	ercpbl 17604*	Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)    &   ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))
Theorem	erlecpbl 17605*	Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))
Theorem	qusaddvallem 17606*	Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	qusaddflem 17607*	The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    ⇒   ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))
Theorem	qusaddval 17608*	The addition in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (+g‘𝑅)    &   ⊢ ∙ = (+g‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	qusaddf 17609*	The addition in a quotient structure as a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (+g‘𝑅)    &   ⊢ ∙ = (+g‘𝑈)    ⇒   ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))
Theorem	qusmulval 17610*	The multiplication in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	qusmulf 17611*	The multiplication in a quotient structure as a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    ⇒   ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))
Theorem	fnpr2o 17612	Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
Theorem	fnpr2ob 17613	Biconditional version of fnpr2o 17612. (Contributed by Jim Kingdon, 27-Sep-2023.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
Theorem	fvpr0o 17614	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
⊢ (𝐴 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
Theorem	fvpr1o 17615	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
⊢ (𝐵 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
Theorem	fvprif 17616	The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝐶) = if(𝐶 = ∅, 𝐴, 𝐵))
Theorem	xpsfrnel 17617*	Elementhood in the target space of the function 𝐹 appearing in xpsval 17625. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
Theorem	xpsfeq 17618	A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)
Theorem	xpsfrnel2 17619*	Elementhood in the target space of the function 𝐹 appearing in xpsval 17625. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ ({⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
Theorem	xpscf 17620	Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ({⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
Theorem	xpsfval 17621*	The value of the function appearing in xpsval 17625. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
Theorem	xpsff1o 17622*	The function appearing in xpsval 17625 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {∅, 1o}. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Theorem	xpsfrn 17623*	A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Theorem	xpsff1o2 17624*	The function appearing in xpsval 17625 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {∅, 1o}. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹
Theorem	xpsval 17625*	Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   ⊢ 𝐺 = (Scalar‘𝑅)    &   ⊢ 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    ⇒   ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))
Theorem	xpsrnbas 17626*	The indexed structure product that appears in xpsval 17625 has the same base as the target of the function 𝐹. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   ⊢ 𝐺 = (Scalar‘𝑅)    &   ⊢ 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    ⇒   ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
Theorem	xpsbas 17627	The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇))
Theorem	xpsaddlem 17628*	Lemma for xpsadd 17629 and xpsmul 17630. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &   ⊢ · = (𝐸‘𝑅)    &   ⊢ × = (𝐸‘𝑆)    &   ⊢ ∙ = (𝐸‘𝑇)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   ⊢ 𝑈 = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    &   ⊢ ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹) → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))    &   ⊢ (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Theorem	xpsadd 17629	Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &   ⊢ · = (+g‘𝑅)    &   ⊢ × = (+g‘𝑆)    &   ⊢ ∙ = (+g‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Theorem	xpsmul 17630	Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ ∙ = (.r‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Theorem	xpssca 17631	Value of the scalar field of a binary structure product. For concreteness, we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝐺 = (Scalar‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐺 = (Scalar‘𝑇))
Theorem	xpsvsca 17632	Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝐺 = (Scalar‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ · = ( ·𝑠 ‘𝑅)    &   ⊢ × = ( ·𝑠 ‘𝑆)    &   ⊢ ∙ = ( ·𝑠 ‘𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)    ⇒   ⊢ (𝜑 → (𝐴 ∙ ⟨𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
Theorem	xpsless 17633	Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ ≤ = (le‘𝑇)    ⇒   ⊢ (𝜑 → ≤ ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌)))
Theorem	xpsle 17634	Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ ≤ = (le‘𝑇)    &   ⊢ 𝑀 = (le‘𝑅)    &   ⊢ 𝑁 = (le‘𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ≤ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷)))
7.2  Moore spaces
Syntax	cmre 17635	The class of Moore systems.
class Moore
Syntax	cmrc 17636	The class function generating Moore closures.
class mrCls
Syntax	cmri 17637	mrInd is a class function which takes a Moore system to its set of independent sets.
class mrInd
Syntax	cacs 17638	The class of algebraic closure (Moore) systems.
class ACS
Definition	df-mre 17639*	Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 23216) and vector spaces (lssmre 21068) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76. See ismre 17643, mresspw 17645, mre1cl 17647 and mreintcl 17648 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17653); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17654. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)
⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
Definition	df-mrc 17640*	Define the Moore closure of a generating set, which is the smallest closed set containing all generating elements. Definition of Moore closure in [Schechter] p. 79. This generalizes topological closure (mrccls 23217) and linear span (mrclsp 21091). A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcssid 17674), (2) isotone, i.e., 𝑥 ⊆ 𝑦 implies that (𝑁‘𝑥) ⊆ (𝑁‘𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 17673), and (3) idempotent, i.e., (𝑁‘(𝑁‘𝑥)) = (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcidm 17676.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation 𝑁 on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.)
⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
Definition	df-mri 17641*	In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.)
⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
Definition	df-acs 17642*	An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 9613 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
Theorem	ismre 17643*	Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)))
Theorem	fnmre 17644	The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 23062 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ Moore Fn V
Theorem	mresspw 17645	A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
Theorem	mress 17646	A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
Theorem	mre1cl 17647	In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
Theorem	mreintcl 17648	A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶)
Theorem	mreiincl 17649*	A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
Theorem	mrerintcl 17650	The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
Theorem	mreriincl 17651*	The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
Theorem	mreincl 17652	Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)
Theorem	mreuni 17653	Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
Theorem	mreunirn 17654	Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶))
Theorem	ismred 17655*	Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋))
Theorem	ismred2 17656*	Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋))
Theorem	mremre 17657	The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))
Theorem	submre 17658	The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))
Theorem	xrsle 17659	The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ≤ = (le‘ℝ*𝑠)
Theorem	xrge0le 17660	The "less than or equal to" relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.)
⊢ ≤ = (le‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrsbas 17661	The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ* = (Base‘ℝ*𝑠)
Theorem	xrge0base 17662	The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞)))
7.2.1  Moore closures
Theorem	mrcflem 17663*	The domain and codomain of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶)
Theorem	fnmrc 17664	Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ mrCls Fn ∪ ran Moore
Theorem	mrcfval 17665*	Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
Theorem	mrcf 17666	The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
Theorem	mrcval 17667*	Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
Theorem	mrccl 17668	The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
Theorem	mrcsncl 17669	The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)
Theorem	mrcid 17670	The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈)
Theorem	mrcssv 17671	The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)
Theorem	mrcidb 17672	A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))
Theorem	mrcss 17673	Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
Theorem	mrcssid 17674	The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝐹‘𝑈))
Theorem	mrcidb2 17675	A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) ⊆ 𝑈))
Theorem	mrcidm 17676	The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘(𝐹‘𝑈)) = (𝐹‘𝑈))
Theorem	mrcsscl 17677	The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝑉)
Theorem	mrcuni 17678	Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) = (𝐹‘∪ (𝐹 “ 𝑈)))
Theorem	mrcun 17679	Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))
Theorem	mrcssvd 17680	The Moore closure of a set is a subset of the base. Deduction form of mrcssv 17671. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    ⇒   ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋)
Theorem	mrcssd 17681	Moore closure preserves subset ordering. Deduction form of mrcss 17673. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁‘𝑈) ⊆ (𝑁‘𝑉))
Theorem	mrcssidd 17682	A set is contained in its Moore closure. Deduction form of mrcssid 17674. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈))
Theorem	mrcidmd 17683	Moore closure is idempotent. Deduction form of mrcidm 17676. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈))
Theorem	mressmrcd 17684	In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
Theorem	submrc 17685	In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    &   ⊢ 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))
Theorem	mrieqvlemd 17686	In a Moore system, if 𝑌 is a member of 𝑆, (𝑆 ∖ {𝑌}) and 𝑆 have the same closure if and only if 𝑌 is in the closure of (𝑆 ∖ {𝑌}). Used in the proof of mrieqvd 17695 and mrieqv2d 17696. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)))
7.2.2  Independent sets in a Moore system
Theorem	mrisval 17687*	Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    ⇒   ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
Theorem	ismri 17688*	Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    ⇒   ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
Theorem	ismri2 17689*	Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    ⇒   ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Theorem	ismri2d 17690*	Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Theorem	ismri2dd 17691*	Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐼)
Theorem	mriss 17692	An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐼 = (mrInd‘𝐴)    ⇒   ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋)
Theorem	mrissd 17693	An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
Theorem	ismri2dad 17694	Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
Theorem	mrieqvd 17695*	In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))
Theorem	mrieqv2d 17696*	In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆))))
Theorem	mrissmrcd 17697	In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 17684, and so are equal by mrieqv2d 17696.) (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    ⇒   ⊢ (𝜑 → 𝑆 = 𝑇)
Theorem	mrissmrid 17698	In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝑇 ∈ 𝐼)
Theorem	mreexd 17699*	In a Moore system, the closure operator is said to have the exchange property if, for all elements 𝑦 and 𝑧 of the base set and subsets 𝑆 of the base set such that 𝑧 is in the closure of (𝑆 ∪ {𝑦}) but not in the closure of 𝑆, 𝑦 is in the closure of (𝑆 ∪ {𝑧}) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))    &   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘𝑆))    ⇒   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))
Theorem	mreexmrid 17700*	In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘𝑆))    ⇒   ⊢ (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼)