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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremecdmn0m 6401* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
(𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)

Theoremereldm 6402 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)       (𝜑 → (𝐴𝑋𝐵𝑋))

Theoremerth 6403 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑋)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Theoremerth2 6404 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Theoremerthi 6405 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Theoremecidsn 6406 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
[𝐴] I = {𝐴}

Theoremqseq1 6407 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))

Theoremqseq2 6408 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))

Theoremelqsg 6409* Closed form of elqs 6410. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))

Theoremelqs 6410* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
𝐵 ∈ V       (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅)

Theoremelqsi 6411* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥𝐴 𝐵 = [𝑥]𝑅)

Theoremecelqsg 6412 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Theoremecelqsi 6413 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝑅 ∈ V       (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Theoremecopqsi 6414 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
𝑅 ∈ V    &   𝑆 = ((𝐴 × 𝐴) / 𝑅)       ((𝐵𝐴𝐶𝐴) → [⟨𝐵, 𝐶⟩]𝑅𝑆)

Theoremqsexg 6415 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 / 𝑅) ∈ V)

Theoremqsex 6416 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
𝐴 ∈ V       (𝐴 / 𝑅) ∈ V

Theoremuniqs 6417 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
(𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))

Theoremqsss 6418 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝐴)       (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)

Theoremuniqs2 6419 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
(𝜑𝑅 Er 𝐴)    &   (𝜑𝑅𝑉)       (𝜑 (𝐴 / 𝑅) = 𝐴)

Theoremsnec 6420 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐴 ∈ V       {[𝐴]𝑅} = ({𝐴} / 𝑅)

Theoremecqs 6421 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
𝑅 ∈ V       [𝐴]𝑅 = ({𝐴} / 𝑅)

Theoremecid 6422 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐴 ∈ V       [𝐴] E = 𝐴

Theoremecidg 6423 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
(𝐴𝑉 → [𝐴] E = 𝐴)

Theoremqsid 6424 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝐴 / E ) = 𝐴

Theoremectocld 6425* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑆 = (𝐵 / 𝑅)    &   ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))    &   ((𝜒𝑥𝐵) → 𝜑)       ((𝜒𝐴𝑆) → 𝜓)

Theoremectocl 6426* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝑆 = (𝐵 / 𝑅)    &   ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))    &   (𝑥𝐵𝜑)       (𝐴𝑆𝜓)

Theoremelqsn0m 6427* An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥𝐵)

Theoremelqsn0 6428 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)

Theoremecelqsdm 6429 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵𝐴)

Theoremxpider 6430 A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝐴 × 𝐴) Er 𝐴

Theoremiinerm 6431* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → 𝑥𝐴 𝑅 Er 𝐵)

Theoremriinerm 6432* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵)

Theoremerinxp 6433 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Theoremecinxp 6434 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
(((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))

Theoremqsinxp 6435 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
((𝑅𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))

Theoremqsel 6436 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)

Theoremqliftlem 6437* 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))

Theoremqliftrel 6438* 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑𝐹 ⊆ ((𝑋 / 𝑅) × 𝑌))

Theoremqliftel 6439* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴)))

Theoremqliftel1 6440* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       ((𝜑𝑥𝑋) → [𝑥]𝑅𝐹𝐴)

Theoremqliftfun 6441* The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)    &   (𝑥 = 𝑦𝐴 = 𝐵)       (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))

Theoremqliftfund 6442* The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   ((𝜑𝑥𝑅𝑦) → 𝐴 = 𝐵)       (𝜑 → Fun 𝐹)

Theoremqliftfuns 6443* The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))

Theoremqliftf 6444* The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)       (𝜑 → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))

Theoremqliftval 6445* The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   (𝜑𝑅 Er 𝑋)    &   (𝜑𝑋 ∈ V)    &   (𝑥 = 𝐶𝐴 = 𝐵)    &   (𝜑 → Fun 𝐹)       ((𝜑𝐶𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)

Theoremecoptocl 6446* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
𝑆 = ((𝐵 × 𝐶) / 𝑅)    &   ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑𝜓))    &   ((𝑥𝐵𝑦𝐶) → 𝜑)       (𝐴𝑆𝜓)

Theorem2ecoptocl 6447* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
𝑆 = ((𝐶 × 𝐷) / 𝑅)    &   ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑𝜓))    &   ([⟨𝑧, 𝑤⟩]𝑅 = 𝐵 → (𝜓𝜒))    &   (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)       ((𝐴𝑆𝐵𝑆) → 𝜒)

Theorem3ecoptocl 6448* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
𝑆 = ((𝐷 × 𝐷) / 𝑅)    &   ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑𝜓))    &   ([⟨𝑧, 𝑤⟩]𝑅 = 𝐵 → (𝜓𝜒))    &   ([⟨𝑣, 𝑢⟩]𝑅 = 𝐶 → (𝜒𝜃))    &   (((𝑥𝐷𝑦𝐷) ∧ (𝑧𝐷𝑤𝐷) ∧ (𝑣𝐷𝑢𝐷)) → 𝜑)       ((𝐴𝑆𝐵𝑆𝐶𝑆) → 𝜃)

Theorembrecop 6449* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
∈ V    &    Er (𝐺 × 𝐺)    &   𝐻 = ((𝐺 × 𝐺) / )    &    = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] 𝑦 = [⟨𝑣, 𝑢⟩] ) ∧ 𝜑))}    &   ((((𝑧𝐺𝑤𝐺) ∧ (𝐴𝐺𝐵𝐺)) ∧ ((𝑣𝐺𝑢𝐺) ∧ (𝐶𝐺𝐷𝐺))) → (([⟨𝑧, 𝑤⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝑣, 𝑢⟩] = [⟨𝐶, 𝐷⟩] ) → (𝜑𝜓)))       (((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)) → ([⟨𝐴, 𝐵⟩] [⟨𝐶, 𝐷⟩] 𝜓))

Theoremeroveu 6450* Lemma for eroprf 6452. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐽 = (𝐴 / 𝑅)    &   𝐾 = (𝐵 / 𝑆)    &   (𝜑𝑇𝑍)    &   (𝜑𝑅 Er 𝑈)    &   (𝜑𝑆 Er 𝑉)    &   (𝜑𝑇 Er 𝑊)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑+ :(𝐴 × 𝐵)⟶𝐶)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))       ((𝜑 ∧ (𝑋𝐽𝑌𝐾)) → ∃!𝑧𝑝𝐴𝑞𝐵 ((𝑋 = [𝑝]𝑅𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))

Theoremerovlem 6451* Lemma for eroprf 6452. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
𝐽 = (𝐴 / 𝑅)    &   𝐾 = (𝐵 / 𝑆)    &   (𝜑𝑇𝑍)    &   (𝜑𝑅 Er 𝑈)    &   (𝜑𝑆 Er 𝑉)    &   (𝜑𝑇 Er 𝑊)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑+ :(𝐴 × 𝐵)⟶𝐶)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}       (𝜑 = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))

Theoremeroprf 6452* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
𝐽 = (𝐴 / 𝑅)    &   𝐾 = (𝐵 / 𝑆)    &   (𝜑𝑇𝑍)    &   (𝜑𝑅 Er 𝑈)    &   (𝜑𝑆 Er 𝑉)    &   (𝜑𝑇 Er 𝑊)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑+ :(𝐴 × 𝐵)⟶𝐶)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)    &   𝐿 = (𝐶 / 𝑇)       (𝜑 :(𝐽 × 𝐾)⟶𝐿)

Theoremeroprf2 6453* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐽 = (𝐴 / )    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐴 ((𝑥 = [𝑝] 𝑦 = [𝑞] ) ∧ 𝑧 = [(𝑝 + 𝑞)] )}    &   (𝜑𝑋)    &   (𝜑 Er 𝑈)    &   (𝜑𝐴𝑈)    &   (𝜑+ :(𝐴 × 𝐴)⟶𝐴)    &   ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐴𝑢𝐴))) → ((𝑟 𝑠𝑡 𝑢) → (𝑟 + 𝑡) (𝑠 + 𝑢)))       (𝜑 :(𝐽 × 𝐽)⟶𝐽)

Theoremecopoveq 6454* This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation (specified by the hypothesis) in terms of its operation 𝐹. (Contributed by NM, 16-Aug-1995.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Theoremecopovsym 6455* Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   (𝑥 + 𝑦) = (𝑦 + 𝑥)       (𝐴 𝐵𝐵 𝐴)

Theoremecopovtrn 6456* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   (𝑥 + 𝑦) = (𝑦 + 𝑥)    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))    &   ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))       ((𝐴 𝐵𝐵 𝐶) → 𝐴 𝐶)

Theoremecopover 6457* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   (𝑥 + 𝑦) = (𝑦 + 𝑥)    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))    &   ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))        Er (𝑆 × 𝑆)

Theoremecopovsymg 6458* Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝐴 𝐵𝐵 𝐴)

Theoremecopovtrng 6459* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by Jim Kingdon, 1-Sep-2019.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))       ((𝐴 𝐵𝐵 𝐶) → 𝐴 𝐶)

Theoremecopoverg 6460* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by Jim Kingdon, 1-Sep-2019.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))        Er (𝑆 × 𝑆)

Theoremth3qlem1 6461* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Er 𝑆    &   (((𝑦𝑆𝑤𝑆) ∧ (𝑧𝑆𝑣𝑆)) → ((𝑦 𝑤𝑧 𝑣) → (𝑦 + 𝑧) (𝑤 + 𝑣)))       ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ∃*𝑥𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ))

Theoremth3qlem2 6462* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
∈ V    &    Er (𝑆 × 𝑆)    &   ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))       ((𝐴 ∈ ((𝑆 × 𝑆) / ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] 𝐵 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))

Theoremth3qcor 6463* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
∈ V    &    Er (𝑆 × 𝑆)    &   ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))    &   𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))}       Fun 𝐺

Theoremth3q 6464* Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
∈ V    &    Er (𝑆 × 𝑆)    &   ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))    &   𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))}       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )

Theoremoviec 6465* Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → 𝐻 ∈ (𝑆 × 𝑆))    &   (((𝑎𝑆𝑏𝑆) ∧ (𝑔𝑆𝑆)) → 𝐾 ∈ (𝑆 × 𝑆))    &   (((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆)) → 𝐿 ∈ (𝑆 × 𝑆))    &    ∈ V    &    Er (𝑆 × 𝑆)    &    = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}    &   (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝜑𝜓))    &   (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝜑𝜒))    &    + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝐽))}    &   (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝐽 = 𝐾)    &   (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝐽 = 𝐿)    &   (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝐽 = 𝐻)    &    = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑄𝑦𝑄) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] 𝑦 = [⟨𝑐, 𝑑⟩] ) ∧ 𝑧 = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}    &   𝑄 = ((𝑆 × 𝑆) / )    &   ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((𝜓𝜒) → 𝐾 𝐿))       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] [⟨𝐶, 𝐷⟩] ) = [𝐻] )

Theoremecovcom 6466* Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6467 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐶 = ((𝑆 × 𝑆) / )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )    &   𝐷 = 𝐻    &   𝐺 = 𝐽       ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremecovicom 6467* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
𝐶 = ((𝑆 × 𝑆) / )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐷 = 𝐻)    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐺 = 𝐽)       ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremecovass 6468* Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6469 will be more useful. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )    &   (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))    &   𝐽 = 𝐿    &   𝐾 = 𝑀       ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremecoviass 6469* Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )    &   (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐾 = 𝑀)       ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremecovdi 6470* Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6471 will be more helpful. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑀𝑆𝑁𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = [⟨𝑊, 𝑋⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = [⟨𝑌, 𝑍⟩] )    &   (((𝑊𝑆𝑋𝑆) ∧ (𝑌𝑆𝑍𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑀𝑆𝑁𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝑊𝑆𝑋𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑌𝑆𝑍𝑆))    &   𝐻 = 𝐾    &   𝐽 = 𝐿       ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremecovidi 6471* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑀𝑆𝑁𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = [⟨𝑊, 𝑋⟩] )    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = [⟨𝑌, 𝑍⟩] )    &   (((𝑊𝑆𝑋𝑆) ∧ (𝑌𝑆𝑍𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )    &   (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑀𝑆𝑁𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝑊𝑆𝑋𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑌𝑆𝑍𝑆))    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐻 = 𝐾)    &   (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)       ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

2.6.25  The mapping operation

Syntaxcmap 6472 Extend the definition of a class to include the mapping operation. (Read for 𝐴𝑚 𝐵, "the set of all functions that map from 𝐵 to 𝐴.)
class 𝑚

Syntaxcpm 6473 Extend the definition of a class to include the partial mapping operation. (Read for 𝐴pm 𝐵, "the set of all partial functions that map from 𝐵 to 𝐴.)
class pm

Definitiondf-map 6474* Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴𝑚 𝐵) (see mapval 6484). Many authors write 𝐴 followed by 𝐵 as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g., Definition 10.42 of [TakeutiZaring] p. 95). Other authors show 𝐵 as a prefixed superscript, which is read "𝐴 pre 𝐵 " (e.g., definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map(𝐵, 𝐴) for our (𝐴𝑚 𝐵). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 8-Dec-2003.)
𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})

Definitiondf-pm 6475* Define the partial mapping operation. A partial function from 𝐵 to 𝐴 is a function from a subset of 𝐵 to 𝐴. The set of all partial functions from 𝐵 to 𝐴 is written (𝐴pm 𝐵) (see pmvalg 6483). A notation for this operation apparently does not appear in the literature. We use pm to distinguish it from the less general set exponentiation operation 𝑚 (df-map 6474) . See mapsspm 6506 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})

Theoremmapprc 6476* When 𝐴 is a proper class, the class of all functions mapping 𝐴 to 𝐵 is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)

Theorempmex 6477* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)
((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (Fun 𝑓𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)

Theoremmapex 6478* The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)
((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)

Theoremfnmap 6479 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝑚 Fn (V × V)

Theoremfnpm 6480 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
pm Fn (V × V)

Theoremreldmmap 6481 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Rel dom ↑𝑚

Theoremmapvalg 6482* The value of set exponentiation. (𝐴𝑚 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})

Theorempmvalg 6483* The value of the partial mapping operation. (𝐴pm 𝐵) is the set of all partial functions that map from 𝐵 to 𝐴. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴𝐶𝐵𝐷) → (𝐴pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})

Theoremmapval 6484* The value of set exponentiation (inference version). (𝐴𝑚 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}

Theoremelmapg 6485 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴𝑚 𝐵) ↔ 𝐶:𝐵𝐴))

Theoremelmapd 6486 Deduction form of elmapg 6485. (Contributed by BJ, 11-Apr-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐶 ∈ (𝐴𝑚 𝐵) ↔ 𝐶:𝐵𝐴))

Theoremmapdm0 6487 The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.)
(𝐵𝑉 → (𝐵𝑚 ∅) = {∅})

Theoremelpmg 6488 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))

Theoremelpm2g 6489 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊) → (𝐹 ∈ (𝐴pm 𝐵) ↔ (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵)))

Theoremelpm2r 6490 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐹:𝐶𝐴𝐶𝐵)) → 𝐹 ∈ (𝐴pm 𝐵))

Theoremelpmi 6491 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))

Theorempmfun 6492 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐹 ∈ (𝐴pm 𝐵) → Fun 𝐹)

Theoremelmapex 6493 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)
(𝐴 ∈ (𝐵𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))

Theoremelmapi 6494 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴:𝐶𝐵)

Theoremelmapfn 6495 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
(𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴 Fn 𝐶)

Theoremelmapfun 6496 A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
(𝐴 ∈ (𝐵𝑚 𝐶) → Fun 𝐴)

Theoremelmapssres 6497 A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷) ∈ (𝐵𝑚 𝐷))

Theoremfpmg 6498 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐴𝑉𝐵𝑊𝐹:𝐴𝐵) → 𝐹 ∈ (𝐵pm 𝐴))

Theorempmss12g 6499 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
(((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))

Theorempmresg 6500 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹𝐵) ∈ (𝐴pm 𝐵))

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