Home | Metamath
Proof Explorer Theorem List (p. 84 of 449) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28623) |
Hilbert Space Explorer
(28624-30146) |
Users' Mathboxes
(30147-44804) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | errn 8301 | The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | ||
Theorem | erssxp 8302 | An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) | ||
Theorem | erex 8303 | An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V)) | ||
Theorem | erexb 8304 | An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V)) | ||
Theorem | iserd 8305* | A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) & ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) ⇒ ⊢ (𝜑 → 𝑅 Er 𝐴) | ||
Theorem | iseri 8306* | A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8305, which avoids the need to provide a "dummy antecedent" 𝜑 if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.) |
⊢ Rel 𝑅 & ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥) & ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) & ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥) ⇒ ⊢ 𝑅 Er 𝐴 | ||
Theorem | iseriALT 8307* | Alternate proof of iseri 8306, avoiding the usage of mptru 1535 and ⊤ as antecedent by using ax-mp 5 and one of the hypotheses as antecedent. This results, however, in a slightly longer proof. (Contributed by AV, 30-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Rel 𝑅 & ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥) & ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) & ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥) ⇒ ⊢ 𝑅 Er 𝐴 | ||
Theorem | brdifun 8308 | Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | swoer 8309* | Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))) ⇒ ⊢ (𝜑 → 𝑅 Er 𝑋) | ||
Theorem | swoord1 8310* | The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.) |
⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶)) | ||
Theorem | swoord2 8311* | The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.) |
⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐶 < 𝐴 ↔ 𝐶 < 𝐵)) | ||
Theorem | swoso 8312* | If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦) ⇒ ⊢ (𝜑 → < Or 𝑌) | ||
Theorem | eqerlem 8313* | Lemma for eqer 8314. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} ⇒ ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | ||
Theorem | eqer 8314* | Equivalence relation involving equality of dependent classes 𝐴(𝑥) and 𝐵(𝑦). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof shortened by AV, 1-May-2021.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} ⇒ ⊢ 𝑅 Er V | ||
Theorem | ider 8315 | The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.) |
⊢ I Er V | ||
Theorem | 0er 8316 | The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 1-May-2021.) |
⊢ ∅ Er ∅ | ||
Theorem | eceq1 8317 | Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) | ||
Theorem | eceq1d 8318 | Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) | ||
Theorem | eceq2 8319 | Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) | ||
Theorem | eceq2i 8320 | Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, inference version. (Contributed by Peter Mazsa, 11-May-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ [𝐶]𝐴 = [𝐶]𝐵 | ||
Theorem | eceq2d 8321 | Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵) | ||
Theorem | elecg 8322 | Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | ||
Theorem | elec 8323 | Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴) | ||
Theorem | relelec 8324 | Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | ||
Theorem | ecss 8325 | An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) ⇒ ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋) | ||
Theorem | ecdmn0 8326 | A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) | ||
Theorem | ereldm 8327 | Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | ||
Theorem | erth 8328 | Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) | ||
Theorem | erth2 8329 | 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.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) | ||
Theorem | erthi 8330 | 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.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | ||
Theorem | erdisj 8331 | Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)) | ||
Theorem | ecidsn 8332 | An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.) |
⊢ [𝐴] I = {𝐴} | ||
Theorem | qseq1 8333 | Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | ||
Theorem | qseq2 8334 | Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) | ||
Theorem | qseq2i 8335 | Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 / 𝐴) = (𝐶 / 𝐵) | ||
Theorem | qseq2d 8336 | Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) | ||
Theorem | qseq12 8337 | Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷)) | ||
Theorem | elqsg 8338* | Closed form of elqs 8339. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) | ||
Theorem | elqs 8339* | Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) | ||
Theorem | elqsi 8340* | Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) | ||
Theorem | elqsecl 8341* | Membership in a quotient set by an equivalence class according to ∼. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦})) | ||
Theorem | ecelqsg 8342 | Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | ||
Theorem | ecelqsi 8343 | Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝑅 ∈ V ⇒ ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | ||
Theorem | ecopqsi 8344 | "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.) |
⊢ 𝑅 ∈ V & ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅) ⇒ ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ 𝑆) | ||
Theorem | qsexg 8345 | A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) | ||
Theorem | qsex 8346 | A quotient set exists. (Contributed by NM, 14-Aug-1995.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 / 𝑅) ∈ V | ||
Theorem | uniqs 8347 | The union of a quotient set. (Contributed by NM, 9-Dec-2008.) |
⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) | ||
Theorem | qsss 8348 | 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.) |
⊢ (𝜑 → 𝑅 Er 𝐴) ⇒ ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴) | ||
Theorem | uniqs2 8349 | The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) | ||
Theorem | snec 8350 | 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) | ||
Theorem | ecqs 8351 | Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.) |
⊢ 𝑅 ∈ V ⇒ ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) | ||
Theorem | ecid 8352 | A set is equal to its coset under the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ [𝐴]◡ E = 𝐴 | ||
Theorem | qsid 8353 | A set is equal to its quotient set modulo the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 / ◡ E ) = 𝐴 | ||
Theorem | ectocld 8354* | Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝑆 = (𝐵 / 𝑅) & ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑) ⇒ ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓) | ||
Theorem | ectocl 8355* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ 𝐵 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑆 → 𝜓) | ||
Theorem | elqsn0 8356 | A quotient set does not contain the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) | ||
Theorem | ecelqsdm 8357 | Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.) |
⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵 ∈ 𝐴) | ||
Theorem | xpider 8358 | A Cartesian square is an equivalence relation (in general, it is not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝐴 × 𝐴) Er 𝐴 | ||
Theorem | iiner 8359* | The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) | ||
Theorem | riiner 8360* | The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) | ||
Theorem | erinxp 8361 | A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵) | ||
Theorem | ecinxp 8362 | Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.) |
⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴))) | ||
Theorem | qsinxp 8363 | Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴)))) | ||
Theorem | qsdisj 8364 | Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅)) ⇒ ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) | ||
Theorem | qsdisj2 8365* | A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.) |
⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) | ||
Theorem | qsel 8366 | 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 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅) | ||
Theorem | uniinqs 8367 | Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4852. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝑅 Er 𝑋 ⇒ ⊢ ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → ∪ (𝐵 ∩ 𝐶) = (∪ 𝐵 ∩ ∪ 𝐶)) | ||
Theorem | qliftlem 8368* | 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) | ||
Theorem | qliftrel 8369* | 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → 𝐹 ⊆ ((𝑋 / 𝑅) × 𝑌)) | ||
Theorem | qliftel 8370* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴))) | ||
Theorem | qliftel1 8371* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅𝐹𝐴) | ||
Theorem | qliftfun 8372* | 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 (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵))) | ||
Theorem | qliftfund 8373* | 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 (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | qliftfuns 8374* | 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 (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴))) | ||
Theorem | qliftf 8375* | The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌)) | ||
Theorem | qliftval 8376* | The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵) | ||
Theorem | ecoptocl 8377* | Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅) & ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑆 → 𝜓) | ||
Theorem | 2ecoptocl 8378* | Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝑆 = ((𝐶 × 𝐷) / 𝑅) & ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ([〈𝑧, 𝑤〉]𝑅 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝜒) | ||
Theorem | 3ecoptocl 8379* | Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.) |
⊢ 𝑆 = ((𝐷 × 𝐷) / 𝑅) & ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ([〈𝑧, 𝑤〉]𝑅 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ ([〈𝑣, 𝑢〉]𝑅 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐷)) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝜃) | ||
Theorem | brecop 8380* | 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〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))} & ⊢ ((((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺)) ∧ ((𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (𝜑 ↔ 𝜓))) ⇒ ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈𝐴, 𝐵〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔ 𝜓)) | ||
Theorem | brecop2 8381 | Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.) (Revised by AV, 12-Jul-2022.) |
⊢ dom ∼ = (𝐺 × 𝐺) & ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ ) & ⊢ 𝑅 ⊆ (𝐻 × 𝐻) & ⊢ ≤ ⊆ (𝐺 × 𝐺) & ⊢ ¬ ∅ ∈ 𝐺 & ⊢ dom + = (𝐺 × 𝐺) & ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈𝐴, 𝐵〉] ∼ 𝑅[〈𝐶, 𝐷〉] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶))) ⇒ ⊢ ([〈𝐴, 𝐵〉] ∼ 𝑅[〈𝐶, 𝐷〉] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶)) | ||
Theorem | eroveu 8382* | Lemma for erov 8384 and eroprf 8385. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝐽 = (𝐴 / 𝑅) & ⊢ 𝐾 = (𝐵 / 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) | ||
Theorem | erovlem 8383* | Lemma for erov 8384 and eroprf 8385. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ 𝐽 = (𝐴 / 𝑅) & ⊢ 𝐾 = (𝐵 / 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} ⇒ ⊢ (𝜑 → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))) | ||
Theorem | erov 8384* | The value 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) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ([𝑃]𝑅 ⨣ [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇) | ||
Theorem | eroprf 8385* | 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) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ 𝐿 = (𝐶 / 𝑇) ⇒ ⊢ (𝜑 → ⨣ :(𝐽 × 𝐾)⟶𝐿) | ||
Theorem | erov2 8386* | The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐽 = (𝐴 / ∼ ) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑥 = [𝑝] ∼ ∧ 𝑦 = [𝑞] ∼ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∼ )} & ⊢ (𝜑 → ∼ ∈ 𝑋) & ⊢ (𝜑 → ∼ Er 𝑈) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → + :(𝐴 × 𝐴)⟶𝐴) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴))) → ((𝑟 ∼ 𝑠 ∧ 𝑡 ∼ 𝑢) → (𝑟 + 𝑡) ∼ (𝑠 + 𝑢))) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ([𝑃] ∼ ⨣ [𝑄] ∼ ) = [(𝑃 + 𝑄)] ∼ ) | ||
Theorem | eroprf2 8387* | Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐽 = (𝐴 / ∼ ) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑥 = [𝑝] ∼ ∧ 𝑦 = [𝑞] ∼ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∼ )} & ⊢ (𝜑 → ∼ ∈ 𝑋) & ⊢ (𝜑 → ∼ Er 𝑈) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → + :(𝐴 × 𝐴)⟶𝐴) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴))) → ((𝑟 ∼ 𝑠 ∧ 𝑡 ∼ 𝑢) → (𝑟 + 𝑡) ∼ (𝑠 + 𝑢))) ⇒ ⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽) | ||
Theorem | ecopoveq 8388* | 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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⇒ ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (〈𝐴, 𝐵〉 ∼ 〈𝐶, 𝐷〉 ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) | ||
Theorem | ecopovsym 8389* | 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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) ⇒ ⊢ (𝐴 ∼ 𝐵 → 𝐵 ∼ 𝐴) | ||
Theorem | ecopovtrn 8390* | 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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶) | ||
Theorem | ecopover 8391* | 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.) (Proof shortened by AV, 1-May-2021.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ∼ Er (𝑆 × 𝑆) | ||
Theorem | eceqoveq 8392* | Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ Er (𝑆 × 𝑆) & ⊢ dom + = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (〈𝐴, 𝐵〉 ∼ 〈𝐶, 𝐷〉 ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ([〈𝐴, 𝐵〉] ∼ = [〈𝐶, 𝐷〉] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) | ||
Theorem | ecovcom 8393* | Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐷, 𝐺〉] ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) & ⊢ 𝐷 = 𝐻 & ⊢ 𝐺 = 𝐽 ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | ecovass 8394* | Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐺, 𝐻〉] ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑣, 𝑢〉] ∼ ) = [〈𝑁, 𝑄〉] ∼ ) & ⊢ (((𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝐺, 𝐻〉] ∼ + [〈𝑣, 𝑢〉] ∼ ) = [〈𝐽, 𝐾〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑁, 𝑄〉] ∼ ) = [〈𝐿, 𝑀〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆)) & ⊢ 𝐽 = 𝐿 & ⊢ 𝐾 = 𝑀 ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | ecovdi 8395* | Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑣, 𝑢〉] ∼ ) = [〈𝑀, 𝑁〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ · [〈𝑀, 𝑁〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ · [〈𝑧, 𝑤〉] ∼ ) = [〈𝑊, 𝑋〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ · [〈𝑣, 𝑢〉] ∼ ) = [〈𝑌, 𝑍〉] ∼ ) & ⊢ (((𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆)) → ([〈𝑊, 𝑋〉] ∼ + [〈𝑌, 𝑍〉] ∼ ) = [〈𝐾, 𝐿〉] ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆)) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆)) & ⊢ 𝐻 = 𝐾 & ⊢ 𝐽 = 𝐿 ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Syntax | cmap 8396 | Extend the definition of a class to include the mapping operation. (Read for 𝐴 ↑m 𝐵, "the set of all functions that map from 𝐵 to 𝐴.) |
class ↑m | ||
Syntax | cpm 8397 | Extend the definition of a class to include the partial mapping operation. (Read for 𝐴 ↑pm 𝐵, "the set of all partial functions that map from 𝐵 to 𝐴.) |
class ↑pm | ||
Definition | df-map 8398* | Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴 ↑m 𝐵) (see mapval 8408). 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 (𝐴 ↑m 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 ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | ||
Definition | df-pm 8399* | 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 (𝐴 ↑pm 𝐵) (see pmvalg 8407). 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 8398). See mapsspm 8430 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.) |
⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | ||
Theorem | mapprc 8400* | 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 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |