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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | erdm 8301 | The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | ||
Theorem | ercl 8302 | Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑋) | ||
Theorem | ersym 8303 | An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐵𝑅𝐴) | ||
Theorem | ercl2 8304 | Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑋) | ||
Theorem | ersymb 8305 | An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) | ||
Theorem | ertr 8306 | An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) ⇒ ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) | ||
Theorem | ertrd 8307 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ertr2d 8308 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐶𝑅𝐴) | ||
Theorem | ertr3d 8309 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐵𝑅𝐴) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ertr4d 8310 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | erref 8311 | An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐴) | ||
Theorem | ercnv 8312 | The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) | ||
Theorem | errn 8313 | 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 8314 | 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 8315 | 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 8316 | 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 8317* | 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 8318* | A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8317, 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 8319* | Alternate proof of iseri 8318, avoiding the usage of mptru 1544 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 8320 | 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 8321* | 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 8322* | 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 8323* | 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 8324* | 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 8325* | Lemma for eqer 8326. (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 8326* | 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 8327 | 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 8328 | 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 8329 | Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) | ||
Theorem | eceq1d 8330 | Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) | ||
Theorem | eceq2 8331 | Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) | ||
Theorem | eceq2i 8332 | 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 8333 | 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 8334 | 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 8335 | 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 8336 | 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 8337 | 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 8338 | 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 8339 | 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 8340 | 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 8341 | 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 8342 | 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 8343 | 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 8344 | An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.) |
⊢ [𝐴] I = {𝐴} | ||
Theorem | qseq1 8345 | Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | ||
Theorem | qseq2 8346 | Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) | ||
Theorem | qseq2i 8347 | Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 / 𝐴) = (𝐶 / 𝐵) | ||
Theorem | qseq2d 8348 | Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) | ||
Theorem | qseq12 8349 | Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷)) | ||
Theorem | elqsg 8350* | Closed form of elqs 8351. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) | ||
Theorem | elqs 8351* | Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) | ||
Theorem | elqsi 8352* | Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) | ||
Theorem | elqsecl 8353* | 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 8354 | 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 8355 | 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 8356 | "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 8357 | A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) | ||
Theorem | qsex 8358 | A quotient set exists. (Contributed by NM, 14-Aug-1995.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 / 𝑅) ∈ V | ||
Theorem | uniqs 8359 | The union of a quotient set. (Contributed by NM, 9-Dec-2008.) |
⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) | ||
Theorem | qsss 8360 | 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 8361 | 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 8362 | 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 8363 | Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.) |
⊢ 𝑅 ∈ V ⇒ ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) | ||
Theorem | ecid 8364 | 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 8365 | 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 8366* | 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 8367* | 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 8368 | A quotient set does not contain the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) | ||
Theorem | ecelqsdm 8369 | 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 8370 | 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 8371* | 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 8372* | 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 8373 | 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 8374 | 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 8375 | 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 8376 | 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 8377* | A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.) |
⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) | ||
Theorem | qsel 8378 | 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 8379 | Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4864. (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 8380* | 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 8381* | 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 8382* | 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 8383* | 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 8384* | 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 8385* | 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 8386* | 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 8387* | 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 8388* | 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 8389* | 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 8390* | 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 8391* | 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 8392* | 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 8393 | 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 8394* | Lemma for erov 8396 and eroprf 8397. (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 8395* | Lemma for erov 8396 and eroprf 8397. (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 8396* | 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 8397* | 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 8398* | 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 8399* | 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 8400* | 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))) |
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