Home | Metamath
Proof Explorer Theorem List (p. 78 of 426) | < 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
(1-27775) |
Hilbert Space Explorer
(27776-29300) |
Users' Mathboxes
(29301-42551) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nnawordi 7701 | Adding to both sides of an inequality in ω. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))) | ||
Theorem | nnaass 7702 | Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))) | ||
Theorem | nndi 7703 | Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶))) | ||
Theorem | nnmass 7704 | Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶))) | ||
Theorem | nnmsucr 7705 | Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)) | ||
Theorem | nnmcom 7706 | Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) | ||
Theorem | nnaword 7707 | Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))) | ||
Theorem | nnacan 7708 | Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | nnaword1 7709 | Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝐵)) | ||
Theorem | nnaword2 7710 | Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +𝑜 𝐴)) | ||
Theorem | nnmordi 7711 | Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))) | ||
Theorem | nnmord 7712 | Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))) | ||
Theorem | nnmword 7713 | Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) | ||
Theorem | nnmcan 7714 | Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | nnmwordi 7715 | Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) | ||
Theorem | nnmwordri 7716 | Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶))) | ||
Theorem | nnawordex 7717* | Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)) | ||
Theorem | nnaordex 7718* | Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))) | ||
Theorem | 1onn 7719 | One is a natural number. (Contributed by NM, 29-Oct-1995.) |
⊢ 1𝑜 ∈ ω | ||
Theorem | 2onn 7720 | The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.) |
⊢ 2𝑜 ∈ ω | ||
Theorem | 3onn 7721 | The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 3𝑜 ∈ ω | ||
Theorem | 4onn 7722 | The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 4𝑜 ∈ ω | ||
Theorem | oaabslem 7723 | Lemma for oaabs 7724. (Contributed by NM, 9-Dec-2004.) |
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) = ω) | ||
Theorem | oaabs 7724 | Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +𝑜 𝐵) = 𝐵) | ||
Theorem | oaabs2 7725 | The absorption law oaabs 7724 is also a property of higher powers of ω. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ (ω ↑𝑜 𝐶) ∧ 𝐵 ∈ On) ∧ (ω ↑𝑜 𝐶) ⊆ 𝐵) → (𝐴 +𝑜 𝐵) = 𝐵) | ||
Theorem | omabslem 7726 | Lemma for omabs 7727. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω) | ||
Theorem | omabs 7727 | Ordinal multiplication is also absorbed by powers of ω. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵)) | ||
Theorem | nnm1 7728 | Multiply an element of ω by 1𝑜. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 1𝑜) = 𝐴) | ||
Theorem | nnm2 7729 | Multiply an element of ω by 2𝑜. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴)) | ||
Theorem | nn2m 7730 | Multiply an element of ω by 2𝑜. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (2𝑜 ·𝑜 𝐴) = (𝐴 +𝑜 𝐴)) | ||
Theorem | nnneo 7731 | If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ suc 𝐶 = (2𝑜 ·𝑜 𝐵)) | ||
Theorem | nneob 7732* | A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))) | ||
Theorem | omsmolem 7733* | Lemma for omsmo 7734. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.) |
⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) | ||
Theorem | omsmo 7734* | A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.) |
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1→𝐴) | ||
Theorem | omopthlem1 7735 | Lemma for omopthi 7737. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐶 ∈ ω ⇒ ⊢ (𝐴 ∈ 𝐶 → ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶)) | ||
Theorem | omopthlem2 7736 | Lemma for omopthi 7737. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω & ⊢ 𝐶 ∈ ω & ⊢ 𝐷 ∈ ω ⇒ ⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵)) | ||
Theorem | omopthi 7737 | An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 13057. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω & ⊢ 𝐶 ∈ ω & ⊢ 𝐷 ∈ ω ⇒ ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | omopth 7738 | An ordered pair theorem for finite integers. Analogous to nn0opthi 13057. (Contributed by Scott Fenton, 1-May-2012.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ ω)) → ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (((𝐶 +𝑜 𝐷) ·𝑜 (𝐶 +𝑜 𝐷)) +𝑜 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Syntax | wer 7739 | Extend the definition of a wff to include the equivalence predicate. |
wff 𝑅 Er 𝐴 | ||
Syntax | cec 7740 | Extend the definition of a class to include equivalence class. |
class [𝐴]𝑅 | ||
Syntax | cqs 7741 | Extend the definition of a class to include quotient set. |
class (𝐴 / 𝑅) | ||
Definition | df-er 7742 | Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 7743 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 7762, ersymb 7756, and ertr 7757. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.) |
⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | ||
Theorem | dfer2 7743* | Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))) | ||
Definition | df-ec 7744 | Define the 𝑅-coset of 𝐴. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of 𝐴 modulo 𝑅 when 𝑅 is an equivalence relation (i.e. when Er 𝑅; see dfer2 7743). In this case, 𝐴 is a representative (member) of the equivalence class [𝐴]𝑅, which contains all sets that are equivalent to 𝐴. Definition of [Enderton] p. 57 uses the notation [𝐴] (subscript) 𝑅, although we simply follow the brackets by 𝑅 since we don't have subscripted expressions. For an alternate definition, see dfec2 7745. (Contributed by NM, 23-Jul-1995.) |
⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | ||
Theorem | dfec2 7745* | Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦}) | ||
Theorem | ecexg 7746 | An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.) |
⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) | ||
Theorem | ecexr 7747 | A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) | ||
Definition | df-qs 7748* | Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | ||
Theorem | ereq1 7749 | Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴)) | ||
Theorem | ereq2 7750 | Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝐴 = 𝐵 → (𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵)) | ||
Theorem | errel 7751 | An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → Rel 𝑅) | ||
Theorem | erdm 7752 | The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | ||
Theorem | ercl 7753 | 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 7754 | 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 7755 | 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 7756 | 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 7757 | 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 7758 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ertr2d 7759 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐶𝑅𝐴) | ||
Theorem | ertr3d 7760 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐵𝑅𝐴) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ertr4d 7761 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | erref 7762 | 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 7763 | The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) | ||
Theorem | errn 7764 | 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 7765 | 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 7766 | 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 7767 | 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 7768* | 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 7769* | A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 7768, 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 7770* | Alternate proof of iseri 7769, avoiding the usage of trud 1493 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 7771 | 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 7772* | 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 7773* | 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 7774* | 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 7775* | 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 7776* | Lemma for eqer 7777. (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 7777* | 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 | eqerOLD 7778* | Obsolete proof of eqer 7777 as of 1-May-2021. 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 modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} ⇒ ⊢ 𝑅 Er V | ||
Theorem | ider 7779 | 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 7780 | 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 | 0erOLD 7781 | Obsolete proof of 0er 7780 as of 1-May-2021. The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∅ Er ∅ | ||
Theorem | eceq1 7782 | Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) | ||
Theorem | eceq1d 7783 | Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) | ||
Theorem | eceq2 7784 | Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) | ||
Theorem | elecg 7785 | 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 7786 | 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 7787 | 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 7788 | 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 7789 | 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 7790 | 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 7791 | 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 7792 | 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 7793 | 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 7794 | 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 7795 | An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.) |
⊢ [𝐴] I = {𝐴} | ||
Theorem | qseq1 7796 | Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | ||
Theorem | qseq2 7797 | Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) | ||
Theorem | elqsg 7798* | Closed form of elqs 7799. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) | ||
Theorem | elqs 7799* | Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) | ||
Theorem | elqsi 7800* | Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |