Theorem List for Intuitionistic Logic Explorer - 6201-6300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | swoord1 6201* |
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 6202* |
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 | eqerlem 6203* |
Lemma for eqer 6204. (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 6204* |
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.)
|
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
& ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} ⇒ ⊢ 𝑅 Er V |
|
Theorem | ider 6205 |
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 6206 |
The empty set is an equivalence relation on the empty set. (Contributed
by Mario Carneiro, 5-Sep-2015.)
|
⊢ ∅ Er ∅ |
|
Theorem | eceq1 6207 |
Equality theorem for equivalence class. (Contributed by NM,
23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
|
Theorem | eceq1d 6208 |
Equality theorem for equivalence class (deduction form). (Contributed
by Jim Kingdon, 31-Dec-2019.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) |
|
Theorem | eceq2 6209 |
Equality theorem for equivalence class. (Contributed by NM,
23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
|
Theorem | elecg 6210 |
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 6211 |
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 6212 |
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 6213 |
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 | ecdmn0m 6214* |
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) |
|
Theorem | ereldm 6215 |
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 6216 |
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 6217 |
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 6218 |
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 | ecidsn 6219 |
An equivalence class modulo the identity relation is a singleton.
(Contributed by NM, 24-Oct-2004.)
|
⊢ [𝐴] I = {𝐴} |
|
Theorem | qseq1 6220 |
Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) |
|
Theorem | qseq2 6221 |
Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) |
|
Theorem | elqsg 6222* |
Closed form of elqs 6223. (Contributed by Rodolfo Medina,
12-Oct-2010.)
|
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
|
Theorem | elqs 6223* |
Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
|
Theorem | elqsi 6224* |
Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
|
Theorem | ecelqsg 6225 |
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 6226 |
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 6227 |
"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 6228 |
A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by
Mario Carneiro, 9-Jul-2014.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) |
|
Theorem | qsex 6229 |
A quotient set exists. (Contributed by NM, 14-Aug-1995.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 / 𝑅) ∈ V |
|
Theorem | uniqs 6230 |
The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
|
⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
|
Theorem | qsss 6231 |
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 6232 |
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 6233 |
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 6234 |
Equivalence class in terms of quotient set. (Contributed by NM,
29-Jan-1999.)
|
⊢ 𝑅 ∈ V ⇒ ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) |
|
Theorem | ecid 6235 |
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 |
|
Theorem | ecidg 6236 |
A set is equal to its converse epsilon coset. (Note: converse epsilon
is not an equivalence relation.) (Contributed by Jim Kingdon,
8-Jan-2020.)
|
⊢ (𝐴 ∈ 𝑉 → [𝐴]◡
E = 𝐴) |
|
Theorem | qsid 6237 |
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 |
|
Theorem | ectocld 6238* |
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 6239* |
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 | elqsn0m 6240* |
An element of a quotient set is inhabited. (Contributed by Jim Kingdon,
21-Aug-2019.)
|
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ 𝐵) |
|
Theorem | elqsn0 6241 |
A quotient set doesn't contain the empty set. (Contributed by NM,
24-Aug-1995.)
|
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) |
|
Theorem | ecelqsdm 6242 |
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 | xpiderm 6243* |
A square Cartesian product is an equivalence relation (in general it's
not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)
|
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 × 𝐴) Er 𝐴) |
|
Theorem | iinerm 6244* |
The intersection of a nonempty family of equivalence relations is an
equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
|
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) |
|
Theorem | riinerm 6245* |
The relative intersection of a family of equivalence relations is an
equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
|
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ ∩
𝑥 ∈ 𝐴 𝑅) Er 𝐵) |
|
Theorem | erinxp 6246 |
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 6247 |
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 6248 |
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 | qsel 6249 |
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 | qliftlem 6250* |
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 6251* |
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 6252* |
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 6253* |
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 6254* |
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 6255* |
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 6256* |
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 6257* |
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 6258* |
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 6259* |
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 6260* |
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 6261* |
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 6262* |
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 | eroveu 6263* |
Lemma for eroprf 6265. (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 6264* |
Lemma for eroprf 6265. (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 | eroprf 6265* |
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 | eroprf2 6266* |
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 6267* |
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 6268* |
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 6269* |
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 6270* |
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)) ∧ ∃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 | ecopovsymg 6271* |
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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) ⇒ ⊢ (𝐴 ∼ 𝐵 → 𝐵 ∼ 𝐴) |
|
Theorem | ecopovtrng 6272* |
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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
& ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
& ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶) |
|
Theorem | ecopoverg 6273* |
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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
& ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
& ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ∼ Er (𝑆 × 𝑆) |
|
Theorem | th3qlem1 6274* |
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 & ⊢ (((𝑦 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ((𝑦 ∼ 𝑤 ∧ 𝑧 ∼ 𝑣) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣))) ⇒ ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) →
∃*𝑥∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼
)) |
|
Theorem | th3qlem2 6275* |
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)
& ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((〈𝑤, 𝑣〉 ∼ 〈𝑢, 𝑡〉 ∧ 〈𝑠, 𝑓〉 ∼ 〈𝑔, ℎ〉) → (〈𝑤, 𝑣〉 + 〈𝑠, 𝑓〉) ∼ (〈𝑢, 𝑡〉 + 〈𝑔, ℎ〉))) ⇒ ⊢ ((𝐴 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / ∼ )) →
∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [〈𝑤, 𝑣〉] ∼ ∧ 𝐵 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼
)) |
|
Theorem | th3qcor 6276* |
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)
& ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((〈𝑤, 𝑣〉 ∼ 〈𝑢, 𝑡〉 ∧ 〈𝑠, 𝑓〉 ∼ 〈𝑔, ℎ〉) → (〈𝑤, 𝑣〉 + 〈𝑠, 𝑓〉) ∼ (〈𝑢, 𝑡〉 + 〈𝑔, ℎ〉))) & ⊢ 𝐺 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧
∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼
))} ⇒ ⊢ Fun 𝐺 |
|
Theorem | th3q 6277* |
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)
& ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((〈𝑤, 𝑣〉 ∼ 〈𝑢, 𝑡〉 ∧ 〈𝑠, 𝑓〉 ∼ 〈𝑔, ℎ〉) → (〈𝑤, 𝑣〉 + 〈𝑠, 𝑓〉) ∼ (〈𝑢, 𝑡〉 + 〈𝑔, ℎ〉))) & ⊢ 𝐺 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧
∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼
))} ⇒ ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([〈𝐴, 𝐵〉] ∼ 𝐺[〈𝐶, 𝐷〉] ∼ ) = [(〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉)] ∼ ) |
|
Theorem | oviec 6278* |
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.)
|
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐻 ∈ (𝑆 × 𝑆)) & ⊢ (((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → 𝐾 ∈ (𝑆 × 𝑆)) & ⊢ (((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → 𝐿 ∈ (𝑆 × 𝑆)) & ⊢ ∼
∈ V
& ⊢ ∼ Er (𝑆 × 𝑆)
& ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ 𝜑))} & ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝜑 ↔ 𝜓)) & ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝜑 ↔ 𝜒)) & ⊢ + =
{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝐽))} & ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝐽 = 𝐾)
& ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝐽 = 𝐿)
& ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝐽 = 𝐻)
& ⊢ ⨣ =
{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [〈𝑎, 𝑏〉] ∼ ∧ 𝑦 = [〈𝑐, 𝑑〉] ∼ ) ∧ 𝑧 = [(〈𝑎, 𝑏〉 + 〈𝑐, 𝑑〉)] ∼ ))} & ⊢ 𝑄 = ((𝑆 × 𝑆) / ∼ ) & ⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((𝜓 ∧ 𝜒) → 𝐾 ∼ 𝐿)) ⇒ ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([〈𝐴, 𝐵〉] ∼ ⨣ [〈𝐶, 𝐷〉] ∼ ) = [𝐻] ∼ ) |
|
Theorem | ecovcom 6279* |
Lemma used to transfer a commutative law via an equivalence relation.
Most uses will want ecovicom 6280 instead. (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 | ecovicom 6280* |
Lemma used to transfer a commutative law via an equivalence relation.
(Contributed by Jim Kingdon, 15-Sep-2019.)
|
⊢ 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〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐷 = 𝐻)
& ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐺 = 𝐽) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
|
Theorem | ecovass 6281* |
Lemma used to transfer an associative law via an equivalence relation.
In most cases ecoviass 6282 will be more useful. (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 | ecoviass 6282* |
Lemma used to transfer an associative law via an equivalence relation.
(Contributed by Jim Kingdon, 16-Sep-2019.)
|
⊢ 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)) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐽 = 𝐿)
& ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐾 = 𝑀) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
|
Theorem | ecovdi 6283* |
Lemma used to transfer a distributive law via an equivalence relation.
Most likely ecovidi 6284 will be more helpful. (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))) |
|
Theorem | ecovidi 6284* |
Lemma used to transfer a distributive law via an equivalence relation.
(Contributed by Jim Kingdon, 17-Sep-2019.)
|
⊢ 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)) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐻 = 𝐾)
& ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐽 = 𝐿) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
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2.6.25 Equinumerosity
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Syntax | cen 6285 |
Extend class definition to include the equinumerosity relation
("approximately equals" symbol)
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class ≈ |
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Syntax | cdom 6286 |
Extend class definition to include the dominance relation (curly
less-than-or-equal)
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class ≼ |
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Syntax | cfn 6287 |
Extend class definition to include the class of all finite sets.
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class Fin |
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Definition | df-en 6288* |
Define the equinumerosity relation. Definition of [Enderton] p. 129.
We define ≈ to be a binary relation rather
than a connective, so
its arguments must be sets to be meaningful. This is acceptable because
we do not consider equinumerosity for proper classes. We derive the
usual definition as bren 6294. (Contributed by NM, 28-Mar-1998.)
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⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
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Definition | df-dom 6289* |
Define the dominance relation. Compare Definition of [Enderton] p. 145.
Typical textbook definitions are derived as brdom 6297 and domen 6298.
(Contributed by NM, 28-Mar-1998.)
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⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} |
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Definition | df-fin 6290* |
Define the (proper) class of all finite sets. Similar to Definition
10.29 of [TakeutiZaring] p. 91,
whose "Fin(a)" corresponds to
our "𝑎 ∈ Fin". This definition is
meaningful whether or not we
accept the Axiom of Infinity ax-inf2 10929. (Contributed by NM,
22-Aug-2008.)
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⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} |
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Theorem | relen 6291 |
Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
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⊢ Rel ≈ |
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Theorem | reldom 6292 |
Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
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⊢ Rel ≼ |
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Theorem | encv 6293 |
If two classes are equinumerous, both classes are sets. (Contributed by
AV, 21-Mar-2019.)
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⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
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Theorem | bren 6294* |
Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
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⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
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Theorem | brdomg 6295* |
Dominance relation. (Contributed by NM, 15-Jun-1998.)
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⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
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Theorem | brdomi 6296* |
Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
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⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
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Theorem | brdom 6297* |
Dominance relation. (Contributed by NM, 15-Jun-1998.)
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⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
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Theorem | domen 6298* |
Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146.
(Contributed by NM, 15-Jun-1998.)
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⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
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Theorem | domeng 6299* |
Dominance in terms of equinumerosity, with the sethood requirement
expressed as an antecedent. Example 1 of [Enderton] p. 146.
(Contributed by NM, 24-Apr-2004.)
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⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
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Theorem | ctex 6300 |
A countable set is a set. (Contributed by Thierry Arnoux,
29-Dec-2016.)
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⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |