P. 145 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

hash2pwpr 14401

If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.)

⊢ (((♯‘𝑃) = 2 ∧ 𝑃 ∈ 𝒫 {𝑋, 𝑌}) → 𝑃 = {𝑋, 𝑌})

Theorem

hashle2pr 14402*

A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.)

⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((♯‘𝑃) ≤ 2 ↔ ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))

Theorem

hashle2prv 14403*

A nonempty subset of a powerset of a class 𝑉 has size less than or equal to two iff it is an unordered pair of elements of 𝑉. (Contributed by AV, 24-Nov-2021.)

⊢ (𝑃 ∈ (𝒫 𝑉 ∖ {∅}) → ((♯‘𝑃) ≤ 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑃 = {𝑎, 𝑏}))

Theorem

pr2pwpr 14404*

The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝑝 ∈ 𝒫 {𝐴, 𝐵} ∣ 𝑝 ≈ 2_o} = {{𝐴, 𝐵}})

Theorem

hashge2el2dif 14405*

A set with size at least 2 has at least 2 different elements. (Contributed by AV, 18-Mar-2019.)

⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦)

Theorem

hashge2el2difr 14406*

A set with at least 2 different elements has size at least 2. (Contributed by AV, 14-Oct-2020.)

⊢ ((𝐷 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦) → 2 ≤ (♯‘𝐷))

Theorem

hashge2el2difb 14407*

A set has size at least 2 iff it has at least 2 different elements. (Contributed by AV, 14-Oct-2020.)

⊢ (𝐷 ∈ 𝑉 → (2 ≤ (♯‘𝐷) ↔ ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦))

Theorem

hashdmpropge2 14408

The size of the domain of a class which contains two ordered pairs with different first components is greater than or equal to 2. (Contributed by AV, 12-Nov-2021.)

⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ 𝐹)    ⇒   ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐹))

Theorem

hashtplei 14409

An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

⊢ ({𝐴, 𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐵, 𝐶}) ≤ 3)

Theorem

hashtpg 14410

The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.)

⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (♯‘{𝐴, 𝐵, 𝐶}) = 3))

Theorem

hash7g 14411

The size of an unordered set of seven different elements. (Contributed by AV, 2-Aug-2025.)

⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐷 ∈ 𝑉 ∧ (𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐴 ≠ 𝐸 ∧ 𝐴 ≠ 𝐹 ∧ 𝐴 ≠ 𝐺)) ∧ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ (𝐵 ≠ 𝐸 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐺)) ∧ (𝐶 ≠ 𝐷 ∧ (𝐶 ≠ 𝐸 ∧ 𝐶 ≠ 𝐹 ∧ 𝐶 ≠ 𝐺))) ∧ ((𝐷 ≠ 𝐸 ∧ 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐺) ∧ (𝐸 ≠ 𝐹 ∧ 𝐸 ≠ 𝐺 ∧ 𝐹 ≠ 𝐺)))) → (♯‘(({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})) = 7)

Theorem

hashge3el3dif 14412*

A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 14405, which has an elementary proof, the dominance relation and 1-1 functions from a set with three elements which are known to be different are used to prove this theorem. Although there is also an elementary proof for this theorem, it might be much longer. After all, this proof should be kept because it can be used as template for proofs for higher cardinalities. (Contributed by AV, 20-Mar-2019.) (Proof modification is discouraged.)

⊢ ((𝐷 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐷)) → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 (𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧))

Theorem

elss2prb 14413*

An element of the set of subsets with two elements is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)

⊢ (𝑃 ∈ {𝑧 ∈ 𝒫 𝑉 ∣ (♯‘𝑧) = 2} ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}))

Theorem

hash2sspr 14414*

A subset of size two is an unordered pair of elements of its superset. (Contributed by Alexander van der Vekens, 12-Jul-2017.) (Proof shortened by AV, 4-Nov-2020.)

⊢ ((𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑃 = {𝑎, 𝑏})

Theorem

exprelprel 14415*

If there is an element of the set of subsets with two elements in a set, an unordered pair of sets is in the set. (Contributed by Alexander van der Vekens, 12-Jul-2018.)

⊢ (∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}𝑝 ∈ 𝑋 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋)

Theorem

hash3tr 14416*

A set of size three is an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.)

⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐})

Theorem

hash1to3 14417*

If the size of a set is between 1 and 3 (inclusively), the set is a singleton or an unordered pair or an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.)

⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))

Theorem

hash3tpde 14418*

A set of size three is an unordered triple of three different elements. (Contributed by AV, 21-Jul-2025.)

⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}))

Theorem

hash3tpexb 14419*

A set of size three is an unordered triple if and only if it contains three different elements. (Contributed by AV, 21-Jul-2025.)

⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 3 ↔ ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})))

Theorem

hash3tpb 14420*

A set of size three is a proper unordered triple. (Contributed by AV, 21-Jul-2025.)

⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 3 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})))

Theorem

tpf1ofv0 14421*

The value of a one-to-one function onto a triple at 0. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹‘0) = 𝐴)

Theorem

tpf1ofv1 14422*

The value of a one-to-one function onto a triple at 1. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   ⊢ (𝐵 ∈ 𝑉 → (𝐹‘1) = 𝐵)

Theorem

tpf1ofv2 14423*

The value of a one-to-one function onto a triple at 2. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   ⊢ (𝐶 ∈ 𝑉 → (𝐹‘2) = 𝐶)

Theorem

tpf 14424*

A function into a (proper) triple. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   ⊢ 𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)⟶𝑇)

Theorem

tpfo 14425*

A function onto a (proper) triple. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   ⊢ 𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)–onto→𝑇)

Theorem

tpf1o 14426*

A bijection onto a (proper) triple. (Contributed by AV, 21-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   ⊢ 𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (♯‘𝑇) = 3) → 𝐹:(0..^3)–1-1-onto→𝑇)

5.6.11.2  Functions with a domain containing at least two different elements

Theorem

fundmge2nop0 14427

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 14428 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 17080. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)

⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))

Theorem

fundmge2nop 14428

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) (Proof shortened by AV, 9-Jun-2021.)

⊢ ((Fun 𝐺 ∧ 2 ≤ (♯‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))

Theorem

fun2dmnop0 14429

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 14430 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 17080. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fun2dmnop 14430

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 9-Jun-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun 𝐺 ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

5.6.11.3  Finite induction on the size of the first component of a binary relation

Theorem

hashdifsnp1 14431

If the size of a set is a nonnegative integer increased by 1, the size of the set with one of its elements removed is this nonnegative integer. (Contributed by Alexander van der Vekens, 7-Jan-2018.)

⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ₀) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌))

Theorem

fi1uzind 14432*

Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as ordered pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 14437) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

⊢ 𝐹 ∈ V    &   ⊢ 𝐿 ∈ ℕ₀    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = 𝐿) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Theorem

brfi1uzind 14433*

Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with 𝐿 = 0 (see brfi1ind 14434) or 𝐿 = 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Proof shortened by AV, 23-Oct-2020.) (Revised by AV, 28-Mar-2021.)

⊢ Rel 𝐺    &   ⊢ 𝐹 ∈ V    &   ⊢ 𝐿 ∈ ℕ₀    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝐿) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Theorem

brfi1ind 14434*

Properties of a binary relation with a finite first component, proven by finite induction on the size of the first component. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Revised by AV, 28-Mar-2021.)

⊢ Rel 𝐺    &   ⊢ 𝐹 ∈ V    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Theorem

brfi1indALT 14435*

Alternate proof of brfi1ind 14434, which does not use brfi1uzind 14433. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ Rel 𝐺    &   ⊢ 𝐹 ∈ V    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Theorem

opfi1uzind 14436*

Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as ordered pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 14437) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    &   ⊢ 𝐿 ∈ ℕ₀    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Theorem

opfi1ind 14437*

Properties of an ordered pair with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as ordered pairs of vertices and edges) with a finite number of vertices, e.g., fusgrfis 29293. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 0) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin) → 𝜑)

5.7  Words over a set

This section is about words (or strings) over a set (alphabet) defined as finite sequences of symbols (or characters) being elements of the alphabet. Although it is often required that the underlying set/alphabet be nonempty, finite and not a proper class, these restrictions are not made in the current definition df-word 14439, see, for example, s1cli 14530: ⟨“𝐴”⟩ ∈ Word V. Note that the empty word ∅ (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 14465. The set Word 𝑆 of words over 𝑆 is the free monoid over 𝑆, where the monoid law is concatenation and the monoid unit is the empty word. Besides the definition of words themselves, several operations on words are defined in this section:

Name	Reference	Explanation	Example	Remarks
Length (or size) of a word	df-hash 14256: (♯‘𝑊)	Operation which determines the number of the symbols within the word.	𝑊:(0..^𝑁)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 𝑁	This is not a special definition for words, but for arbitrary sets.
First symbol of a word	df-fv 6494: (𝑊‘0)	Operation which extracts the first symbol of a word. The result is the symbol at the first place in the sequence representing the word.	𝑊:(0..^1)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊‘0) ∈ 𝑆	This is not a special definition for words, but for arbitrary functions with a half-open range of nonnegative integers as domain.
Last symbol of a word	df-lsw 14488: (lastS‘𝑊)	Operation which extracts the last symbol of a word. The result is the symbol at the last place in the sequence representing the word.	𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (lastS‘𝑊) = (𝑊‘2)	Note that the index of the last symbol is less by 1 than the length of the word.
Subword (or substring) of a word	df-substr 14566: (𝑊 substr ⟨𝐼, 𝐽⟩)	Operation which extracts a portion of a word. The result is a consecutive, reindexed part of the sequence representing the word.	𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊 substr ⟨1, 2⟩) ∈ Word 𝑆 ∧ (♯‘(𝑊 substr ⟨1, 2⟩)) = 1	Note that the last index of the range defining the subword is greater by 1 than the index of the last symbol of the subword in the sequence of the original word.
Concatenation of two words	df-concat 14496: (𝑊 ++ 𝑈)	Operation combining two words to one new word. The result is a combined, reindexed sequence build from the sequences representing the two words.	(𝑊 ∈ Word 𝑆 ∧ 𝑈 ∈ Word 𝑆) → (♯‘(𝑊 ++ 𝑈)) = ((♯‘𝑊) + (♯‘𝑈))	Note that the index of the first symbol of the second concatenated word is the length of the first word in the concatenation.
Reversal of a word	df-reverse 14683: (reverse‘𝑊)	Operation which reverses the order of symbols in a word.	(𝑊 ∈ Word 𝑉 → (♯‘(reverse‘𝑊)) = (♯‘𝑊))
Cyclical shift of a word	df-csh 14713: (𝑊 cyclShift 𝑁)	Operation cyclically shifting the symbols by a number of positions.	(𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊)
Splicing of a word	df-splice 14674: (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)	Operation which replaces portions of words.	((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
Singleton word	df-s1 14521: ⟨“𝑆”⟩	Constructor building a word of length 1 from a symbol.	(♯‘⟨“𝑆”⟩) = 1

Conventions:

• Words are usually represented by class variable 𝑊, or if two words are involved, by 𝑊 and 𝑈 or by 𝐴 and 𝐵.
• The alphabets are usually represented by class variable 𝑉 (because any arbitrary set can be an alphabet), sometimes also by 𝑆 (especially as codomain of the function representing a word, because the alphabet is the set of symbols).
• The symbols are usually represented by class variables 𝑆 or 𝐴, or if two symbols are involved, by 𝑆 and 𝑇 or by 𝐴 and 𝐵.
• The indices of the sequence representing a word are usually represented by class variable 𝐼, if two indices are involved (especially for subwords) by 𝐼 and 𝐽, or by 𝑀 and 𝑁.
• The length of a word is usually represented by class variables 𝑁 or 𝐿.
• The number of positions by which to cyclically shift a word is usually represented by class variables 𝑁 or 𝐿.

5.7.1  Definitions and basic theorems

Syntax

cword 14438

Syntax for the Word operator.

class Word 𝑆

Definition

df-word 14439*

Define the class of words over a set. A word (sometimes also called a string) is a finite sequence of symbols from a set (alphabet) 𝑆. Definition in Section 9.1 of [AhoHopUll] p. 318. The domain is forced to be an initial segment of ℕ₀ so that two words with the same symbols in the same order be equal. The set Word 𝑆 is sometimes denoted by S*, using the Kleene star, although the Kleene star, or Kleene closure, is sometimes reserved to denote an operation on languages. The set Word 𝑆 equipped with concatenation is the free monoid over 𝑆, and the monoid unit is the empty word (see frmdval 18743). (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}

Theorem

iswrd 14440*

Property of being a word over a set with an existential quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ₀ 𝑊:(0..^𝑙)⟶𝑆)

Theorem

wrdval 14441*

Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)))

Theorem

iswrdi 14442

A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑊:(0..^𝐿)⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrdf 14443

A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Theorem

wrdfd 14444

A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021.)

⊢ (𝜑 → 𝑁 = (♯‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   ⊢ (𝜑 → 𝑊:(0..^𝑁)⟶𝑆)

Theorem

iswrdb 14445

A word over an alphabet is a function from an open range of nonnegative integers (of length equal to the length of the word) into the alphabet. (Contributed by Alexander van der Vekens, 30-Jul-2018.)

⊢ (𝑊 ∈ Word 𝑆 ↔ 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Theorem

wrddm 14446

The indices of a word (i.e. its domain regarded as function) are elements of an open range of nonnegative integers (of length equal to the length of the word). (Contributed by AV, 2-May-2020.)

⊢ (𝑊 ∈ Word 𝑆 → dom 𝑊 = (0..^(♯‘𝑊)))

Theorem

sswrd 14447

The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

⊢ (𝑆 ⊆ 𝑇 → Word 𝑆 ⊆ Word 𝑇)

Theorem

snopiswrd 14448

A singleton of an ordered pair (with 0 as first component) is a word. (Contributed by AV, 23-Nov-2018.) (Proof shortened by AV, 18-Apr-2021.)

⊢ (𝑆 ∈ 𝑉 → {⟨0, 𝑆⟩} ∈ Word 𝑉)

Theorem

wrdexg 14449

The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V)

Theorem

wrdexb 14450

The set of words over a set is a set, bidirectional version. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.)

⊢ (𝑆 ∈ V ↔ Word 𝑆 ∈ V)

Theorem

wrdexi 14451

The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)

⊢ 𝑆 ∈ V    ⇒   ⊢ Word 𝑆 ∈ V

Theorem

wrdsymbcl 14452

A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝐼) ∈ 𝑉)

Theorem

wrdfn 14453

A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊 Fn (0..^(♯‘𝑊)))

Theorem

wrdv 14454

A word over an alphabet is a word over the universal class. (Contributed by AV, 8-Feb-2021.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑊 ∈ Word 𝑉 → 𝑊 ∈ Word V)

Theorem

wrdlndm 14455

The length of a word is not in the domain of the word (regarded as a function). (Contributed by AV, 3-Mar-2021.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∉ dom 𝑊)

Theorem

iswrdsymb 14456*

An arbitrary word is a word over an alphabet if all of its symbols belong to the alphabet. (Contributed by AV, 23-Jan-2021.)

⊢ ((𝑊 ∈ Word V ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ∈ 𝑉) → 𝑊 ∈ Word 𝑉)

Theorem

wrdfin 14457

A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊 ∈ Fin)

Theorem

lencl 14458

The length of a word is a nonnegative integer. This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 27-Aug-2015.)

⊢ (𝑊 ∈ Word 𝑆 → (♯‘𝑊) ∈ ℕ₀)

Theorem

lennncl 14459

The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)

⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ)

Theorem

wrdffz 14460

A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0...((♯‘𝑊) − 1))⟶𝑆)

Theorem

wrdeq 14461

Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑆 = 𝑇 → Word 𝑆 = Word 𝑇)

Theorem

wrdeqi 14462

Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)

⊢ 𝑆 = 𝑇    ⇒   ⊢ Word 𝑆 = Word 𝑇

Theorem

iswrddm0 14463

A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)

⊢ (𝑊:∅⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrd0 14464

The empty set is a word (the empty word, frequently denoted ε in this context). This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 13-May-2020.)

⊢ ∅ ∈ Word 𝑆

Theorem

0wrd0 14465

The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)

⊢ (𝑊 ∈ Word ∅ ↔ 𝑊 = ∅)

Theorem

ffz0iswrd 14466

A sequence with zero-based indices is a word. (Contributed by AV, 31-Jan-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑊:(0...𝐿)⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrdsymb 14467

A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.)

⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ Word (𝑆 “ (0..^(♯‘𝑆))))

Theorem

nfwrd 14468

Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)

⊢ Ⅎ𝑥𝑆    ⇒   ⊢ Ⅎ𝑥Word 𝑆

Theorem

csbwrdg 14469*

Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆)

Theorem

wrdnval 14470*

Words of a fixed length are mappings from a fixed half-open integer interval. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Proof shortened by AV, 13-May-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} = (𝑉 ↑_m (0..^𝑁)))

Theorem

wrdmap 14471

Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ↔ 𝑊 ∈ (𝑉 ↑_m (0..^𝑁))))

Theorem

hashwrdn 14472*

If there is only a finite number of symbols, the number of words of a fixed length over these sysmbols is the number of these symbols raised to the power of the length. (Contributed by Alexander van der Vekens, 25-Mar-2018.)

⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ₀) → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = ((♯‘𝑉)↑𝑁))

Theorem

wrdnfi 14473*

If there is only a finite number of symbols, the number of words of a fixed length over these symbols is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) Remove unnecessary antecedent. (Revised by JJ, 18-Nov-2022.)

⊢ (𝑉 ∈ Fin → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ Fin)

Theorem

wrdsymb0 14474

A symbol at a position "outside" of a word. (Contributed by Alexander van der Vekens, 26-May-2018.) (Proof shortened by AV, 2-May-2020.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝑊) ≤ 𝐼) → (𝑊‘𝐼) = ∅))

Theorem

wrdlenge1n0 14475

A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)

⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (♯‘𝑊)))

Theorem

len0nnbi 14476

The length of a word is a positive integer iff the word is not empty. (Contributed by AV, 22-Mar-2022.)

⊢ (𝑊 ∈ Word 𝑆 → (𝑊 ≠ ∅ ↔ (♯‘𝑊) ∈ ℕ))

Theorem

wrdlenge2n0 14477

A word with length at least 2 is not empty. (Contributed by AV, 18-Jun-2018.) (Proof shortened by AV, 14-Oct-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 𝑊 ≠ ∅)

Theorem

wrdsymb1 14478

The first symbol of a nonempty word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑊)) → (𝑊‘0) ∈ 𝑉)

Theorem

wrdlen1 14479*

A word of length 1 starts with a symbol. (Contributed by AV, 20-Jul-2018.) (Proof shortened by AV, 19-Oct-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ 𝑉 (𝑊‘0) = 𝑣)

Theorem

fstwrdne 14480

The first symbol of a nonempty word is an element of the alphabet for the word. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊‘0) ∈ 𝑉)

Theorem

fstwrdne0 14481

The first symbol of a nonempty word is an element of the alphabet for the word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)

⊢ ((𝑁 ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) → (𝑊‘0) ∈ 𝑉)

Theorem

eqwrd 14482*

Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)

⊢ ((𝑈 ∈ Word 𝑆 ∧ 𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈‘𝑖) = (𝑊‘𝑖))))

Theorem

elovmpowrd 14483*

Implications for the value of an operation defined by the maps-to notation with a class abstraction of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑})    ⇒   ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))

Theorem

elovmptnn0wrd 14484*

Implications for the value of an operation defined by the maps-to notation with a function of nonnegative integers into a class abstraction of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦 and 𝑛. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)

⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑}))    ⇒   ⊢ (𝑍 ∈ ((𝑉𝑂𝑌)‘𝑁) → ((𝑉 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑍 ∈ Word 𝑉)))

Theorem

wrdred1 14485

A word truncated by a symbol is a word. (Contributed by AV, 29-Jan-2021.)

⊢ (𝐹 ∈ Word 𝑆 → (𝐹 ↾ (0..^((♯‘𝐹) − 1))) ∈ Word 𝑆)

Theorem

wrdred1hash 14486

The length of a word truncated by a symbol. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)

⊢ ((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (♯‘𝐹)) → (♯‘(𝐹 ↾ (0..^((♯‘𝐹) − 1)))) = ((♯‘𝐹) − 1))

5.7.2  Last symbol of a word

Syntax

clsw 14487

Extend class notation with the Last Symbol of a word.

class lastS

Definition

df-lsw 14488

Extract the last symbol of a word. May be not meaningful for other sets which are not words. The name lastS (as abbreviation of "lastSymbol") is a compromise between usually used names for corresponding functions in computer programs (as last() or lastChar()), the terminology used for words in set.mm ("symbol" instead of "character") and brevity ("lastS" is shorter than "lastChar" and "lastSymbol"). Labels of theorems about last symbols of a word will contain the abbreviation "lsw" (Last Symbol of a Word). (Contributed by Alexander van der Vekens, 18-Mar-2018.)

⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))

Theorem

lsw 14489

Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018.)

⊢ (𝑊 ∈ 𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))

Theorem

lsw0 14490

The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 19-Mar-2018.) (Proof shortened by AV, 2-May-2020.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = ∅)

Theorem

lsw0g 14491

The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.)

⊢ (lastS‘∅) = ∅

Theorem

lsw1 14492

The last symbol of a word of length 1 is the first symbol of this word. (Contributed by Alexander van der Vekens, 19-Mar-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → (lastS‘𝑊) = (𝑊‘0))

Theorem

lswcl 14493

Closure of the last symbol: the last symbol of a nonempty word belongs to the alphabet for the word. (Contributed by AV, 2-Aug-2018.) (Proof shortened by AV, 29-Apr-2020.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (lastS‘𝑊) ∈ 𝑉)

Theorem

lswlgt0cl 14494

The last symbol of a nonempty word is an element of the alphabet for the word. (Contributed by Alexander van der Vekens, 1-Oct-2018.) (Proof shortened by AV, 29-Apr-2020.)

⊢ ((𝑁 ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) → (lastS‘𝑊) ∈ 𝑉)

5.7.3  Concatenations of words

Syntax

cconcat 14495

Syntax for the concatenation operator.

class ++

Definition

df-concat 14496*

Define the concatenation operator which combines two words. Definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)

⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))))

Theorem

ccatfn 14497

The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.) (Proof shortened by AV, 29-Apr-2020.)

⊢ ++ Fn (V × V)

Theorem

ccatfval 14498*

Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))))

Theorem

ccatcl 14499

The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 29-Apr-2020.)

⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)

Theorem

ccatlen 14500

The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.)

⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))