Home | Metamath
Proof Explorer Theorem List (p. 145 of 461) | < 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
(1-28865) |
Hilbert Space Explorer
(28866-30388) |
Users' Mathboxes
(30389-46009) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | s3eqs2s1eq 14401 | Two length 3 words are equal iff the corresponding length 2 words and singleton words consisting of their symbols are equal. (Contributed by AV, 4-Jan-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 = 〈“𝐷𝐸𝐹”〉 ↔ (〈“𝐴𝐵”〉 = 〈“𝐷𝐸”〉 ∧ 〈“𝐶”〉 = 〈“𝐹”〉))) | ||
Theorem | s3eq3seq 14402 | Two length 3 words are equal iff the corresponding symbols are equal. (Contributed by AV, 4-Jan-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 = 〈“𝐷𝐸𝐹”〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | ||
Theorem | swrds2 14403 | Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) | ||
Theorem | swrds2m 14404 | Extract two adjacent symbols from a word in reverse direction. (Contributed by AV, 11-May-2022.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr 〈(𝑁 − 2), 𝑁〉) = 〈“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”〉) | ||
Theorem | wrdlen2i 14405 | Implications of a word of length two. (Contributed by AV, 27-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑉) → (𝑊 = {〈0, 𝑆〉, 〈1, 𝑇〉} → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇)))) | ||
Theorem | wrd2pr2op 14406 | A word of length two represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018.) (Proof shortened by AV, 26-Jan-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = {〈0, (𝑊‘0)〉, 〈1, (𝑊‘1)〉}) | ||
Theorem | wrdlen2 14407 | A word of length two. (Contributed by AV, 20-Oct-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑉) → (𝑊 = {〈0, 𝑆〉, 〈1, 𝑇〉} ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇)))) | ||
Theorem | wrdlen2s2 14408 | A word of length two as doubleton word. (Contributed by AV, 20-Oct-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = 〈“(𝑊‘0)(𝑊‘1)”〉) | ||
Theorem | wrdl2exs2 14409* | A word of length two is a doubleton word. (Contributed by AV, 25-Jan-2021.) |
⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → ∃𝑠 ∈ 𝑆 ∃𝑡 ∈ 𝑆 𝑊 = 〈“𝑠𝑡”〉) | ||
Theorem | pfx2 14410 | A prefix of length two. (Contributed by AV, 15-May-2020.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix 2) = 〈“(𝑊‘0)(𝑊‘1)”〉) | ||
Theorem | wrd3tpop 14411 | A word of length three represented as triple of ordered pairs. (Contributed by AV, 26-Jan-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → 𝑊 = {〈0, (𝑊‘0)〉, 〈1, (𝑊‘1)〉, 〈2, (𝑊‘2)〉}) | ||
Theorem | wrdlen3s3 14412 | A word of length three as length 3 string. (Contributed by AV, 26-Jan-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → 𝑊 = 〈“(𝑊‘0)(𝑊‘1)(𝑊‘2)”〉) | ||
Theorem | repsw2 14413 | The "repeated symbol word" of length two. (Contributed by AV, 6-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 2) = 〈“𝑆𝑆”〉) | ||
Theorem | repsw3 14414 | The "repeated symbol word" of length three. (Contributed by AV, 6-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 3) = 〈“𝑆𝑆𝑆”〉) | ||
Theorem | swrd2lsw 14415 | Extract the last two symbols from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 < (♯‘𝑊)) → (𝑊 substr 〈((♯‘𝑊) − 2), (♯‘𝑊)〉) = 〈“(𝑊‘((♯‘𝑊) − 2))(lastS‘𝑊)”〉) | ||
Theorem | 2swrd2eqwrdeq 14416 | Two words of length at least two are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by AV, 12-Oct-2022.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (♯‘𝑊)) → (𝑊 = 𝑈 ↔ ((♯‘𝑊) = (♯‘𝑈) ∧ ((𝑊 prefix ((♯‘𝑊) − 2)) = (𝑈 prefix ((♯‘𝑊) − 2)) ∧ (𝑊‘((♯‘𝑊) − 2)) = (𝑈‘((♯‘𝑊) − 2)) ∧ (lastS‘𝑊) = (lastS‘𝑈))))) | ||
Theorem | ccatw2s1ccatws2 14417 | The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 29-Jan-2024.) |
⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝑊 ++ 〈“𝑋𝑌”〉)) | ||
Theorem | ccatw2s1ccatws2OLD 14418 | Obsolete version of ccatw2s1ccatws2 14417 as of 29-Jan-2024. The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝑊 ++ 〈“𝑋𝑌”〉)) | ||
Theorem | ccat2s1fvwALT 14419 | Alternate proof of ccat2s1fvw 14099 using words of length 2, see df-s2 14311. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 28-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | ccat2s1fvwALTOLD 14420 | Obsolete version of ccat2s1fvwALT 14419 as of 28-Jan-2024. Alternate proof of ccat2s1fvwOLD 14100 using words of length 2, see df-s2 14311. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | wwlktovf 14421* | Lemma 1 for wrd2f1tovbij 14425. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ 𝐹:𝐷⟶𝑅 | ||
Theorem | wwlktovf1 14422* | Lemma 2 for wrd2f1tovbij 14425. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ 𝐹:𝐷–1-1→𝑅 | ||
Theorem | wwlktovfo 14423* | Lemma 3 for wrd2f1tovbij 14425. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–onto→𝑅) | ||
Theorem | wwlktovf1o 14424* | Lemma 4 for wrd2f1tovbij 14425. (Contributed by Alexander van der Vekens, 28-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–1-1-onto→𝑅) | ||
Theorem | wrd2f1tovbij 14425* | There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.) |
⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}) | ||
Theorem | eqwrds3 14426 | A word is equal with a length 3 string iff it has length 3 and the same symbol at each position. (Contributed by AV, 12-May-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊 = 〈“𝐴𝐵𝐶”〉 ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = 𝐵 ∧ (𝑊‘2) = 𝐶)))) | ||
Theorem | wrdl3s3 14427* | A word of length 3 is a length 3 string. (Contributed by AV, 18-May-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) | ||
Theorem | s3sndisj 14428* | The singletons consisting of length 3 strings which have distinct third symbols are disjunct. (Contributed by AV, 17-May-2021.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑍 {〈“𝐴𝐵𝑐”〉}) | ||
Theorem | s3iunsndisj 14429* | The union of singletons consisting of length 3 strings which have distinct first and third symbols are disjunct. (Contributed by AV, 17-May-2021.) |
⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑌 ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){〈“𝑎𝐵𝑐”〉}) | ||
Theorem | ofccat 14430 | Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
⊢ (𝜑 → 𝐸 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑇) & ⊢ (𝜑 → 𝐻 ∈ Word 𝑇) & ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐺)) & ⊢ (𝜑 → (♯‘𝐹) = (♯‘𝐻)) ⇒ ⊢ (𝜑 → ((𝐸 ++ 𝐹) ∘f 𝑅(𝐺 ++ 𝐻)) = ((𝐸 ∘f 𝑅𝐺) ++ (𝐹 ∘f 𝑅𝐻))) | ||
Theorem | ofs1 14431 | Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f 𝑅〈“𝐵”〉) = 〈“(𝐴𝑅𝐵)”〉) | ||
Theorem | ofs2 14432 | Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴𝐵”〉 ∘f 𝑅〈“𝐶𝐷”〉) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉) | ||
A relation, 𝑅, has the reflexive property if 𝐴𝑅𝐴 holds whenever 𝐴 is an element which could be related by the relation, namely, an element of its domain or range. Eliminating dummy variables, we see that a segment of the identity relation must be a subset of the relation, or ( I ↾ (ran 𝑅 ∪ dom 𝑅)) ⊆ 𝑅. See idref 6930. A relation, 𝑅, has the transitive property if 𝐴𝑅𝐶 holds whenever there exists an intermediate value 𝐵 such that both 𝐴𝑅𝐵 and 𝐵𝑅𝐶 hold. This can be expressed without dummy variables as (𝑅 ∘ 𝑅) ⊆ 𝑅. See cotr 5956. The transitive closure of a relation, (t+‘𝑅), is the smallest superset of the relation which has the transitive property. Likewise, the reflexive-transitive closure, (t*‘𝑅), is the smallest superset which has both the reflexive and transitive properties. Not to be confused with the transitive closure of a set, trcl 9255, which is a closure relative to a different transitive property, df-tr 5147. | ||
Theorem | coss12d 14433 | Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷)) | ||
Theorem | trrelssd 14434 | The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) & ⊢ (𝜑 → 𝑆 ⊆ 𝑅) & ⊢ (𝜑 → 𝑇 ⊆ 𝑅) ⇒ ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅) | ||
Theorem | xpcogend 14435 | The most interesting case of the composition of two Cartesian products. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅) ⇒ ⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷)) | ||
Theorem | xpcoidgend 14436 | If two classes are not disjoint, then the composition of their Cartesian product with itself is idempotent. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) ⇒ ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) | ||
Theorem | cotr2g 14437* | Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 14438 for the main application. (Contributed by RP, 22-Mar-2020.) |
⊢ dom 𝐵 ⊆ 𝐷 & ⊢ (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸 & ⊢ ran 𝐴 ⊆ 𝐹 ⇒ ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | ||
Theorem | cotr2 14438* | Two ways of saying a relation is transitive. Special instance of cotr2g 14437. (Contributed by RP, 22-Mar-2020.) |
⊢ dom 𝑅 ⊆ 𝐴 & ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 & ⊢ ran 𝑅 ⊆ 𝐶 ⇒ ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | ||
Theorem | cotr3 14439* | Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.) |
⊢ 𝐴 = dom 𝑅 & ⊢ 𝐵 = (𝐴 ∩ 𝐶) & ⊢ 𝐶 = ran 𝑅 ⇒ ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | ||
Theorem | coemptyd 14440 | Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) | ||
Theorem | xptrrel 14441 | The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) | ||
Theorem | 0trrel 14442 | The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (∅ ∘ ∅) ⊆ ∅ | ||
Theorem | cleq1lem 14443 | Equality implies bijection. (Contributed by RP, 9-May-2020.) |
⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑))) | ||
Theorem | cleq1 14444* | Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) | ||
Theorem | clsslem 14445* | The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.) |
⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) | ||
Syntax | ctcl 14446 | Extend class notation to include the transitive closure symbol. |
class t+ | ||
Syntax | crtcl 14447 | Extend class notation with reflexive-transitive closure. |
class t* | ||
Definition | df-trcl 14448* | Transitive closure of a relation. This is the smallest superset which has the transitive property. (Contributed by FL, 27-Jun-2011.) |
⊢ t+ = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) | ||
Definition | df-rtrcl 14449* | Reflexive-transitive closure of a relation. This is the smallest superset which is reflexive property over all elements of its domain and range and has the transitive property. (Contributed by FL, 27-Jun-2011.) |
⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) | ||
Theorem | trcleq1 14450* | Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | ||
Theorem | trclsslem 14451* | The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.) |
⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | ||
Theorem | trcleq2lem 14452 | Equality implies bijection. (Contributed by RP, 5-May-2020.) |
⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) | ||
Theorem | cvbtrcl 14453* | Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.) |
⊢ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)} | ||
Theorem | trcleq12lem 14454 | Equality implies bijection. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) | ||
Theorem | trclexlem 14455 | Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) | ||
Theorem | trclublem 14456* | If a relation exists then the class of transitive relations which are supersets of that relation is not empty. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}) | ||
Theorem | trclubi 14457* | The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 2-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.) |
⊢ Rel 𝑅 & ⊢ 𝑅 ∈ V ⇒ ⊢ ∩ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅) | ||
Theorem | trclubgi 14458* | The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure. (Contributed by RP, 3-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.) |
⊢ 𝑅 ∈ V ⇒ ⊢ ∩ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) | ||
Theorem | trclub 14459* | The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 17-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ (dom 𝑅 × ran 𝑅)) | ||
Theorem | trclubg 14460* | The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure (as a relation). (Contributed by RP, 17-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | ||
Theorem | trclfv 14461* | The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}) | ||
Theorem | brintclab 14462* | Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.) |
⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑥)) | ||
Theorem | brtrclfv 14463* | Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) | ||
Theorem | brcnvtrclfv 14464* | Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) | ||
Theorem | brtrclfvcnv 14465* | Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘◡𝑅)𝐵 ↔ ∀𝑟((◡𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) | ||
Theorem | brcnvtrclfvcnv 14466* | Two ways of expressing the transitive closure of the converse of the converse of a binary relation. (Contributed by RP, 10-May-2020.) |
⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘◡𝑅)𝐵 ↔ ∀𝑟((◡𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) | ||
Theorem | trclfvss 14467 | The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑅 ⊆ 𝑆) → (t+‘𝑅) ⊆ (t+‘𝑆)) | ||
Theorem | trclfvub 14468 | The transitive closure of a relation has an upper bound. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | ||
Theorem | trclfvlb 14469 | The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) | ||
Theorem | trclfvcotr 14470 | The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | ||
Theorem | trclfvlb2 14471 | The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ 𝑅) ⊆ (t+‘𝑅)) | ||
Theorem | trclfvlb3 14472 | The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ (t+‘𝑅)) | ||
Theorem | cotrtrclfv 14473 | The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅) | ||
Theorem | trclidm 14474 | The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅)) | ||
Theorem | trclun 14475 | Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆)))) | ||
Theorem | trclfvg 14476 | The value of the transitive closure of a relation is a superset or (for proper classes) the empty set. (Contributed by RP, 8-May-2020.) |
⊢ (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅) | ||
Theorem | trclfvcotrg 14477 | The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.) |
⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | ||
Theorem | reltrclfv 14478 | The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) | ||
Theorem | dmtrclfv 14479 | The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅) | ||
Syntax | crelexp 14480 | Extend class notation to include relation exponentiation. |
class ↑𝑟 | ||
Definition | df-relexp 14481* | Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 22-May-2020.) |
⊢ ↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) | ||
Theorem | reldmrelexp 14482 | The domain of the repeated composition of a relation is a relation. (Contributed by AV, 12-Jul-2024.) |
⊢ Rel dom ↑𝑟 | ||
Theorem | relexp0g 14483 | A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | ||
Theorem | relexp0 14484 | A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | relexp0d 14485 | A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | relexpsucnnr 14486 | A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexp1g 14487 | A relation composed once is itself. (Contributed by RP, 22-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | dfid5 14488 | Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ (𝑥↑𝑟1)) | ||
Theorem | dfid6 14489* | Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ ∪ 𝑛 ∈ {1} (𝑥↑𝑟𝑛)) | ||
Theorem | relexp1d 14490 | A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | relexpsucnnl 14491 | A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpsucl 14492 | A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpsucr 14493 | A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexpsucrd 14494 | A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexpsucld 14495 | A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpcnv 14496 | Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)) | ||
Theorem | relexpcnvd 14497 | Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)) | ||
Theorem | relexp0rel 14498 | The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) | ||
Theorem | relexprelg 14499 | The exponentiation of a class is a relation except when the exponent is one and the class is not a relation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexprel 14500 | The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (𝑅↑𝑟𝑁)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |