HomeHome Metamath Proof Explorer
Theorem List (p. 146 of 482)
< 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  Metamath Proof Explorer
(1-30715)
  Hilbert Space Explorer  Hilbert Space Explorer
(30716-32238)
  Users' Mathboxes  Users' Mathboxes
(32239-48161)
 

Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwrdsymbcl 14501 A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(β™―β€˜π‘Š))) β†’ (π‘Šβ€˜πΌ) ∈ 𝑉)
 
Theoremwrdfn 14502 A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
(π‘Š ∈ Word 𝑆 β†’ π‘Š Fn (0..^(β™―β€˜π‘Š)))
 
Theoremwrdv 14503 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.)
(π‘Š ∈ Word 𝑉 β†’ π‘Š ∈ Word V)
 
Theoremwrdlndm 14504 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.)
(π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) βˆ‰ dom π‘Š)
 
Theoremiswrdsymb 14505* An arbitrary word is a word over an alphabet if all of its symbols belong to the alphabet. (Contributed by AV, 23-Jan-2021.)
((π‘Š ∈ Word V ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(π‘Šβ€˜π‘–) ∈ 𝑉) β†’ π‘Š ∈ Word 𝑉)
 
Theoremwrdfin 14506 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.)
(π‘Š ∈ Word 𝑆 β†’ π‘Š ∈ Fin)
 
Theoremlencl 14507 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.)
(π‘Š ∈ Word 𝑆 β†’ (β™―β€˜π‘Š) ∈ β„•0)
 
Theoremlennncl 14508 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
((π‘Š ∈ Word 𝑆 ∧ π‘Š β‰  βˆ…) β†’ (β™―β€˜π‘Š) ∈ β„•)
 
Theoremwrdffz 14509 A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.)
(π‘Š ∈ Word 𝑆 β†’ π‘Š:(0...((β™―β€˜π‘Š) βˆ’ 1))βŸΆπ‘†)
 
Theoremwrdeq 14510 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝑆 = 𝑇 β†’ Word 𝑆 = Word 𝑇)
 
Theoremwrdeqi 14511 Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)
𝑆 = 𝑇    β‡’   Word 𝑆 = Word 𝑇
 
Theoremiswrddm0 14512 A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)
(π‘Š:βˆ…βŸΆπ‘† β†’ π‘Š ∈ Word 𝑆)
 
Theoremwrd0 14513 The empty set is a word (the empty word, frequently denoted Ξ΅ 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.)
βˆ… ∈ Word 𝑆
 
Theorem0wrd0 14514 The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)
(π‘Š ∈ Word βˆ… ↔ π‘Š = βˆ…)
 
Theoremffz0iswrd 14515 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.)
(π‘Š:(0...𝐿)βŸΆπ‘† β†’ π‘Š ∈ Word 𝑆)
 
Theoremwrdsymb 14516 A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.)
(𝑆 ∈ Word 𝐴 β†’ 𝑆 ∈ Word (𝑆 β€œ (0..^(β™―β€˜π‘†))))
 
Theoremnfwrd 14517 Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)
β„²π‘₯𝑆    β‡’   β„²π‘₯Word 𝑆
 
Theoremcsbwrdg 14518* Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑆 ∈ 𝑉 β†’ ⦋𝑆 / π‘₯⦌Word π‘₯ = Word 𝑆)
 
Theoremwrdnval 14519* 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 ∈ β„•0) β†’ {𝑀 ∈ Word 𝑉 ∣ (β™―β€˜π‘€) = 𝑁} = (𝑉 ↑m (0..^𝑁)))
 
Theoremwrdmap 14520 Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)
((𝑉 ∈ 𝑋 ∧ 𝑁 ∈ β„•0) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 𝑁) ↔ π‘Š ∈ (𝑉 ↑m (0..^𝑁))))
 
Theoremhashwrdn 14521* 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 ∈ β„•0) β†’ (β™―β€˜{𝑀 ∈ Word 𝑉 ∣ (β™―β€˜π‘€) = 𝑁}) = ((β™―β€˜π‘‰)↑𝑁))
 
Theoremwrdnfi 14522* 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 β†’ {𝑀 ∈ Word 𝑉 ∣ (β™―β€˜π‘€) = 𝑁} ∈ Fin)
 
Theoremwrdsymb0 14523 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝐼 ∈ β„€) β†’ ((𝐼 < 0 ∨ (β™―β€˜π‘Š) ≀ 𝐼) β†’ (π‘Šβ€˜πΌ) = βˆ…))
 
Theoremwrdlenge1n0 14524 A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)
(π‘Š ∈ Word 𝑉 β†’ (π‘Š β‰  βˆ… ↔ 1 ≀ (β™―β€˜π‘Š)))
 
Theoremlen0nnbi 14525 The length of a word is a positive integer iff the word is not empty. (Contributed by AV, 22-Mar-2022.)
(π‘Š ∈ Word 𝑆 β†’ (π‘Š β‰  βˆ… ↔ (β™―β€˜π‘Š) ∈ β„•))
 
Theoremwrdlenge2n0 14526 A word with length at least 2 is not empty. (Contributed by AV, 18-Jun-2018.) (Proof shortened by AV, 14-Oct-2018.)
((π‘Š ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘Š)) β†’ π‘Š β‰  βˆ…)
 
Theoremwrdsymb1 14527 The first symbol of a nonempty word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
((π‘Š ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘Š)) β†’ (π‘Šβ€˜0) ∈ 𝑉)
 
Theoremwrdlen1 14528* A word of length 1 starts with a symbol. (Contributed by AV, 20-Jul-2018.) (Proof shortened by AV, 19-Oct-2018.)
((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 1) β†’ βˆƒπ‘£ ∈ 𝑉 (π‘Šβ€˜0) = 𝑣)
 
Theoremfstwrdne 14529 The first symbol of a nonempty word is element of the alphabet for the word. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)
((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…) β†’ (π‘Šβ€˜0) ∈ 𝑉)
 
Theoremfstwrdne0 14530 The first symbol of a nonempty word is element of the alphabet for the word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑁 ∈ β„• ∧ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 𝑁)) β†’ (π‘Šβ€˜0) ∈ 𝑉)
 
Theoremeqwrd 14531* 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.)
((π‘ˆ ∈ Word 𝑆 ∧ π‘Š ∈ Word 𝑇) β†’ (π‘ˆ = π‘Š ↔ ((β™―β€˜π‘ˆ) = (β™―β€˜π‘Š) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘ˆ))(π‘ˆβ€˜π‘–) = (π‘Šβ€˜π‘–))))
 
Theoremelovmpowrd 14532* 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 πœ‘ 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 ∣ πœ‘})    β‡’   (𝑍 ∈ (π‘‰π‘‚π‘Œ) β†’ (𝑉 ∈ V ∧ π‘Œ ∈ V ∧ 𝑍 ∈ Word 𝑉))
 
Theoremelovmptnn0wrd 14533* 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 πœ‘ 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 ∈ β„•0 ↦ {𝑧 ∈ Word 𝑣 ∣ πœ‘}))    β‡’   (𝑍 ∈ ((π‘‰π‘‚π‘Œ)β€˜π‘) β†’ ((𝑉 ∈ V ∧ π‘Œ ∈ V) ∧ (𝑁 ∈ β„•0 ∧ 𝑍 ∈ Word 𝑉)))
 
Theoremwrdred1 14534 A word truncated by a symbol is a word. (Contributed by AV, 29-Jan-2021.)
(𝐹 ∈ Word 𝑆 β†’ (𝐹 β†Ύ (0..^((β™―β€˜πΉ) βˆ’ 1))) ∈ Word 𝑆)
 
Theoremwrdred1hash 14535 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 ≀ (β™―β€˜πΉ)) β†’ (β™―β€˜(𝐹 β†Ύ (0..^((β™―β€˜πΉ) βˆ’ 1)))) = ((β™―β€˜πΉ) βˆ’ 1))
 
5.7.2  Last symbol of a word
 
Syntaxclsw 14536 Extend class notation with the Last Symbol of a word.
class lastS
 
Definitiondf-lsw 14537 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 = (𝑀 ∈ V ↦ (π‘€β€˜((β™―β€˜π‘€) βˆ’ 1)))
 
Theoremlsw 14538 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.)
(π‘Š ∈ 𝑋 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
 
Theoremlsw0 14539 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.)
((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 0) β†’ (lastSβ€˜π‘Š) = βˆ…)
 
Theoremlsw0g 14540 The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.)
(lastSβ€˜βˆ…) = βˆ…
 
Theoremlsw1 14541 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.)
((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 1) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜0))
 
Theoremlswcl 14542 Closure of the last symbol: the last symbol of a not empty word belongs to the alphabet for the word. (Contributed by AV, 2-Aug-2018.) (Proof shortened by AV, 29-Apr-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…) β†’ (lastSβ€˜π‘Š) ∈ 𝑉)
 
Theoremlswlgt0cl 14543 The last symbol of a nonempty word is element of the alphabet for the word. (Contributed by Alexander van der Vekens, 1-Oct-2018.) (Proof shortened by AV, 29-Apr-2020.)
((𝑁 ∈ β„• ∧ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 𝑁)) β†’ (lastSβ€˜π‘Š) ∈ 𝑉)
 
5.7.3  Concatenations of words
 
Syntaxcconcat 14544 Syntax for the concatenation operator.
class ++
 
Definitiondf-concat 14545* 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, 𝑑 ∈ V ↦ (π‘₯ ∈ (0..^((β™―β€˜π‘ ) + (β™―β€˜π‘‘))) ↦ if(π‘₯ ∈ (0..^(β™―β€˜π‘ )), (π‘ β€˜π‘₯), (π‘‘β€˜(π‘₯ βˆ’ (β™―β€˜π‘ ))))))
 
Theoremccatfn 14546 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.) (Proof shortened by AV, 29-Apr-2020.)
++ Fn (V Γ— V)
 
Theoremccatfval 14547* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ π‘Š) β†’ (𝑆 ++ 𝑇) = (π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ↦ if(π‘₯ ∈ (0..^(β™―β€˜π‘†)), (π‘†β€˜π‘₯), (π‘‡β€˜(π‘₯ βˆ’ (β™―β€˜π‘†))))))
 
Theoremccatcl 14548 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 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (𝑆 ++ 𝑇) ∈ Word 𝐡)
 
Theoremccatlen 14549 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 𝐡) β†’ (β™―β€˜(𝑆 ++ 𝑇)) = ((β™―β€˜π‘†) + (β™―β€˜π‘‡)))
 
Theoremccat0 14550 The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐡) β†’ ((𝑆 ++ 𝑇) = βˆ… ↔ (𝑆 = βˆ… ∧ 𝑇 = βˆ…)))
 
Theoremccatval1 14551 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐡 ∧ 𝐼 ∈ (0..^(β™―β€˜π‘†))) β†’ ((𝑆 ++ 𝑇)β€˜πΌ) = (π‘†β€˜πΌ))
 
Theoremccatval2 14552 Value of a symbol in the right half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡 ∧ 𝐼 ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((𝑆 ++ 𝑇)β€˜πΌ) = (π‘‡β€˜(𝐼 βˆ’ (β™―β€˜π‘†))))
 
Theoremccatval3 14553 Value of a symbol in the right half of a concatenated word, using an index relative to the subword. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof shortened by AV, 30-Apr-2020.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡 ∧ 𝐼 ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜(𝐼 + (β™―β€˜π‘†))) = (π‘‡β€˜πΌ))
 
Theoremelfzelfzccat 14554 An element of a finite set of sequential integers up to the length of a word is an element of an extended finite set of sequential integers up to the length of a concatenation of this word with another word. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (𝑁 ∈ (0...(β™―β€˜π΄)) β†’ 𝑁 ∈ (0...(β™―β€˜(𝐴 ++ 𝐡)))))
 
Theoremccatvalfn 14555 The concatenation of two words is a function over the half-open integer range having the sum of the lengths of the word as length. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (𝐴 ++ 𝐡) Fn (0..^((β™―β€˜π΄) + (β™―β€˜π΅))))
 
Theoremccatsymb 14556 The symbol at a given position in a concatenated word. (Contributed by AV, 26-May-2018.) (Proof shortened by AV, 24-Nov-2018.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐼 ∈ β„€) β†’ ((𝐴 ++ 𝐡)β€˜πΌ) = if(𝐼 < (β™―β€˜π΄), (π΄β€˜πΌ), (π΅β€˜(𝐼 βˆ’ (β™―β€˜π΄)))))
 
Theoremccatfv0 14557 The first symbol of a concatenation of two words is the first symbol of the first word if the first word is not empty. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 0 < (β™―β€˜π΄)) β†’ ((𝐴 ++ 𝐡)β€˜0) = (π΄β€˜0))
 
Theoremccatval1lsw 14558 The last symbol of the left (nonempty) half of a concatenated word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Proof shortened by AV, 1-May-2020.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐴 β‰  βˆ…) β†’ ((𝐴 ++ 𝐡)β€˜((β™―β€˜π΄) βˆ’ 1)) = (lastSβ€˜π΄))
 
Theoremccatval21sw 14559 The first symbol of the right (nonempty) half of a concatenated word. (Contributed by AV, 23-Apr-2022.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ ((𝐴 ++ 𝐡)β€˜(β™―β€˜π΄)) = (π΅β€˜0))
 
Theoremccatlid 14560 Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
(𝑆 ∈ Word 𝐡 β†’ (βˆ… ++ 𝑆) = 𝑆)
 
Theoremccatrid 14561 Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
(𝑆 ∈ Word 𝐡 β†’ (𝑆 ++ βˆ…) = 𝑆)
 
Theoremccatass 14562 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡 ∧ π‘ˆ ∈ Word 𝐡) β†’ ((𝑆 ++ 𝑇) ++ π‘ˆ) = (𝑆 ++ (𝑇 ++ π‘ˆ)))
 
Theoremccatrn 14563 The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ran (𝑆 ++ 𝑇) = (ran 𝑆 βˆͺ ran 𝑇))
 
Theoremccatidid 14564 Concatenation of the empty word by the empty word. (Contributed by AV, 26-Mar-2022.)
(βˆ… ++ βˆ…) = βˆ…
 
Theoremlswccatn0lsw 14565 The last symbol of a word concatenated with a nonempty word is the last symbol of the nonempty word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (lastSβ€˜(𝐴 ++ 𝐡)) = (lastSβ€˜π΅))
 
Theoremlswccat0lsw 14566 The last symbol of a word concatenated with the empty word is the last symbol of the word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)
(π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜(π‘Š ++ βˆ…)) = (lastSβ€˜π‘Š))
 
Theoremccatalpha 14567 A concatenation of two arbitrary words is a word over an alphabet iff the symbols of both words belong to the alphabet. (Contributed by AV, 28-Feb-2021.)
((𝐴 ∈ Word V ∧ 𝐡 ∈ Word V) β†’ ((𝐴 ++ 𝐡) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝐡 ∈ Word 𝑆)))
 
Theoremccatrcl1 14568 Reverse closure of a concatenation: If the concatenation of two arbitrary words is a word over an alphabet then the symbols of the first word belong to the alphabet. (Contributed by AV, 3-Mar-2021.)
((𝐴 ∈ Word 𝑋 ∧ 𝐡 ∈ Word π‘Œ ∧ (π‘Š = (𝐴 ++ 𝐡) ∧ π‘Š ∈ Word 𝑆)) β†’ 𝐴 ∈ Word 𝑆)
 
5.7.4  Singleton words
 
Syntaxcs1 14569 Syntax for the singleton word constructor.
class βŸ¨β€œπ΄β€βŸ©
 
Definitiondf-s1 14570 Define the canonical injection from symbols to words. Although not required, 𝐴 should usually be a set. Otherwise, the singleton word βŸ¨β€œπ΄β€βŸ© would be the singleton word consisting of the empty set, see s1prc 14578, and not, as maybe expected, the empty word, see also s1nz 14581. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
βŸ¨β€œπ΄β€βŸ© = {⟨0, ( I β€˜π΄)⟩}
 
Theoremids1 14571 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œ( I β€˜π΄)β€βŸ©
 
Theorems1val 14572 Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴 ∈ 𝑉 β†’ βŸ¨β€œπ΄β€βŸ© = {⟨0, 𝐴⟩})
 
Theorems1rn 14573 The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.)
(𝐴 ∈ 𝑉 β†’ ran βŸ¨β€œπ΄β€βŸ© = {𝐴})
 
Theorems1eq 14574 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝐴 = 𝐡 β†’ βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œπ΅β€βŸ©)
 
Theorems1eqd 14575 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œπ΅β€βŸ©)
 
Theorems1cl 14576 A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.)
(𝐴 ∈ 𝐡 β†’ βŸ¨β€œπ΄β€βŸ© ∈ Word 𝐡)
 
Theorems1cld 14577 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄β€βŸ© ∈ Word 𝐡)
 
Theorems1prc 14578 Value of a singleton word if the symbol is a proper class. (Contributed by AV, 26-Mar-2022.)
(Β¬ 𝐴 ∈ V β†’ βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œβˆ…β€βŸ©)
 
Theorems1cli 14579 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
βŸ¨β€œπ΄β€βŸ© ∈ Word V
 
Theorems1len 14580 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(β™―β€˜βŸ¨β€œπ΄β€βŸ©) = 1
 
Theorems1nz 14581 A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.)
βŸ¨β€œπ΄β€βŸ© β‰  βˆ…
 
Theorems1dm 14582 The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.)
dom βŸ¨β€œπ΄β€βŸ© = {0}
 
Theorems1dmALT 14583 Alternate version of s1dm 14582, having a shorter proof, but requiring that 𝐴 is a set. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 𝑆 β†’ dom βŸ¨β€œπ΄β€βŸ© = {0})
 
Theorems1fv 14584 Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴 ∈ 𝐡 β†’ (βŸ¨β€œπ΄β€βŸ©β€˜0) = 𝐴)
 
Theoremlsws1 14585 The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.)
(𝐴 ∈ 𝑉 β†’ (lastSβ€˜βŸ¨β€œπ΄β€βŸ©) = 𝐴)
 
Theoremeqs1 14586 A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
((π‘Š ∈ Word 𝐴 ∧ (β™―β€˜π‘Š) = 1) β†’ π‘Š = βŸ¨β€œ(π‘Šβ€˜0)β€βŸ©)
 
Theoremwrdl1exs1 14587* A word of length 1 is a singleton word. (Contributed by AV, 24-Jan-2021.)
((π‘Š ∈ Word 𝑆 ∧ (β™―β€˜π‘Š) = 1) β†’ βˆƒπ‘  ∈ 𝑆 π‘Š = βŸ¨β€œπ‘ β€βŸ©)
 
Theoremwrdl1s1 14588 A word of length 1 is a singleton word consisting of the first symbol of the word. (Contributed by AV, 22-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
(𝑆 ∈ 𝑉 β†’ (π‘Š = βŸ¨β€œπ‘†β€βŸ© ↔ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 1 ∧ (π‘Šβ€˜0) = 𝑆)))
 
Theorems111 14589 The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (βŸ¨β€œπ‘†β€βŸ© = βŸ¨β€œπ‘‡β€βŸ© ↔ 𝑆 = 𝑇))
 
5.7.5  Concatenations with singleton words
 
Theoremccatws1cl 14590 The concatenation of a word with a singleton word is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ∈ Word 𝑉)
 
Theoremccatws1clv 14591 The concatenation of a word with a singleton word (which can be over a different alphabet) is a word. (Contributed by AV, 5-Mar-2022.)
(π‘Š ∈ Word 𝑉 β†’ (π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ∈ Word V)
 
Theoremccat2s1cl 14592 The concatenation of two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ Word 𝑉)
 
Theoremccats1alpha 14593 A concatenation of a word with a singleton word is a word over an alphabet 𝑆 iff the symbols of both words belong to the alphabet 𝑆. (Contributed by AV, 27-Mar-2022.)
((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ π‘ˆ) β†’ ((𝐴 ++ βŸ¨β€œπ‘‹β€βŸ©) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆)))
 
Theoremccatws1len 14594 The length of the concatenation of a word with a singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 4-Mar-2022.)
(π‘Š ∈ Word 𝑉 β†’ (β™―β€˜(π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)) = ((β™―β€˜π‘Š) + 1))
 
Theoremccatws1lenp1b 14595 The length of a word is 𝑁 iff the length of the concatenation of the word with a singleton word is 𝑁 + 1. (Contributed by AV, 4-Mar-2022.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜(π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)) = (𝑁 + 1) ↔ (β™―β€˜π‘Š) = 𝑁))
 
Theoremwrdlenccats1lenm1 14596 The length of a word is the length of the word concatenated with a singleton word minus 1. (Contributed by AV, 28-Jun-2018.) (Revised by AV, 5-Mar-2022.)
(π‘Š ∈ Word 𝑉 β†’ ((β™―β€˜(π‘Š ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1) = (β™―β€˜π‘Š))
 
Theoremccat2s1len 14597 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 14-Jan-2024.)
(β™―β€˜(βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©)) = 2
 
Theoremccatw2s1cl 14598 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ Word 𝑉)
 
Theoremccatw2s1len 14599 The length of the concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 5-Mar-2022.)
(π‘Š ∈ Word 𝑉 β†’ (β™―β€˜((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)) = ((β™―β€˜π‘Š) + 2))
 
Theoremccats1val1 14600 Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by JJ, 20-Jan-2024.)
((π‘Š ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(β™―β€˜π‘Š))) β†’ ((π‘Š ++ βŸ¨β€œπ‘†β€βŸ©)β€˜πΌ) = (π‘Šβ€˜πΌ))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48161
  Copyright terms: Public domain < Previous  Next >