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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempfxfn 14501 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (𝑆 prefix 𝐿) Fn (0..^𝐿))
 
Theorempfxfv 14502 A symbol in a prefix of a word, indexed using the prefix' indices. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ 𝐼 ∈ (0..^𝐿)) β†’ ((π‘Š prefix 𝐿)β€˜πΌ) = (π‘Šβ€˜πΌ))
 
Theorempfxlen 14503 Length of a prefix. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (β™―β€˜(𝑆 prefix 𝐿)) = 𝐿)
 
Theorempfxid 14504 A word is a prefix of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 β†’ (𝑆 prefix (β™―β€˜π‘†)) = 𝑆)
 
Theorempfxrn 14505 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ran (π‘Š prefix 𝐿) βŠ† 𝑉)
 
Theorempfxn0 14506 A prefix consisting of at least one symbol is not empty. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 2-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ β„• ∧ 𝐿 ≀ (β™―β€˜π‘Š)) β†’ (π‘Š prefix 𝐿) β‰  βˆ…)
 
Theorempfxnd 14507 The value of a prefix operation for a length argument larger than the word length is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6873). (Contributed by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ β„•0 ∧ (β™―β€˜π‘Š) < 𝐿) β†’ (π‘Š prefix 𝐿) = βˆ…)
 
Theorempfxnd0 14508 The value of a prefix operation for a length argument not in the range of the word length is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6873). (Contributed by AV, 3-Dec-2022.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 βˆ‰ (0...(β™―β€˜π‘Š))) β†’ (π‘Š prefix 𝐿) = βˆ…)
 
Theorempfxwrdsymb 14509 A prefix of a word is a word over the symbols it consists of. (Contributed by AV, 3-Dec-2022.)
(𝑆 ∈ Word 𝐴 β†’ (𝑆 prefix 𝐿) ∈ Word (𝑆 β€œ (0..^𝐿)))
 
Theoremaddlenrevpfx 14510 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.) (Revised by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(β™―β€˜π‘Š))) β†’ ((β™―β€˜(π‘Š substr βŸ¨π‘€, (β™―β€˜π‘Š)⟩)) + (β™―β€˜(π‘Š prefix 𝑀))) = (β™―β€˜π‘Š))
 
Theoremaddlenpfx 14511 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(β™―β€˜π‘Š))) β†’ ((β™―β€˜(π‘Š prefix 𝑀)) + (β™―β€˜(π‘Š substr βŸ¨π‘€, (β™―β€˜π‘Š)⟩))) = (β™―β€˜π‘Š))
 
Theorempfxfv0 14512 The first symbol of a prefix is the first symbol of the word. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝐿)β€˜0) = (π‘Šβ€˜0))
 
Theorempfxtrcfv 14513 A symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ… ∧ 𝐼 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1))β€˜πΌ) = (π‘Šβ€˜πΌ))
 
Theorempfxtrcfv0 14514 The first symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘Š)) β†’ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1))β€˜0) = (π‘Šβ€˜0))
 
Theorempfxfvlsw 14515 The last symbol in a nonempty prefix of a word. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 3-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(β™―β€˜π‘Š))) β†’ (lastSβ€˜(π‘Š prefix 𝐿)) = (π‘Šβ€˜(𝐿 βˆ’ 1)))
 
Theorempfxeq 14516* The prefixes of two words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 4-May-2020.)
(((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉) ∧ (𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ (𝑀 ≀ (β™―β€˜π‘Š) ∧ 𝑁 ≀ (β™―β€˜π‘ˆ))) β†’ ((π‘Š prefix 𝑀) = (π‘ˆ prefix 𝑁) ↔ (𝑀 = 𝑁 ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘Šβ€˜π‘–) = (π‘ˆβ€˜π‘–))))
 
Theorempfxtrcfvl 14517 The last symbol in a word truncated by one symbol. (Contributed by AV, 16-Jun-2018.) (Revised by AV, 5-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘Š)) β†’ (lastSβ€˜(π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1))) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)))
 
Theorempfxsuffeqwrdeq 14518 Two words are equal if and only if they have the same prefix and the same suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by AV, 5-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(β™―β€˜π‘Š))) β†’ (π‘Š = 𝑆 ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘†) ∧ ((π‘Š prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (π‘Š substr ⟨𝐼, (β™―β€˜π‘Š)⟩) = (𝑆 substr ⟨𝐼, (β™―β€˜π‘Š)⟩)))))
 
Theorempfxsuff1eqwrdeq 14519 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by AV, 6-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
 
Theoremdisjwrdpfx 14520* Sets of words are disjoint if each set contains exactly the extensions of distinct words of a fixed length. Remark: A word π‘Š is called an "extension" of a word 𝑃 if 𝑃 is a prefix of π‘Š. (Contributed by AV, 29-Jul-2018.) (Revised by AV, 6-May-2020.)
Disj 𝑦 ∈ π‘Š {π‘₯ ∈ Word 𝑉 ∣ (π‘₯ prefix 𝑁) = 𝑦}
 
Theoremccatpfx 14521 Concatenating a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
((𝑆 ∈ Word 𝐴 ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ ((𝑆 prefix π‘Œ) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) = (𝑆 prefix 𝑍))
 
Theorempfxccat1 14522 Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by AV, 6-May-2020.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ((𝑆 ++ 𝑇) prefix (β™―β€˜π‘†)) = 𝑆)
 
Theorempfx1 14523 The prefix of length one of a nonempty word expressed as a singleton word. (Contributed by AV, 15-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…) β†’ (π‘Š prefix 1) = βŸ¨β€œ(π‘Šβ€˜0)β€βŸ©)
 
5.7.8  Subwords of subwords
 
Theoremswrdswrdlem 14524 Lemma for swrdswrd 14525. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
(((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)) ∧ 𝑀 ∈ (0...𝑁)) ∧ (𝐾 ∈ (0...(𝑁 βˆ’ 𝑀)) ∧ 𝐿 ∈ (𝐾...(𝑁 βˆ’ 𝑀)))) β†’ (π‘Š ∈ Word 𝑉 ∧ (𝑀 + 𝐾) ∈ (0...(𝑀 + 𝐿)) ∧ (𝑀 + 𝐿) ∈ (0...(β™―β€˜π‘Š))))
 
Theoremswrdswrd 14525 A subword of a subword is a subword. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)) ∧ 𝑀 ∈ (0...𝑁)) β†’ ((𝐾 ∈ (0...(𝑁 βˆ’ 𝑀)) ∧ 𝐿 ∈ (𝐾...(𝑁 βˆ’ 𝑀))) β†’ ((π‘Š substr βŸ¨π‘€, π‘βŸ©) substr ⟨𝐾, 𝐿⟩) = (π‘Š substr ⟨(𝑀 + 𝐾), (𝑀 + 𝐿)⟩)))
 
Theorempfxswrd 14526 A prefix of a subword is a subword. (Contributed by AV, 2-Apr-2018.) (Revised by AV, 8-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)) ∧ 𝑀 ∈ (0...𝑁)) β†’ (𝐿 ∈ (0...(𝑁 βˆ’ 𝑀)) β†’ ((π‘Š substr βŸ¨π‘€, π‘βŸ©) prefix 𝐿) = (π‘Š substr βŸ¨π‘€, (𝑀 + 𝐿)⟩)))
 
Theoremswrdpfx 14527 A subword of a prefix is a subword. (Contributed by Alexander van der Vekens, 6-Apr-2018.) (Revised by AV, 8-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))) β†’ ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) β†’ ((π‘Š prefix 𝑁) substr ⟨𝐾, 𝐿⟩) = (π‘Š substr ⟨𝐾, 𝐿⟩)))
 
Theorempfxpfx 14528 A prefix of a prefix is a prefix. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by AV, 8-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)) ∧ 𝐿 ∈ (0...𝑁)) β†’ ((π‘Š prefix 𝑁) prefix 𝐿) = (π‘Š prefix 𝐿))
 
Theorempfxpfxid 14529 A prefix of a prefix with the same length is the original prefix. In other words, the operation "prefix of length 𝑁 " is idempotent. (Contributed by AV, 5-Apr-2018.) (Revised by AV, 8-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝑁) prefix 𝑁) = (π‘Š prefix 𝑁))
 
5.7.9  Subwords and concatenations
 
Theorempfxcctswrd 14530 The concatenation of the prefix of a word and the rest of the word yields the word itself. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 9-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝑀) ++ (π‘Š substr βŸ¨π‘€, (β™―β€˜π‘Š)⟩)) = π‘Š)
 
Theoremlenpfxcctswrd 14531 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 9-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜((π‘Š prefix 𝑀) ++ (π‘Š substr βŸ¨π‘€, (β™―β€˜π‘Š)⟩))) = (β™―β€˜π‘Š))
 
Theoremlenrevpfxcctswrd 14532 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.) (Revised by AV, 9-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜((π‘Š substr βŸ¨π‘€, (β™―β€˜π‘Š)⟩) ++ (π‘Š prefix 𝑀))) = (β™―β€˜π‘Š))
 
Theorempfxlswccat 14533 Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 9-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…) β†’ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ©) = π‘Š)
 
Theoremccats1pfxeq 14534 The last symbol of a word concatenated with the word with the last symbol removed results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.) (Revised by AV, 9-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ (β™―β€˜π‘ˆ) = ((β™―β€˜π‘Š) + 1)) β†’ (π‘Š = (π‘ˆ prefix (β™―β€˜π‘Š)) β†’ π‘ˆ = (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)))
 
Theoremccats1pfxeqrex 14535* There exists a symbol such that its concatenation after the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ (β™―β€˜π‘ˆ) = ((β™―β€˜π‘Š) + 1)) β†’ (π‘Š = (π‘ˆ prefix (β™―β€˜π‘Š)) β†’ βˆƒπ‘  ∈ 𝑉 π‘ˆ = (π‘Š ++ βŸ¨β€œπ‘ β€βŸ©)))
 
Theoremccatopth 14536 An opth 5432-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)
(((𝐴 ∈ Word 𝑋 ∧ 𝐡 ∈ Word 𝑋) ∧ (𝐢 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (β™―β€˜π΄) = (β™―β€˜πΆ)) β†’ ((𝐴 ++ 𝐡) = (𝐢 ++ 𝐷) ↔ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
 
Theoremccatopth2 14537 An opth 5432-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
(((𝐴 ∈ Word 𝑋 ∧ 𝐡 ∈ Word 𝑋) ∧ (𝐢 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (β™―β€˜π΅) = (β™―β€˜π·)) β†’ ((𝐴 ++ 𝐡) = (𝐢 ++ 𝐷) ↔ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
 
Theoremccatlcan 14538 Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐴 ∈ Word 𝑋 ∧ 𝐡 ∈ Word 𝑋 ∧ 𝐢 ∈ Word 𝑋) β†’ ((𝐢 ++ 𝐴) = (𝐢 ++ 𝐡) ↔ 𝐴 = 𝐡))
 
Theoremccatrcan 14539 Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐴 ∈ Word 𝑋 ∧ 𝐡 ∈ Word 𝑋 ∧ 𝐢 ∈ Word 𝑋) β†’ ((𝐴 ++ 𝐢) = (𝐡 ++ 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremwrdeqs1cat 14540 Decompose a nonempty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)
((π‘Š ∈ Word 𝐴 ∧ π‘Š β‰  βˆ…) β†’ π‘Š = (βŸ¨β€œ(π‘Šβ€˜0)β€βŸ© ++ (π‘Š substr ⟨1, (β™―β€˜π‘Š)⟩)))
 
Theoremcats1un 14541 Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)
((𝐴 ∈ Word 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴 ++ βŸ¨β€œπ΅β€βŸ©) = (𝐴 βˆͺ {⟨(β™―β€˜π΄), 𝐡⟩}))
 
Theoremwrdind 14542* Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)
(π‘₯ = βˆ… β†’ (πœ‘ ↔ πœ“))    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ’))    &   (π‘₯ = (𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) β†’ (πœ‘ ↔ πœƒ))    &   (π‘₯ = 𝐴 β†’ (πœ‘ ↔ 𝜏))    &   πœ“    &   ((𝑦 ∈ Word 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ (πœ’ β†’ πœƒ))    β‡’   (𝐴 ∈ Word 𝐡 β†’ 𝜏)
 
Theoremwrd2ind 14543* Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.) (Proof shortened by AV, 12-Oct-2022.)
((π‘₯ = βˆ… ∧ 𝑀 = βˆ…) β†’ (πœ‘ ↔ πœ“))    &   ((π‘₯ = 𝑦 ∧ 𝑀 = 𝑒) β†’ (πœ‘ ↔ πœ’))    &   ((π‘₯ = (𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) ∧ 𝑀 = (𝑒 ++ βŸ¨β€œπ‘ β€βŸ©)) β†’ (πœ‘ ↔ πœƒ))    &   (π‘₯ = 𝐴 β†’ (𝜌 ↔ 𝜏))    &   (𝑀 = 𝐡 β†’ (πœ‘ ↔ 𝜌))    &   πœ“    &   (((𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑒 ∈ Word π‘Œ ∧ 𝑠 ∈ π‘Œ) ∧ (β™―β€˜π‘¦) = (β™―β€˜π‘’)) β†’ (πœ’ β†’ πœƒ))    β‡’   ((𝐴 ∈ Word 𝑋 ∧ 𝐡 ∈ Word π‘Œ ∧ (β™―β€˜π΄) = (β™―β€˜π΅)) β†’ 𝜏)
 
5.7.10  Subwords of concatenations
 
Theoremswrdccatfn 14544 The subword of a concatenation as function. (Contributed by Alexander van der Vekens, 27-May-2018.)
(((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((β™―β€˜π΄) + (β™―β€˜π΅))))) β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) Fn (0..^(𝑁 βˆ’ 𝑀)))
 
Theoremswrdccatin1 14545 The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(β™―β€˜π΄))) β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = (𝐴 substr βŸ¨π‘€, π‘βŸ©)))
 
Theorempfxccatin12lem4 14546 Lemma 4 for pfxccatin12 14553. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 23-May-2018.)
((𝐿 ∈ β„•0 ∧ 𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„€) β†’ ((𝐾 ∈ (0..^(𝑁 βˆ’ 𝑀)) ∧ Β¬ 𝐾 ∈ (0..^(𝐿 βˆ’ 𝑀))) β†’ 𝐾 ∈ ((𝐿 βˆ’ 𝑀)..^((𝐿 βˆ’ 𝑀) + (𝑁 βˆ’ 𝐿)))))
 
Theorempfxccatin12lem2a 14547 Lemma for pfxccatin12lem2 14551. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) β†’ ((𝐾 ∈ (0..^(𝑁 βˆ’ 𝑀)) ∧ Β¬ 𝐾 ∈ (0..^(𝐿 βˆ’ 𝑀))) β†’ (𝐾 + 𝑀) ∈ (𝐿..^𝑋)))
 
Theorempfxccatin12lem1 14548 Lemma 1 for pfxccatin12 14553. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) β†’ ((𝐾 ∈ (0..^(𝑁 βˆ’ 𝑀)) ∧ Β¬ 𝐾 ∈ (0..^(𝐿 βˆ’ 𝑀))) β†’ (𝐾 βˆ’ (𝐿 βˆ’ 𝑀)) ∈ (0..^(𝑁 βˆ’ 𝐿))))
 
Theoremswrdccatin2 14549 The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)
𝐿 = (β™―β€˜π΄)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (β™―β€˜π΅)))) β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = (𝐡 substr ⟨(𝑀 βˆ’ 𝐿), (𝑁 βˆ’ 𝐿)⟩)))
 
Theorempfxccatin12lem2c 14550 Lemma for pfxccatin12lem2 14551 and pfxccatin12lem3 14552. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
𝐿 = (β™―β€˜π΄)    β‡’   (((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (β™―β€˜π΅))))) β†’ ((𝐴 ++ 𝐡) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(β™―β€˜(𝐴 ++ 𝐡)))))
 
Theorempfxccatin12lem2 14551 Lemma 2 for pfxccatin12 14553. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.)
𝐿 = (β™―β€˜π΄)    β‡’   (((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (β™―β€˜π΅))))) β†’ ((𝐾 ∈ (0..^(𝑁 βˆ’ 𝑀)) ∧ Β¬ 𝐾 ∈ (0..^(𝐿 βˆ’ 𝑀))) β†’ (((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©)β€˜πΎ) = ((𝐡 prefix (𝑁 βˆ’ 𝐿))β€˜(𝐾 βˆ’ (β™―β€˜(𝐴 substr βŸ¨π‘€, 𝐿⟩))))))
 
Theorempfxccatin12lem3 14552 Lemma 3 for pfxccatin12 14553. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
𝐿 = (β™―β€˜π΄)    β‡’   (((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (β™―β€˜π΅))))) β†’ ((𝐾 ∈ (0..^(𝑁 βˆ’ 𝑀)) ∧ 𝐾 ∈ (0..^(𝐿 βˆ’ 𝑀))) β†’ (((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©)β€˜πΎ) = ((𝐴 substr βŸ¨π‘€, 𝐿⟩)β€˜πΎ)))
 
Theorempfxccatin12 14553 The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by AV, 9-May-2020.)
𝐿 = (β™―β€˜π΄)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (β™―β€˜π΅)))) β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = ((𝐴 substr βŸ¨π‘€, 𝐿⟩) ++ (𝐡 prefix (𝑁 βˆ’ 𝐿)))))
 
Theorempfxccat3 14554 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by AV, 10-May-2020.)
𝐿 = (β™―β€˜π΄)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (β™―β€˜π΅)))) β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = if(𝑁 ≀ 𝐿, (𝐴 substr βŸ¨π‘€, π‘βŸ©), if(𝐿 ≀ 𝑀, (𝐡 substr ⟨(𝑀 βˆ’ 𝐿), (𝑁 βˆ’ 𝐿)⟩), ((𝐴 substr βŸ¨π‘€, 𝐿⟩) ++ (𝐡 prefix (𝑁 βˆ’ 𝐿)))))))
 
Theoremswrdccat 14555 The subword of a concatenation of two words as concatenation of subwords of the two concatenated words. (Contributed by Alexander van der Vekens, 29-May-2018.)
𝐿 = (β™―β€˜π΄)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (β™―β€˜π΅)))) β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = ((𝐴 substr βŸ¨π‘€, if(𝑁 ≀ 𝐿, 𝑁, 𝐿)⟩) ++ (𝐡 substr ⟨if(0 ≀ (𝑀 βˆ’ 𝐿), (𝑀 βˆ’ 𝐿), 0), (𝑁 βˆ’ 𝐿)⟩))))
 
Theorempfxccatpfx1 14556 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
𝐿 = (β™―β€˜π΄)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) β†’ ((𝐴 ++ 𝐡) prefix 𝑁) = (𝐴 prefix 𝑁))
 
Theorempfxccatpfx2 14557 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
𝐿 = (β™―β€˜π΄)    &   π‘€ = (β™―β€˜π΅)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) β†’ ((𝐴 ++ 𝐡) prefix 𝑁) = (𝐴 ++ (𝐡 prefix (𝑁 βˆ’ 𝐿))))
 
Theorempfxccat3a 14558 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by AV, 10-May-2020.)
𝐿 = (β™―β€˜π΄)    &   π‘€ = (β™―β€˜π΅)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (𝑁 ∈ (0...(𝐿 + 𝑀)) β†’ ((𝐴 ++ 𝐡) prefix 𝑁) = if(𝑁 ≀ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐡 prefix (𝑁 βˆ’ 𝐿))))))
 
Theoremswrdccat3blem 14559 Lemma for swrdccat3b 14560. (Contributed by AV, 30-May-2018.)
𝐿 = (β™―β€˜π΄)    β‡’   ((((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (β™―β€˜π΅)))) ∧ (𝐿 + (β™―β€˜π΅)) ≀ 𝐿) β†’ if(𝐿 ≀ 𝑀, (𝐡 substr ⟨(𝑀 βˆ’ 𝐿), (β™―β€˜π΅)⟩), ((𝐴 substr βŸ¨π‘€, 𝐿⟩) ++ 𝐡)) = (𝐴 substr βŸ¨π‘€, (𝐿 + (β™―β€˜π΅))⟩))
 
Theoremswrdccat3b 14560 A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.) (Proof shortened by AV, 14-Oct-2022.)
𝐿 = (β™―β€˜π΄)    β‡’   ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (𝑀 ∈ (0...(𝐿 + (β™―β€˜π΅))) β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, (𝐿 + (β™―β€˜π΅))⟩) = if(𝐿 ≀ 𝑀, (𝐡 substr ⟨(𝑀 βˆ’ 𝐿), (β™―β€˜π΅)⟩), ((𝐴 substr βŸ¨π‘€, 𝐿⟩) ++ 𝐡))))
 
Theorempfxccatid 14561 A prefix of a concatenation of length of the first concatenated word is the first word itself. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 10-May-2020.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝑁 = (β™―β€˜π΄)) β†’ ((𝐴 ++ 𝐡) prefix 𝑁) = 𝐴)
 
Theoremccats1pfxeqbi 14562 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. (Contributed by AV, 24-Oct-2018.) (Revised by AV, 10-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ (β™―β€˜π‘ˆ) = ((β™―β€˜π‘Š) + 1)) β†’ (π‘Š = (π‘ˆ prefix (β™―β€˜π‘Š)) ↔ π‘ˆ = (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)))
 
Theoremswrdccatin1d 14563 The subword of a concatenation of two words within the first of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
(πœ‘ β†’ (β™―β€˜π΄) = 𝐿)    &   (πœ‘ β†’ (𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉))    &   (πœ‘ β†’ 𝑀 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝑁 ∈ (0...𝐿))    β‡’   (πœ‘ β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = (𝐴 substr βŸ¨π‘€, π‘βŸ©))
 
Theoremswrdccatin2d 14564 The subword of a concatenation of two words within the second of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
(πœ‘ β†’ (β™―β€˜π΄) = 𝐿)    &   (πœ‘ β†’ (𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉))    &   (πœ‘ β†’ 𝑀 ∈ (𝐿...𝑁))    &   (πœ‘ β†’ 𝑁 ∈ (𝐿...(𝐿 + (β™―β€˜π΅))))    β‡’   (πœ‘ β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = (𝐡 substr ⟨(𝑀 βˆ’ 𝐿), (𝑁 βˆ’ 𝐿)⟩))
 
Theorempfxccatin12d 14565 The subword of a concatenation of two words within both of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by AV, 10-May-2020.)
(πœ‘ β†’ (β™―β€˜π΄) = 𝐿)    &   (πœ‘ β†’ (𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉))    &   (πœ‘ β†’ 𝑀 ∈ (0...𝐿))    &   (πœ‘ β†’ 𝑁 ∈ (𝐿...(𝐿 + (β™―β€˜π΅))))    β‡’   (πœ‘ β†’ ((𝐴 ++ 𝐡) substr βŸ¨π‘€, π‘βŸ©) = ((𝐴 substr βŸ¨π‘€, 𝐿⟩) ++ (𝐡 prefix (𝑁 βˆ’ 𝐿))))
 
Theoremreuccatpfxs1lem 14566* Lemma for reuccatpfxs1 14567. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
(((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ 𝑋) ∧ βˆ€π‘  ∈ 𝑉 ((π‘Š ++ βŸ¨β€œπ‘ β€βŸ©) ∈ 𝑋 β†’ 𝑆 = 𝑠) ∧ βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ Word 𝑉 ∧ (β™―β€˜π‘₯) = ((β™―β€˜π‘Š) + 1))) β†’ (π‘Š = (π‘ˆ prefix (β™―β€˜π‘Š)) β†’ π‘ˆ = (π‘Š ++ βŸ¨β€œπ‘†β€βŸ©)))
 
Theoremreuccatpfxs1 14567* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 13-Oct-2022.)
Ⅎ𝑣𝑋    β‡’   ((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ Word 𝑉 ∧ (β™―β€˜π‘₯) = ((β™―β€˜π‘Š) + 1))) β†’ (βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ 𝑋 β†’ βˆƒ!π‘₯ ∈ 𝑋 π‘Š = (π‘₯ prefix (β™―β€˜π‘Š))))
 
Theoremreuccatpfxs1v 14568* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 10-May-2022.) (Proof shortened by AV, 13-Oct-2022.)
((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ Word 𝑉 ∧ (β™―β€˜π‘₯) = ((β™―β€˜π‘Š) + 1))) β†’ (βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ 𝑋 β†’ βˆƒ!π‘₯ ∈ 𝑋 π‘Š = (π‘₯ prefix (β™―β€˜π‘Š))))
 
5.7.11  Splicing words (substring replacement)
 
Syntaxcsplice 14569 Syntax for the word splicing operator.
class splice
 
Definitiondf-splice 14570* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 14-Oct-2022.)
splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st β€˜(1st β€˜π‘))) ++ (2nd β€˜π‘)) ++ (𝑠 substr ⟨(2nd β€˜(1st β€˜π‘)), (β™―β€˜π‘ )⟩)))
 
Theoremsplval 14571 Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 11-May-2020.) (Revised by AV, 15-Oct-2022.)
((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ π‘Š ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ π‘Œ)) β†’ (𝑆 splice ⟨𝐹, 𝑇, π‘…βŸ©) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr βŸ¨π‘‡, (β™―β€˜π‘†)⟩)))
 
Theoremsplcl 14572 Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) β†’ (𝑆 splice ⟨𝐹, 𝑇, π‘…βŸ©) ∈ Word 𝐴)
 
Theoremsplid 14573 Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.) (Proof shortened by AV, 14-Oct-2022.)
((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑆 splice βŸ¨π‘‹, π‘Œ, (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)⟩) = 𝑆)
 
Theoremspllen 14574 The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
(πœ‘ β†’ 𝑆 ∈ Word 𝐴)    &   (πœ‘ β†’ 𝐹 ∈ (0...𝑇))    &   (πœ‘ β†’ 𝑇 ∈ (0...(β™―β€˜π‘†)))    &   (πœ‘ β†’ 𝑅 ∈ Word 𝐴)    β‡’   (πœ‘ β†’ (β™―β€˜(𝑆 splice ⟨𝐹, 𝑇, π‘…βŸ©)) = ((β™―β€˜π‘†) + ((β™―β€˜π‘…) βˆ’ (𝑇 βˆ’ 𝐹))))
 
Theoremsplfv1 14575 Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
(πœ‘ β†’ 𝑆 ∈ Word 𝐴)    &   (πœ‘ β†’ 𝐹 ∈ (0...𝑇))    &   (πœ‘ β†’ 𝑇 ∈ (0...(β™―β€˜π‘†)))    &   (πœ‘ β†’ 𝑅 ∈ Word 𝐴)    &   (πœ‘ β†’ 𝑋 ∈ (0..^𝐹))    β‡’   (πœ‘ β†’ ((𝑆 splice ⟨𝐹, 𝑇, π‘…βŸ©)β€˜π‘‹) = (π‘†β€˜π‘‹))
 
Theoremsplfv2a 14576 Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
(πœ‘ β†’ 𝑆 ∈ Word 𝐴)    &   (πœ‘ β†’ 𝐹 ∈ (0...𝑇))    &   (πœ‘ β†’ 𝑇 ∈ (0...(β™―β€˜π‘†)))    &   (πœ‘ β†’ 𝑅 ∈ Word 𝐴)    &   (πœ‘ β†’ 𝑋 ∈ (0..^(β™―β€˜π‘…)))    β‡’   (πœ‘ β†’ ((𝑆 splice ⟨𝐹, 𝑇, π‘…βŸ©)β€˜(𝐹 + 𝑋)) = (π‘…β€˜π‘‹))
 
Theoremsplval2 14577 Value of a splice, assuming the input word 𝑆 has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
(πœ‘ β†’ 𝐴 ∈ Word 𝑋)    &   (πœ‘ β†’ 𝐡 ∈ Word 𝑋)    &   (πœ‘ β†’ 𝐢 ∈ Word 𝑋)    &   (πœ‘ β†’ 𝑅 ∈ Word 𝑋)    &   (πœ‘ β†’ 𝑆 = ((𝐴 ++ 𝐡) ++ 𝐢))    &   (πœ‘ β†’ 𝐹 = (β™―β€˜π΄))    &   (πœ‘ β†’ 𝑇 = (𝐹 + (β™―β€˜π΅)))    β‡’   (πœ‘ β†’ (𝑆 splice ⟨𝐹, 𝑇, π‘…βŸ©) = ((𝐴 ++ 𝑅) ++ 𝐢))
 
5.7.12  Reversing words
 
Syntaxcreverse 14578 Syntax for the word reverse operator.
class reverse
 
Definitiondf-reverse 14579* Define an operation which reverses the order of symbols in a word. This operation is also known as "word reversal" and "word mirroring". (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse = (𝑠 ∈ V ↦ (π‘₯ ∈ (0..^(β™―β€˜π‘ )) ↦ (π‘ β€˜(((β™―β€˜π‘ ) βˆ’ 1) βˆ’ π‘₯))))
 
Theoremrevval 14580* Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(π‘Š ∈ 𝑉 β†’ (reverseβ€˜π‘Š) = (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))))
 
Theoremrevcl 14581 The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(π‘Š ∈ Word 𝐴 β†’ (reverseβ€˜π‘Š) ∈ Word 𝐴)
 
Theoremrevlen 14582 The reverse of a word has the same length as the original. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(π‘Š ∈ Word 𝐴 β†’ (β™―β€˜(reverseβ€˜π‘Š)) = (β™―β€˜π‘Š))
 
Theoremrevfv 14583 Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
((π‘Š ∈ Word 𝐴 ∧ 𝑋 ∈ (0..^(β™―β€˜π‘Š))) β†’ ((reverseβ€˜π‘Š)β€˜π‘‹) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ 𝑋)))
 
Theoremrev0 14584 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(reverseβ€˜βˆ…) = βˆ…
 
Theoremrevs1 14585 Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(reverseβ€˜βŸ¨β€œπ‘†β€βŸ©) = βŸ¨β€œπ‘†β€βŸ©
 
Theoremrevccat 14586 Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (reverseβ€˜(𝑆 ++ 𝑇)) = ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))
 
Theoremrevrev 14587 Reversal is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
(π‘Š ∈ Word 𝐴 β†’ (reverseβ€˜(reverseβ€˜π‘Š)) = π‘Š)
 
5.7.13  Repeated symbol words
 
Syntaxcreps 14588 Extend class notation with words consisting of one repeated symbol.
class repeatS
 
Definitiondf-reps 14589* Definition to construct a word consisting of one repeated symbol, often called "repeated symbol word" for short in the following. (Contributed by Alexander van der Vekens, 4-Nov-2018.)
repeatS = (𝑠 ∈ V, 𝑛 ∈ β„•0 ↦ (π‘₯ ∈ (0..^𝑛) ↦ 𝑠))
 
Theoremreps 14590* Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.)
((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑆 repeatS 𝑁) = (π‘₯ ∈ (0..^𝑁) ↦ 𝑆))
 
Theoremrepsundef 14591 A function mapping a half-open range of nonnegative integers with an upper bound not being a nonnegative integer to a constant is the empty set (in the meaning of "undefined"). (Contributed by AV, 5-Nov-2018.)
(𝑁 βˆ‰ β„•0 β†’ (𝑆 repeatS 𝑁) = βˆ…)
 
Theoremrepsconst 14592 Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 5691. (Contributed by AV, 4-Nov-2018.)
((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑆 repeatS 𝑁) = ((0..^𝑁) Γ— {𝑆}))
 
Theoremrepsf 14593 The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.)
((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑆 repeatS 𝑁):(0..^𝑁)βŸΆπ‘‰)
 
Theoremrepswsymb 14594 The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0 ∧ 𝐼 ∈ (0..^𝑁)) β†’ ((𝑆 repeatS 𝑁)β€˜πΌ) = 𝑆)
 
Theoremrepsw 14595 A function mapping a half-open range of nonnegative integers to a constant is a word consisting of one symbol repeated several times ("repeated symbol word"). (Contributed by AV, 4-Nov-2018.)
((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑆 repeatS 𝑁) ∈ Word 𝑉)
 
Theoremrepswlen 14596 The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (β™―β€˜(𝑆 repeatS 𝑁)) = 𝑁)
 
Theoremrepsw0 14597 The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.)
(𝑆 ∈ 𝑉 β†’ (𝑆 repeatS 0) = βˆ…)
 
Theoremrepsdf2 14598* Alternative definition of a "repeated symbol word". (Contributed by AV, 7-Nov-2018.)
((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (π‘Š = (𝑆 repeatS 𝑁) ↔ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 𝑁 ∧ βˆ€π‘– ∈ (0..^𝑁)(π‘Šβ€˜π‘–) = 𝑆)))
 
Theoremrepswsymball 14599* All the symbols of a "repeated symbol word" are the same. (Contributed by AV, 10-Nov-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) β†’ (π‘Š = (𝑆 repeatS (β™―β€˜π‘Š)) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(π‘Šβ€˜π‘–) = 𝑆))
 
Theoremrepswsymballbi 14600* A word is a "repeated symbol word" iff each of its symbols equals the first symbol of the word. (Contributed by AV, 10-Nov-2018.)
(π‘Š ∈ Word 𝑉 β†’ (π‘Š = ((π‘Šβ€˜0) repeatS (β™―β€˜π‘Š)) ↔ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(π‘Šβ€˜π‘–) = (π‘Šβ€˜0)))
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