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Type | Label | Description |
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Statement | ||
Theorem | elovmptnn0wrd 13901* | 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 13902 | 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 13903 | 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)) | ||
Syntax | clsw 13904 | Extend class notation with the Last Symbol of a word. |
class lastS | ||
Definition | df-lsw 13905 | 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 13906 | 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 13907 | 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 13908 | The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.) |
⊢ (lastS‘∅) = ∅ | ||
Theorem | lsw1 13909 | 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 13910 | 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.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (lastS‘𝑊) ∈ 𝑉) | ||
Theorem | lswlgt0cl 13911 | 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 ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) → (lastS‘𝑊) ∈ 𝑉) | ||
Syntax | cconcat 13912 | Syntax for the concatenation operator. |
class ++ | ||
Definition | df-concat 13913* | 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 13914 | 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 13915* | 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 13916 | 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 13917 | 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))) | ||
Theorem | ccatlenOLD 13918 | Obsolete version of ccatlen 13917 as of 1-Jan-2024. The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇))) | ||
Theorem | ccat0 13919 | 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 𝐵) → ((𝑆 ++ 𝑇) = ∅ ↔ (𝑆 = ∅ ∧ 𝑇 = ∅))) | ||
Theorem | ccatval1 13920 | 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 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆‘𝐼)) | ||
Theorem | ccatval1OLD 13921 | Obsolete version of ccatval1 13920 as of 18-Jan-2024. 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.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆‘𝐼)) | ||
Theorem | ccatval2 13922 | 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 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑇‘(𝐼 − (♯‘𝑆)))) | ||
Theorem | ccatval3 13923 | 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 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(𝐼 + (♯‘𝑆))) = (𝑇‘𝐼)) | ||
Theorem | elfzelfzccat 13924 | 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 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))) | ||
Theorem | ccatvalfn 13925 | 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 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) Fn (0..^((♯‘𝐴) + (♯‘𝐵)))) | ||
Theorem | ccatsymb 13926 | 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 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴))))) | ||
Theorem | ccatfv0 13927 | 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 ∧ 𝐵 ∈ Word 𝑉 ∧ 0 < (♯‘𝐴)) → ((𝐴 ++ 𝐵)‘0) = (𝐴‘0)) | ||
Theorem | ccatval1lsw 13928 | 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 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐴 ≠ ∅) → ((𝐴 ++ 𝐵)‘((♯‘𝐴) − 1)) = (lastS‘𝐴)) | ||
Theorem | ccatval21sw 13929 | The first symbol of the right (nonempty) half of a concatenated word. (Contributed by AV, 23-Apr-2022.) |
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐵 ≠ ∅) → ((𝐴 ++ 𝐵)‘(♯‘𝐴)) = (𝐵‘0)) | ||
Theorem | ccatlid 13930 | 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 𝐵 → (∅ ++ 𝑆) = 𝑆) | ||
Theorem | ccatrid 13931 | 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 𝐵 → (𝑆 ++ ∅) = 𝑆) | ||
Theorem | ccatass 13932 | Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) = (𝑆 ++ (𝑇 ++ 𝑈))) | ||
Theorem | ccatrn 13933 | The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → ran (𝑆 ++ 𝑇) = (ran 𝑆 ∪ ran 𝑇)) | ||
Theorem | ccatidid 13934 | Concatenation of the empty word by the empty word. (Contributed by AV, 26-Mar-2022.) |
⊢ (∅ ++ ∅) = ∅ | ||
Theorem | lswccatn0lsw 13935 | 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 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐵 ≠ ∅) → (lastS‘(𝐴 ++ 𝐵)) = (lastS‘𝐵)) | ||
Theorem | lswccat0lsw 13936 | 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.) |
⊢ (𝑊 ∈ Word 𝑉 → (lastS‘(𝑊 ++ ∅)) = (lastS‘𝑊)) | ||
Theorem | ccatalpha 13937 | 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 ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆))) | ||
Theorem | ccatrcl1 13938 | 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 ∧ 𝐵 ∈ Word 𝑌 ∧ (𝑊 = (𝐴 ++ 𝐵) ∧ 𝑊 ∈ Word 𝑆)) → 𝐴 ∈ Word 𝑆) | ||
Syntax | cs1 13939 | Syntax for the singleton word constructor. |
class 〈“𝐴”〉 | ||
Definition | df-s1 13940 | Define the canonical injection from symbols to words. Although not required, 𝐴 should usually be a set. Otherwise, the singleton word 〈“𝐴”〉 would be the singleton word consisting of the empty set, see s1prc 13948, and not, as maybe expected, the empty word, see also s1nz 13951. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | ||
Theorem | ids1 13941 | Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴”〉 = 〈“( I ‘𝐴)”〉 | ||
Theorem | s1val 13942 | Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | ||
Theorem | s1rn 13943 | The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = {𝐴}) | ||
Theorem | s1eq 13944 | Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) | ||
Theorem | s1eqd 13945 | Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈“𝐴”〉 = 〈“𝐵”〉) | ||
Theorem | s1cl 13946 | 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 ∈ 𝐵 → 〈“𝐴”〉 ∈ Word 𝐵) | ||
Theorem | s1cld 13947 | A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → 〈“𝐴”〉 ∈ Word 𝐵) | ||
Theorem | s1prc 13948 | Value of a singleton word if the symbol is a proper class. (Contributed by AV, 26-Mar-2022.) |
⊢ (¬ 𝐴 ∈ V → 〈“𝐴”〉 = 〈“∅”〉) | ||
Theorem | s1cli 13949 | A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴”〉 ∈ Word V | ||
Theorem | s1len 13950 | Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ (♯‘〈“𝐴”〉) = 1 | ||
Theorem | s1nz 13951 | A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.) |
⊢ 〈“𝐴”〉 ≠ ∅ | ||
Theorem | s1dm 13952 | The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.) |
⊢ dom 〈“𝐴”〉 = {0} | ||
Theorem | s1dmALT 13953 | Alternate version of s1dm 13952, 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 → dom 〈“𝐴”〉 = {0}) | ||
Theorem | s1fv 13954 | Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ (𝐴 ∈ 𝐵 → (〈“𝐴”〉‘0) = 𝐴) | ||
Theorem | lsws1 13955 | The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.) |
⊢ (𝐴 ∈ 𝑉 → (lastS‘〈“𝐴”〉) = 𝐴) | ||
Theorem | eqs1 13956 | A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) | ||
Theorem | wrdl1exs1 13957* | A word of length 1 is a singleton word. (Contributed by AV, 24-Jan-2021.) |
⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 1) → ∃𝑠 ∈ 𝑆 𝑊 = 〈“𝑠”〉) | ||
Theorem | wrdl1s1 13958 | 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 → (𝑊 = 〈“𝑆”〉 ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) | ||
Theorem | s111 13959 | The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ 𝑆 = 𝑇)) | ||
Theorem | ccatws1cl 13960 | The concatenation of a word with a singleton word is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑊 ++ 〈“𝑋”〉) ∈ Word 𝑉) | ||
Theorem | ccatws1clv 13961 | 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.) |
⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ 〈“𝑋”〉) ∈ Word V) | ||
Theorem | ccat2s1cl 13962 | The concatenation of two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (〈“𝑋”〉 ++ 〈“𝑌”〉) ∈ Word 𝑉) | ||
Theorem | ccats1alpha 13963 | 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 ∈ 𝑈) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) | ||
Theorem | ccatws1len 13964 | 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.) |
⊢ (𝑊 ∈ Word 𝑉 → (♯‘(𝑊 ++ 〈“𝑋”〉)) = ((♯‘𝑊) + 1)) | ||
Theorem | ccatws1lenp1b 13965 | 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.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → ((♯‘(𝑊 ++ 〈“𝑋”〉)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁)) | ||
Theorem | wrdlenccats1lenm1 13966 | 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.) |
⊢ (𝑊 ∈ Word 𝑉 → ((♯‘(𝑊 ++ 〈“𝑆”〉)) − 1) = (♯‘𝑊)) | ||
Theorem | ccat2s1len 13967 | The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 14-Jan-2024.) |
⊢ (♯‘(〈“𝑋”〉 ++ 〈“𝑌”〉)) = 2 | ||
Theorem | ccat2s1lenOLD 13968 | Obsolete version of ccat2s1len 13967 as of 14-Jan-2024. The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (♯‘(〈“𝑋”〉 ++ 〈“𝑌”〉)) = 2) | ||
Theorem | ccatw2s1cl 13969 | The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ Word 𝑉) | ||
Theorem | ccatw2s1len 13970 | 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.) |
⊢ (𝑊 ∈ Word 𝑉 → (♯‘((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)) = ((♯‘𝑊) + 2)) | ||
Theorem | ccats1val1 13971 | 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.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | ccats1val1OLD 13972 | Obsolete version of ccats1val1 13971 as of 20-Jan-2024. 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.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | ccats1val2 13973 | Value of the symbol concatenated with a word. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Proof shortened by Alexander van der Vekens, 14-Oct-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = 𝑆) | ||
Theorem | ccat1st1st 13974 | The first symbol of a word concatenated with its first symbol is the first symbol of the word. This theorem holds even if 𝑊 is the empty word. (Contributed by AV, 26-Mar-2022.) |
⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) | ||
Theorem | ccat2s1p1 13975 | Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.) |
⊢ (𝑋 ∈ 𝑉 → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘0) = 𝑋) | ||
Theorem | ccat2s1p2 13976 | Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.) |
⊢ (𝑌 ∈ 𝑉 → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) | ||
Theorem | ccat2s1p1OLD 13977 | Obsolete version of ccat2s1p1 13975 as of 20-Jan-2024. Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘0) = 𝑋) | ||
Theorem | ccat2s1p2OLD 13978 | Obsolete version of ccat2s1p2 13976 as of 20-Jan-2024. Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) | ||
Theorem | ccatw2s1ass 13979 | Associative law for a concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝑊 ++ (〈“𝑋”〉 ++ 〈“𝑌”〉))) | ||
Theorem | ccatw2s1assOLD 13980 | Obsolete version of ccatw2s1ass 13979 as of 29-Jan-2024. Associative law for a concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝑊 ++ (〈“𝑋”〉 ++ 〈“𝑌”〉))) | ||
Theorem | ccatws1n0 13981 | The concatenation of a word with a singleton word is not the empty set. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 5-Mar-2022.) |
⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ 〈“𝑋”〉) ≠ ∅) | ||
Theorem | ccatws1ls 13982 | The last symbol of the concatenation of a word with a singleton word is the symbol of the singleton word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑊 ++ 〈“𝑋”〉)‘(♯‘𝑊)) = 𝑋) | ||
Theorem | lswccats1 13983 | The last symbol of a word concatenated with a singleton word is the symbol of the singleton word. (Contributed by AV, 6-Aug-2018.) (Proof shortened by AV, 22-Oct-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) → (lastS‘(𝑊 ++ 〈“𝑆”〉)) = 𝑆) | ||
Theorem | lswccats1fst 13984 | The last symbol of a nonempty word concatenated with its first symbol is the first symbol. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.) |
⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (lastS‘(𝑃 ++ 〈“(𝑃‘0)”〉)) = ((𝑃 ++ 〈“(𝑃‘0)”〉)‘0)) | ||
Theorem | ccatw2s1p1 13985 | Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = 𝑋) | ||
Theorem | ccatw2s1p1OLD 13986 | Obsolete version of ccatw2s1p1 13985 as of 29-Jan-2024. Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = 𝑋) | ||
Theorem | ccatw2s1p2 13987 | Extract the second of two single symbols concatenated with a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) |
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 + 1)) = 𝑌) | ||
Theorem | ccat2s1fvw 13988 | Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 28-Jan-2024.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | ccat2s1fvwOLD 13989 | Obsolete version of ccat2s1fvw 13988 as of 28-Jan-2024. Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (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 | ccat2s1fst 13990 | The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 28-Jan-2024.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 < (♯‘𝑊)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘0) = (𝑊‘0)) | ||
Theorem | ccat2s1fstOLD 13991 | Obsolete version of ccat2s1fst 13990 as of 28-Jan-2024. The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝑊 ∈ Word 𝑉 ∧ 0 < (♯‘𝑊)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘0) = (𝑊‘0)) | ||
Syntax | csubstr 13992 | Syntax for the subword operator. |
class substr | ||
Definition | df-substr 13993* | Define an operation which extracts portions (called subwords or substrings) of words. Definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅)) | ||
Theorem | swrdnznd 13994 | The value of a subword operation for noninteger arguments is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6694). (Contributed by AV, 2-Dec-2022.) (New usage is discouraged.) |
⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr 〈𝐹, 𝐿〉) = ∅) | ||
Theorem | swrdval 13995* | Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr 〈𝐹, 𝐿〉) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅)) | ||
Theorem | swrd00 13996 | A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ (𝑆 substr 〈𝑋, 𝑋〉) = ∅ | ||
Theorem | swrdcl 13997 | Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr 〈𝐹, 𝐿〉) ∈ Word 𝐴) | ||
Theorem | swrdval2 13998* | Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 2-May-2020.) |
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈𝐹, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹)))) | ||
Theorem | swrdlen 13999 | Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr 〈𝐹, 𝐿〉)) = (𝐿 − 𝐹)) | ||
Theorem | swrdfv 14000 | A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) ∧ 𝑋 ∈ (0..^(𝐿 − 𝐹))) → ((𝑆 substr 〈𝐹, 𝐿〉)‘𝑋) = (𝑆‘(𝑋 + 𝐹))) |
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