P. 147 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ccatval21sw 14601	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 14602	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 14603	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 14604	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 14605	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 14606	Concatenation of the empty word by the empty word. (Contributed by AV, 26-Mar-2022.)
⊢ (∅ ++ ∅) = ∅
Theorem	lswccatn0lsw 14607	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 14608	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 14609	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 14610	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)
5.7.4  Singleton words
Syntax	cs1 14611	Syntax for the singleton word constructor.
class ⟨“𝐴”⟩
Definition	df-s1 14612	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 14620, and not, as maybe expected, the empty word, see also s1nz 14623. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
Theorem	ids1 14613	Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
Theorem	s1val 14614	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 14615	The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ (𝐴 ∈ 𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
Theorem	s1eq 14616	Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
Theorem	s1eqd 14617	Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
Theorem	s1cl 14618	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 14619	A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ⟨“𝐴”⟩ ∈ Word 𝐵)
Theorem	s1prc 14620	Value of a singleton word if the symbol is a proper class. (Contributed by AV, 26-Mar-2022.)
⊢ (¬ 𝐴 ∈ V → ⟨“𝐴”⟩ = ⟨“∅”⟩)
Theorem	s1cli 14621	A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴”⟩ ∈ Word V
Theorem	s1len 14622	Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴”⟩) = 1
Theorem	s1nz 14623	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 14624	The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.)
⊢ dom ⟨“𝐴”⟩ = {0}
Theorem	s1dmALT 14625	Alternate version of s1dm 14624, 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 14626	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 14627	The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → (lastS‘⟨“𝐴”⟩) = 𝐴)
Theorem	eqs1 14628	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 14629*	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 14630	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 14631	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))
5.7.5  Concatenations with singleton words
Theorem	ccatws1cl 14632	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 14633	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 14634	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 14635	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 14636	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 14637	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 14638	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 14639	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	ccatw2s1cl 14640	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 14641	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 14642	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	ccats1val2 14643	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 14644	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 14645	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 14646	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	ccatw2s1ass 14647	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	ccatws1n0 14648	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 14649	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 14650	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 14651	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 14652	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	ccatw2s1p2 14653	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 14654	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	ccat2s1fst 14655	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))
5.7.6  Subwords/substrings
Syntax	csubstr 14656	Syntax for the subword operator.
class substr
Definition	df-substr 14657*	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 14658	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 6910). (Contributed by AV, 2-Dec-2022.) (New usage is discouraged.)
⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
Theorem	swrdval 14659*	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 14660	A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅
Theorem	swrdcl 14661	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 14662*	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 14663	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 14664	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)))
Theorem	swrdfv0 14665	The first symbol in an extracted subword. (Contributed by AV, 27-Apr-2022.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0..^𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘0) = (𝑆‘𝐹))
Theorem	swrdf 14666	A subword of a word is a function from a half-open range of nonnegative integers of the same length as the subword to the set of symbols for the original word. (Contributed by AV, 13-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁 − 𝑀))⟶𝑉)
Theorem	swrdvalfn 14667	Value of the subword extractor as function with domain. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) Fn (0..^(𝐿 − 𝐹)))
Theorem	swrdrn 14668	The range of a subword of a word is a subset of the set of symbols for the word. (Contributed by AV, 13-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ 𝑉)
Theorem	swrdlend 14669	The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 ≤ 𝐹 → (𝑊 substr ⟨𝐹, 𝐿⟩) = ∅))
Theorem	swrdnd 14670	The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑊) < 𝐿) → (𝑊 substr ⟨𝐹, 𝐿⟩) = ∅))
Theorem	swrdnd2 14671	Value of the subword extractor outside its intended domain. (Contributed by Alexander van der Vekens, 24-May-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 𝐴 ∨ (♯‘𝑊) ≤ 𝐴 ∨ 𝐵 ≤ 0) → (𝑊 substr ⟨𝐴, 𝐵⟩) = ∅))
Theorem	swrdnnn0nd 14672	The value of a subword operation for arguments not being nonnegative integers is the empty set. (Contributed by AV, 2-Dec-2022.)
⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
Theorem	swrdnd0 14673	The value of a subword operation for inproper arguments is the empty set. (Contributed by AV, 2-Dec-2022.)
⊢ (𝑆 ∈ Word 𝑉 → (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
Theorem	swrd0 14674	A subword of an empty set is always the empty set. (Contributed by AV, 31-Mar-2018.) (Revised by AV, 20-Oct-2018.) (Proof shortened by AV, 2-May-2020.)
⊢ (∅ substr ⟨𝐹, 𝐿⟩) = ∅
Theorem	swrdrlen 14675	Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐼, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐼))
Theorem	swrdlen2 14676	Length of an extracted subword. (Contributed by AV, 5-May-2020.)
⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ (ℤ≥‘𝐹)) ∧ 𝐿 ≤ (♯‘𝑆)) → (♯‘(𝑆 substr ⟨𝐹, 𝐿⟩)) = (𝐿 − 𝐹))
Theorem	swrdfv2 14677	A symbol in an extracted subword, indexed using the word's indices. (Contributed by AV, 5-May-2020.)
⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ (ℤ≥‘𝐹)) ∧ 𝐿 ≤ (♯‘𝑆)) ∧ 𝑋 ∈ (𝐹..^𝐿)) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘(𝑋 − 𝐹)) = (𝑆‘𝑋))
Theorem	swrdwrdsymb 14678	A subword is a word over the symbols it consists of. (Contributed by AV, 2-Dec-2022.)
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑀, 𝑁⟩) ∈ Word (𝑆 “ (𝑀..^𝑁)))
Theorem	swrdsb0eq 14679	Two subwords with the same bounds are equal if the range is not valid. (Contributed by AV, 4-May-2020.)
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 𝑁 ≤ 𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
Theorem	swrdsbslen 14680	Two subwords with the same bounds have the same length. (Contributed by AV, 4-May-2020.)
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)))
Theorem	swrdspsleq 14681*	Two words have a common subword (starting at the same position with the same length) iff they have the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Proof shortened by AV, 7-May-2020.)
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
Theorem	swrds1 14682	Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 1)⟩) = ⟨“(𝑊‘𝐼)”⟩)
Theorem	swrdlsw 14683	Extract the last single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 substr ⟨((♯‘𝑊) − 1), (♯‘𝑊)⟩) = ⟨“(lastS‘𝑊)”⟩)
Theorem	ccatswrd 14684	Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))
Theorem	swrdccat2 14685	Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) substr ⟨(♯‘𝑆), ((♯‘𝑆) + (♯‘𝑇))⟩) = 𝑇)
5.7.7  Prefixes of a word
Syntax	cpfx 14686	Syntax for the prefix operator.
class prefix
Definition	df-pfx 14687*	Define an operation which extracts prefixes of words, i.e. subwords (or substrings) starting at the beginning of a word (or string). In other words, (𝑆 prefix 𝐿) is the prefix of the word 𝑆 of length 𝐿. Definition in Section 9.1 of [AhoHopUll] p. 318. See also Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix. (Contributed by AV, 2-May-2020.)
⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
Theorem	pfxnndmnd 14688	The value of a prefix operation for out-of-domain arguments. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6910). (Contributed by AV, 3-Dec-2022.) (New usage is discouraged.)
⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)
Theorem	pfxval 14689	Value of a prefix operation. (Contributed by AV, 2-May-2020.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
Theorem	pfx00 14690	The zero length prefix is the empty set. (Contributed by AV, 2-May-2020.)
⊢ (𝑆 prefix 0) = ∅
Theorem	pfx0 14691	A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
⊢ (∅ prefix 𝐿) = ∅
Theorem	pfxval0 14692	Value of a prefix operation. This theorem should only be used in proofs if 𝐿 ∈ ℕ0 is not available. Otherwise (and usually), pfxval 14689 should be used. (Contributed by AV, 3-Dec-2022.) (New usage is discouraged.)
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
Theorem	pfxcl 14693	Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)
Theorem	pfxmpt 14694*	Value of the prefix extractor as a mapping. (Contributed by AV, 2-May-2020.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥)))
Theorem	pfxres 14695	Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by AV, 2-May-2020.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) = (𝑆 ↾ (0..^𝐿)))
Theorem	pfxf 14696	A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. (Contributed by AV, 2-May-2020.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿):(0..^𝐿)⟶𝑉)
Theorem	pfxfn 14697	Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) Fn (0..^𝐿))
Theorem	pfxfv 14698	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.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = (𝑊‘𝐼))
Theorem	pfxlen 14699	Length of a prefix. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by AV, 2-May-2020.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝐿)) = 𝐿)
Theorem	pfxid 14700	A word is a prefix of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by AV, 2-May-2020.)
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix (♯‘𝑆)) = 𝑆)