P. 149 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cshwcsh2id 14801*	A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 29958 and erclwwlkntr 30007. (Contributed by AV, 9-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
⊢ (𝜑 → 𝑧 ∈ Word 𝑉)    &   ⊢ (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))    ⇒   ⊢ (𝜑 → (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
Theorem	cshimadifsn 14802	The image of a cyclically shifted word under its domain without its left bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
Theorem	cshimadifsn0 14803	The image of a cyclically shifted word under its domain without its upper bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
5.7.15  Mapping words by a function
Theorem	wrdco 14804	Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑊) ∈ Word 𝐵)
Theorem	lenco 14805	Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑊)) = (♯‘𝑊))
Theorem	s1co 14806	Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹‘𝑆)”⟩)
Theorem	revco 14807	Mapping of words (i.e., a letterwise mapping) commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (reverse‘𝑊)) = (reverse‘(𝐹 ∘ 𝑊)))
Theorem	ccatco 14808	Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇)))
Theorem	cshco 14809	Mapping of words commutes with the "cyclical shift" operation. (Contributed by AV, 12-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑁 ∈ ℤ ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 cyclShift 𝑁)) = ((𝐹 ∘ 𝑊) cyclShift 𝑁))
Theorem	swrdco 14810	Mapping of words commutes with the substring operation. (Contributed by AV, 11-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) = ((𝐹 ∘ 𝑊) substr ⟨𝑀, 𝑁⟩))
Theorem	pfxco 14811	Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 prefix 𝑁)) = ((𝐹 ∘ 𝑊) prefix 𝑁))
Theorem	lswco 14812	Mapping of (nonempty) words commutes with the "last symbol" operation. This theorem would not hold if 𝑊 = ∅, (𝐹‘∅) ≠ ∅ and ∅ ∈ 𝐴, because then (lastS‘(𝐹 ∘ 𝑊)) = (lastS‘∅) = ∅ ≠ (𝐹‘∅) = (𝐹(lastS‘𝑊)). (Contributed by AV, 11-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅ ∧ 𝐹:𝐴⟶𝐵) → (lastS‘(𝐹 ∘ 𝑊)) = (𝐹‘(lastS‘𝑊)))
Theorem	repsco 14813	Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.)
⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹‘𝑆) repeatS 𝑁))
5.7.16  Longer string literals
Syntax	cs2 14814	Syntax for the length 2 word constructor.
class ⟨“𝐴𝐵”⟩
Syntax	cs3 14815	Syntax for the length 3 word constructor.
class ⟨“𝐴𝐵𝐶”⟩
Syntax	cs4 14816	Syntax for the length 4 word constructor.
class ⟨“𝐴𝐵𝐶𝐷”⟩
Syntax	cs5 14817	Syntax for the length 5 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
Syntax	cs6 14818	Syntax for the length 6 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
Syntax	cs7 14819	Syntax for the length 7 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
Syntax	cs8 14820	Syntax for the length 8 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
Definition	df-s2 14821	Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
Definition	df-s3 14822	Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
Definition	df-s4 14823	Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
Definition	df-s5 14824	Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
Definition	df-s6 14825	Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
Definition	df-s7 14826	Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
Definition	df-s8 14827	Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
Theorem	cats1cld 14828	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
Theorem	cats1co 14829	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ∘ 𝑆) = 𝑈)    &   ⊢ 𝑉 = (𝑈 ++ ⟨“(𝐹‘𝑋)”⟩)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝑇) = 𝑉)
Theorem	cats1cli 14830	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    ⇒   ⊢ 𝑇 ∈ Word V
Theorem	cats1fvn 14831	The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    &   ⊢ (♯‘𝑆) = 𝑀    ⇒   ⊢ (𝑋 ∈ 𝑉 → (𝑇‘𝑀) = 𝑋)
Theorem	cats1fv 14832	A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    &   ⊢ (♯‘𝑆) = 𝑀    &   ⊢ (𝑌 ∈ 𝑉 → (𝑆‘𝑁) = 𝑌)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑁 < 𝑀    ⇒   ⊢ (𝑌 ∈ 𝑉 → (𝑇‘𝑁) = 𝑌)
Theorem	cats1len 14833	The length of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    &   ⊢ (♯‘𝑆) = 𝑀    &   ⊢ (𝑀 + 1) = 𝑁    ⇒   ⊢ (♯‘𝑇) = 𝑁
Theorem	cats1cat 14834	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝐴 ∈ Word V    &   ⊢ 𝑆 ∈ Word V    &   ⊢ 𝐶 = (𝐵 ++ ⟨“𝑋”⟩)    &   ⊢ 𝐵 = (𝐴 ++ 𝑆)    ⇒   ⊢ 𝐶 = (𝐴 ++ 𝑇)
Theorem	cats2cat 14835	Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.)
⊢ 𝐵 ∈ Word V    &   ⊢ 𝐷 ∈ Word V    &   ⊢ 𝐴 = (𝐵 ++ ⟨“𝑋”⟩)    &   ⊢ 𝐶 = (⟨“𝑌”⟩ ++ 𝐷)    ⇒   ⊢ (𝐴 ++ 𝐶) = ((𝐵 ++ ⟨“𝑋𝑌”⟩) ++ 𝐷)
Theorem	s2eqd 14836	Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
Theorem	s3eqd 14837	Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
Theorem	s4eqd 14838	Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)
Theorem	s5eqd 14839	Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅”⟩)
Theorem	s6eqd 14840	Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    &   ⊢ (𝜑 → 𝐹 = 𝑆)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)
Theorem	s7eqd 14841	Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    &   ⊢ (𝜑 → 𝐹 = 𝑆)    &   ⊢ (𝜑 → 𝐺 = 𝑇)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
Theorem	s8eqd 14842	Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    &   ⊢ (𝜑 → 𝐹 = 𝑆)    &   ⊢ (𝜑 → 𝐺 = 𝑇)    &   ⊢ (𝜑 → 𝐻 = 𝑈)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)
Theorem	s3eq2 14843	Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.)
⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)
Theorem	s2cld 14844	A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)
Theorem	s3cld 14845	A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
Theorem	s4cld 14846	A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑋)
Theorem	s5cld 14847	A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word 𝑋)
Theorem	s6cld 14848	A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word 𝑋)
Theorem	s7cld 14849	A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word 𝑋)
Theorem	s8cld 14850	A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word 𝑋)
Theorem	s2cl 14851	A doubleton word is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)
Theorem	s3cl 14852	A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
Theorem	s2cli 14853	A doubleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵”⟩ ∈ Word V
Theorem	s3cli 14854	A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
Theorem	s4cli 14855	A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word V
Theorem	s5cli 14856	A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word V
Theorem	s6cli 14857	A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word V
Theorem	s7cli 14858	A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word V
Theorem	s8cli 14859	A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word V
Theorem	s2fv0 14860	Extract the first symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ (𝐴 ∈ 𝑉 → (⟨“𝐴𝐵”⟩‘0) = 𝐴)
Theorem	s2fv1 14861	Extract the second symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ (𝐵 ∈ 𝑉 → (⟨“𝐴𝐵”⟩‘1) = 𝐵)
Theorem	s2len 14862	The length of a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴𝐵”⟩) = 2
Theorem	s2dm 14863	The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.)
⊢ dom ⟨“𝐴𝐵”⟩ = {0, 1}
Theorem	s3fv0 14864	Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ (𝐴 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
Theorem	s3fv1 14865	Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ (𝐵 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
Theorem	s3fv2 14866	Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ (𝐶 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
Theorem	s3len 14867	The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
Theorem	s4fv0 14868	Extract the first symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
Theorem	s4fv1 14869	Extract the second symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
⊢ (𝐵 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
Theorem	s4fv2 14870	Extract the third symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
⊢ (𝐶 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
Theorem	s4fv3 14871	Extract the fourth symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
⊢ (𝐷 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
Theorem	s4len 14872	The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4
Theorem	s5len 14873	The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴𝐵𝐶𝐷𝐸”⟩) = 5
Theorem	s6len 14874	The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩) = 6
Theorem	s7len 14875	The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩) = 7
Theorem	s8len 14876	The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ (♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩) = 8
Theorem	lsws2 14877	The last symbol of a doubleton word is its second symbol. (Contributed by AV, 8-Feb-2021.)
⊢ (𝐵 ∈ 𝑉 → (lastS‘⟨“𝐴𝐵”⟩) = 𝐵)
Theorem	lsws3 14878	The last symbol of a 3 letter word is its third symbol. (Contributed by AV, 8-Feb-2021.)
⊢ (𝐶 ∈ 𝑉 → (lastS‘⟨“𝐴𝐵𝐶”⟩) = 𝐶)
Theorem	lsws4 14879	The last symbol of a 4 letter word is its fourth symbol. (Contributed by AV, 8-Feb-2021.)
⊢ (𝐷 ∈ 𝑉 → (lastS‘⟨“𝐴𝐵𝐶𝐷”⟩) = 𝐷)
Theorem	s2prop 14880	A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⟨“𝐴𝐵”⟩ = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩})
Theorem	s2dmALT 14881	Alternate version of s2dm 14863, having a shorter proof, but requiring that 𝐴 and 𝐵 are sets. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → dom ⟨“𝐴𝐵”⟩ = {0, 1})
Theorem	s3tpop 14882	A length 3 word is an unordered triple of ordered pairs. (Contributed by AV, 23-Jan-2021.)
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ⟨“𝐴𝐵𝐶”⟩ = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩})
Theorem	s4prop 14883	A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ⟨“𝐴𝐵𝐶𝐷”⟩ = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐷⟩}))
Theorem	s3fn 14884	A length 3 word is a function with a triple as domain. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by AV, 23-Jan-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ⟨“𝐴𝐵𝐶”⟩ Fn {0, 1, 2})
Theorem	funcnvs1 14885	The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.)
⊢ Fun ◡⟨“𝐴”⟩
Theorem	funcnvs2 14886	The converse of a length 2 word is a function if its symbols are different sets. (Contributed by AV, 23-Jan-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → Fun ◡⟨“𝐴𝐵”⟩)
Theorem	funcnvs3 14887	The converse of a length 3 word is a function if its symbols are different sets. (Contributed by Alexander van der Vekens, 31-Jan-2018.) (Revised by AV, 23-Jan-2021.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → Fun ◡⟨“𝐴𝐵𝐶”⟩)
Theorem	funcnvs4 14888	The converse of a length 4 word is a function if its symbols are different sets. (Contributed by AV, 10-Feb-2021.)
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) ∧ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) → Fun ◡⟨“𝐴𝐵𝐶𝐷”⟩)
Theorem	s2f1o 14889	A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐸 = ⟨“𝐴𝐵”⟩ → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵}))
Theorem	f1oun2prg 14890	A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷)) → ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐷⟩}):({0, 1} ∪ {2, 3})–1-1-onto→({𝐴, 𝐵} ∪ {𝐶, 𝐷})))
Theorem	s4f1o 14891	A length 4 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷)) → (𝐸 = ⟨“𝐴𝐵𝐶𝐷”⟩ → 𝐸:dom 𝐸–1-1-onto→({𝐴, 𝐵} ∪ {𝐶, 𝐷}))))
Theorem	s4dom 14892	The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐸 = ⟨“𝐴𝐵𝐶𝐷”⟩ → dom 𝐸 = ({0, 1} ∪ {2, 3})))
Theorem	s2co 14893	Mapping a doubleton word by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵”⟩) = ⟨“(𝐹‘𝐴)(𝐹‘𝐵)”⟩)
Theorem	s3co 14894	Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵𝐶”⟩) = ⟨“(𝐹‘𝐴)(𝐹‘𝐵)(𝐹‘𝐶)”⟩)
Theorem	s0s1 14895	Concatenation of fixed length strings. (This special case of ccatlid 14558 is provided to complete the pattern s0s1 14895, df-s2 14821, df-s3 14822, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
⊢ ⟨“𝐴”⟩ = (∅ ++ ⟨“𝐴”⟩)
Theorem	s1s2 14896	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶”⟩)
Theorem	s1s3 14897	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷”⟩)
Theorem	s1s4 14898	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸”⟩)
Theorem	s1s5 14899	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹”⟩)
Theorem	s1s6 14900	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺”⟩)