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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-s3 14201 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
 
Definitiondf-s4 14202 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
 
Definitiondf-s5 14203 Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
 
Definitiondf-s6 14204 Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
 
Definitiondf-s7 14205 Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
 
Definitiondf-s8 14206 Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
 
Theoremcats1cld 14207 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝑋𝐴)       (𝜑𝑇 ∈ Word 𝐴)
 
Theoremcats1co 14208 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝑋𝐴)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹𝑆) = 𝑈)    &   𝑉 = (𝑈 ++ ⟨“(𝐹𝑋)”⟩)       (𝜑 → (𝐹𝑇) = 𝑉)
 
Theoremcats1cli 14209 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   𝑆 ∈ Word V       𝑇 ∈ Word V
 
Theoremcats1fvn 14210 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       (𝑋𝑉 → (𝑇𝑀) = 𝑋)
 
Theoremcats1fv 14211 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    &   (𝑌𝑉 → (𝑆𝑁) = 𝑌)    &   𝑁 ∈ ℕ0    &   𝑁 < 𝑀       (𝑌𝑉 → (𝑇𝑁) = 𝑌)
 
Theoremcats1len 14212 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
 
Theoremcats1cat 14213 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)
 
Theoremcats2cat 14214 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)
 
Theorems2eqd 14215 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)       (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
 
Theorems3eqd 14216 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
 
Theorems4eqd 14217 Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)
 
Theorems5eqd 14218 Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅”⟩)
 
Theorems6eqd 14219 Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)    &   (𝜑𝐹 = 𝑆)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)
 
Theorems7eqd 14220 Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)    &   (𝜑𝐹 = 𝑆)    &   (𝜑𝐺 = 𝑇)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
 
Theorems8eqd 14221 Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)    &   (𝜑𝐹 = 𝑆)    &   (𝜑𝐺 = 𝑇)    &   (𝜑𝐻 = 𝑈)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)
 
Theorems3eq2 14222 Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.)
(𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)
 
Theorems2cld 14223 A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)
 
Theorems3cld 14224 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
 
Theorems4cld 14225 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑋)
 
Theorems5cld 14226 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word 𝑋)
 
Theorems6cld 14227 A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word 𝑋)
 
Theorems7cld 14228 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)    &   (𝜑𝐺𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word 𝑋)
 
Theorems8cld 14229 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)    &   (𝜑𝐺𝑋)    &   (𝜑𝐻𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word 𝑋)
 
Theorems2cl 14230 A doubleton word is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝐴𝑋𝐵𝑋) → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)
 
Theorems3cl 14231 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
((𝐴𝑋𝐵𝑋𝐶𝑋) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
 
Theorems2cli 14232 A doubleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵”⟩ ∈ Word V
 
Theorems3cli 14233 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ ∈ Word V
 
Theorems4cli 14234 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word V
 
Theorems5cli 14235 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word V
 
Theorems6cli 14236 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word V
 
Theorems7cli 14237 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word V
 
Theorems8cli 14238 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word V
 
Theorems2fv0 14239 Extract the first symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴𝑉 → (⟨“𝐴𝐵”⟩‘0) = 𝐴)
 
Theorems2fv1 14240 Extract the second symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐵𝑉 → (⟨“𝐴𝐵”⟩‘1) = 𝐵)
 
Theorems2len 14241 The length of a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵”⟩) = 2
 
Theorems2dm 14242 The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴𝐵”⟩ = {0, 1}
 
Theorems3fv0 14243 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
 
Theorems3fv1 14244 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
 
Theorems3fv2 14245 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
 
Theorems3len 14246 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶”⟩) = 3
 
Theorems4fv0 14247 Extract the first symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
 
Theorems4fv1 14248 Extract the second symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
 
Theorems4fv2 14249 Extract the third symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
 
Theorems4fv3 14250 Extract the fourth symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐷𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
 
Theorems4len 14251 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4
 
Theorems5len 14252 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸”⟩) = 5
 
Theorems6len 14253 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩) = 6
 
Theorems7len 14254 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩) = 7
 
Theorems8len 14255 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩) = 8
 
Theoremlsws2 14256 The last symbol of a doubleton word is its second symbol. (Contributed by AV, 8-Feb-2021.)
(𝐵𝑉 → (lastS‘⟨“𝐴𝐵”⟩) = 𝐵)
 
Theoremlsws3 14257 The last symbol of a 3 letter word is its third symbol. (Contributed by AV, 8-Feb-2021.)
(𝐶𝑉 → (lastS‘⟨“𝐴𝐵𝐶”⟩) = 𝐶)
 
Theoremlsws4 14258 The last symbol of a 4 letter word is its fourth symbol. (Contributed by AV, 8-Feb-2021.)
(𝐷𝑉 → (lastS‘⟨“𝐴𝐵𝐶𝐷”⟩) = 𝐷)
 
Theorems2prop 14259 A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
((𝐴𝑆𝐵𝑆) → ⟨“𝐴𝐵”⟩ = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩})
 
Theorems2dmALT 14260 Alternate version of s2dm 14242, 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.)
((𝐴𝑆𝐵𝑆) → dom ⟨“𝐴𝐵”⟩ = {0, 1})
 
Theorems3tpop 14261 A length 3 word is an unordered triple of ordered pairs. (Contributed by AV, 23-Jan-2021.)
((𝐴𝑆𝐵𝑆𝐶𝑆) → ⟨“𝐴𝐵𝐶”⟩ = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩})
 
Theorems4prop 14262 A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ⟨“𝐴𝐵𝐶𝐷”⟩ = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐷⟩}))
 
Theorems3fn 14263 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.)
((𝐴𝑉𝐵𝑉𝐶𝑉) → ⟨“𝐴𝐵𝐶”⟩ Fn {0, 1, 2})
 
Theoremfuncnvs1 14264 The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.)
Fun ⟨“𝐴”⟩
 
Theoremfuncnvs2 14265 The converse of a length 2 word is a function if its symbols are different sets. (Contributed by AV, 23-Jan-2021.)
((𝐴𝑉𝐵𝑉𝐴𝐵) → Fun ⟨“𝐴𝐵”⟩)
 
Theoremfuncnvs3 14266 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.)
(((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → Fun ⟨“𝐴𝐵𝐶”⟩)
 
Theoremfuncnvs4 14267 The converse of a length 4 word is a function if its symbols are different sets. (Contributed by AV, 10-Feb-2021.)
((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷)) → Fun ⟨“𝐴𝐵𝐶𝐷”⟩)
 
Theorems2f1o 14268 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.)
((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐸 = ⟨“𝐴𝐵”⟩ → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵}))
 
Theoremf1oun2prg 14269 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.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)) → ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐷⟩}):({0, 1} ∪ {2, 3})–1-1-onto→({𝐴, 𝐵} ∪ {𝐶, 𝐷})))
 
Theorems4f1o 14270 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.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)) → (𝐸 = ⟨“𝐴𝐵𝐶𝐷”⟩ → 𝐸:dom 𝐸1-1-onto→({𝐴, 𝐵} ∪ {𝐶, 𝐷}))))
 
Theorems4dom 14271 The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (𝐸 = ⟨“𝐴𝐵𝐶𝐷”⟩ → dom 𝐸 = ({0, 1} ∪ {2, 3})))
 
Theorems2co 14272 Mapping a doubleton word by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵”⟩) = ⟨“(𝐹𝐴)(𝐹𝐵)”⟩)
 
Theorems3co 14273 Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)       (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵𝐶”⟩) = ⟨“(𝐹𝐴)(𝐹𝐵)(𝐹𝐶)”⟩)
 
Theorems0s1 14274 Concatenation of fixed length strings. (This special case of ccatlid 13930 is provided to complete the pattern s0s1 14274, df-s2 14200, df-s3 14201, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
⟨“𝐴”⟩ = (∅ ++ ⟨“𝐴”⟩)
 
Theorems1s2 14275 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶”⟩)
 
Theorems1s3 14276 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷”⟩)
 
Theorems1s4 14277 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸”⟩)
 
Theorems1s5 14278 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹”⟩)
 
Theorems1s6 14279 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺”⟩)
 
Theorems1s7 14280 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩)
 
Theorems2s2 14281 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷”⟩)
 
Theorems4s2 14282 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹”⟩)
 
Theorems4s3 14283 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺”⟩)
 
Theorems4s4 14284 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺𝐻”⟩)
 
Theorems3s4 14285 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷𝐸𝐹𝐺”⟩)
 
Theorems2s5 14286 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷𝐸𝐹𝐺”⟩)
 
Theorems5s2 14287 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹𝐺”⟩)
 
Theorems2eq2s1eq 14288 Two length 2 words are equal iff the corresponding singleton words consisting of their symbols are equal. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
(((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (⟨“𝐴𝐵”⟩ = ⟨“𝐶𝐷”⟩ ↔ (⟨“𝐴”⟩ = ⟨“𝐶”⟩ ∧ ⟨“𝐵”⟩ = ⟨“𝐷”⟩)))
 
Theorems2eq2seq 14289 Two length 2 words are equal iff the corresponding symbols are equal. (Contributed by AV, 20-Oct-2018.)
(((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (⟨“𝐴𝐵”⟩ = ⟨“𝐶𝐷”⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theorems3eqs2s1eq 14290 Two length 3 words are equal iff the corresponding length 2 words and singleton words consisting of their symbols are equal. (Contributed by AV, 4-Jan-2022.)
(((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐷𝑉𝐸𝑉𝐹𝑉)) → (⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩ ↔ (⟨“𝐴𝐵”⟩ = ⟨“𝐷𝐸”⟩ ∧ ⟨“𝐶”⟩ = ⟨“𝐹”⟩)))
 
Theorems3eq3seq 14291 Two length 3 words are equal iff the corresponding symbols are equal. (Contributed by AV, 4-Jan-2022.)
(((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐷𝑉𝐸𝑉𝐹𝑉)) → (⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
 
Theoremswrds2 14292 Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑊 ∈ Word 𝐴𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)
 
Theoremswrds2m 14293 Extract two adjacent symbols from a word in reverse direction. (Contributed by AV, 11-May-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) = ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩)
 
Theoremwrdlen2i 14294 Implications of a word of length two. (Contributed by AV, 27-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑆𝑉𝑇𝑉) → (𝑊 = {⟨0, 𝑆⟩, ⟨1, 𝑇⟩} → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇))))
 
Theoremwrd2pr2op 14295 A word of length two represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018.) (Proof shortened by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = {⟨0, (𝑊‘0)⟩, ⟨1, (𝑊‘1)⟩})
 
Theoremwrdlen2 14296 A word of length two. (Contributed by AV, 20-Oct-2018.)
((𝑆𝑉𝑇𝑉) → (𝑊 = {⟨0, 𝑆⟩, ⟨1, 𝑇⟩} ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇))))
 
Theoremwrdlen2s2 14297 A word of length two as doubleton word. (Contributed by AV, 20-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
 
Theoremwrdl2exs2 14298* A word of length two is a doubleton word. (Contributed by AV, 25-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → ∃𝑠𝑆𝑡𝑆 𝑊 = ⟨“𝑠𝑡”⟩)
 
Theorempfx2 14299 A prefix of length two. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
 
Theoremwrd3tpop 14300 A word of length three represented as triple of ordered pairs. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → 𝑊 = {⟨0, (𝑊‘0)⟩, ⟨1, (𝑊‘1)⟩, ⟨2, (𝑊‘2)⟩})
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