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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcs7 13301 Syntax for the length 7 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
 
Syntaxcs8 13302 Syntax for the length 8 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
 
Definitiondf-s2 13303 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
 
Definitiondf-s3 13304 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
 
Definitiondf-s4 13305 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
 
Definitiondf-s5 13306 Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
 
Definitiondf-s6 13307 Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
 
Definitiondf-s7 13308 Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
 
Definitiondf-s8 13309 Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
 
Theoremcats1cld 13310 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 13311 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 13312 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   𝑆 ∈ Word V       𝑇 ∈ Word V
 
Theoremcats1fvn 13313 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 13314 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 13315 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 13316 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 13317 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 13318 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)       (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
 
Theorems3eqd 13319 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
 
Theorems4eqd 13320 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 13321 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 13322 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 13323 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 13324 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”⟩)
 
Theorems2cld 13325 A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)
 
Theorems3cld 13326 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
 
Theorems4cld 13327 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑋)
 
Theorems5cld 13328 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word 𝑋)
 
Theorems6cld 13329 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 13330 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 13331 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 13332 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 13333 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
((𝐴𝑋𝐵𝑋𝐶𝑋) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
 
Theorems2cli 13334 A doubleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵”⟩ ∈ Word V
 
Theorems3cli 13335 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ ∈ Word V
 
Theorems4cli 13336 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word V
 
Theorems5cli 13337 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word V
 
Theorems6cli 13338 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word V
 
Theorems7cli 13339 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word V
 
Theorems8cli 13340 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word V
 
Theorems2fv0 13341 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 13342 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 13343 The length of a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵”⟩) = 2
 
Theorems2dm 13344 The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴𝐵”⟩ = {0, 1}
 
Theorems3fv0 13345 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
 
Theorems3fv1 13346 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
 
Theorems3fv2 13347 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
 
Theorems3len 13348 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶”⟩) = 3
 
Theorems4fv0 13349 Extract the first symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
 
Theorems4fv1 13350 Extract the second symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
 
Theorems4fv2 13351 Extract the third symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
 
Theorems4fv3 13352 Extract the fourth symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐷𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
 
Theorems4len 13353 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4
 
Theorems5len 13354 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸”⟩) = 5
 
Theorems6len 13355 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩) = 6
 
Theorems7len 13356 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩) = 7
 
Theorems8len 13357 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩) = 8
 
Theoremlsws2 13358 The last symbol of a doubleton word is its second symbol. (Contributed by AV, 8-Feb-2021.)
(𝐵𝑉 → ( lastS ‘⟨“𝐴𝐵”⟩) = 𝐵)
 
Theoremlsws3 13359 The last symbol of a 3 letter word is its third symbol. (Contributed by AV, 8-Feb-2021.)
(𝐶𝑉 → ( lastS ‘⟨“𝐴𝐵𝐶”⟩) = 𝐶)
 
Theoremlsws4 13360 The last symbol of a 4 letter word is its fourth symbol. (Contributed by AV, 8-Feb-2021.)
(𝐷𝑉 → ( lastS ‘⟨“𝐴𝐵𝐶𝐷”⟩) = 𝐷)
 
Theorems2prop 13361 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 13362 Alternate version of s2dm 13344, 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 13363 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 13364 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 13365 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 13366 The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.)
Fun ⟨“𝐴”⟩
 
Theoremfuncnvs2 13367 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 13368 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 13369 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 13370 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 13371 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 13372 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 13373 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 13374 Mapping a doubleton word by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵”⟩) = ⟨“(𝐹𝐴)(𝐹𝐵)”⟩)
 
Theorems3co 13375 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 13376 Concatenation of fixed length strings. (This special case of ccatlid 13081 is provided to complete the pattern s0s1 13376, df-s2 13303, df-s3 13304, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
⟨“𝐴”⟩ = (∅ ++ ⟨“𝐴”⟩)
 
Theorems1s2 13377 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶”⟩)
 
Theorems1s3 13378 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷”⟩)
 
Theorems1s4 13379 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸”⟩)
 
Theorems1s5 13380 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹”⟩)
 
Theorems1s6 13381 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺”⟩)
 
Theorems1s7 13382 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩)
 
Theorems2s2 13383 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷”⟩)
 
Theorems4s2 13384 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹”⟩)
 
Theorems4s3 13385 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺”⟩)
 
Theorems4s4 13386 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺𝐻”⟩)
 
Theorems3s4 13387 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷𝐸𝐹𝐺”⟩)
 
Theorems2s5 13388 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷𝐸𝐹𝐺”⟩)
 
Theorems5s2 13389 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹𝐺”⟩)
 
Theorems2eq2s1eq 13390 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 13391 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)))
 
Theoremswrds2 13392 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))”⟩)
 
Theoremwrdlen2i 13393 Implications of a word of length 2. (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 13394 A word of length 2 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 13395 A word of length 2. (Contributed by AV, 20-Oct-2018.)
((𝑆𝑉𝑇𝑉) → (𝑊 = {⟨0, 𝑆⟩, ⟨1, 𝑇⟩} ↔ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇))))
 
Theoremwrdlen2s2 13396 A word of length 2 as doubleton word. (Contributed by AV, 20-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 2) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
 
Theoremwrdl2exs2 13397* A word of length 2 is a doubleton word. (Contributed by AV, 25-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (#‘𝑊) = 2) → ∃𝑠𝑆𝑡𝑆 𝑊 = ⟨“𝑠𝑡”⟩)
 
Theoremwrd3tpop 13398 A word of length 3 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)⟩})
 
Theoremwrdlen3s3 13399 A word of length 3 as length 3 string. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)(𝑊‘2)”⟩)
 
Theoremrepsw2 13400 The "repeated symbol word" of length 2. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 2) = ⟨“𝑆𝑆”⟩)
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