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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcats1len 14201 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 14202 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 14203 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 14204 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)       (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Theorems3eqd 14205 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)

Theorems4eqd 14206 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 14207 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 14208 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 14209 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 14210 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 14211 Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.)
(𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

Theorems2cld 14212 A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)

Theorems3cld 14213 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)

Theorems4cld 14214 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑋)

Theorems5cld 14215 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word 𝑋)

Theorems6cld 14216 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 14217 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 14218 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 14219 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 14220 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
((𝐴𝑋𝐵𝑋𝐶𝑋) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)

Theorems2cli 14221 A doubleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵”⟩ ∈ Word V

Theorems3cli 14222 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ ∈ Word V

Theorems4cli 14223 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word V

Theorems5cli 14224 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word V

Theorems6cli 14225 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word V

Theorems7cli 14226 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word V

Theorems8cli 14227 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word V

Theorems2fv0 14228 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 14229 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 14230 The length of a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵”⟩) = 2

Theorems2dm 14231 The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴𝐵”⟩ = {0, 1}

Theorems3fv0 14232 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)

Theorems3fv1 14233 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)

Theorems3fv2 14234 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)

Theorems3len 14235 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶”⟩) = 3

Theorems4fv0 14236 Extract the first symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)

Theorems4fv1 14237 Extract the second symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)

Theorems4fv2 14238 Extract the third symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)

Theorems4fv3 14239 Extract the fourth symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐷𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)

Theorems4len 14240 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4

Theorems5len 14241 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸”⟩) = 5

Theorems6len 14242 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩) = 6

Theorems7len 14243 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩) = 7

Theorems8len 14244 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(♯‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩) = 8

Theoremlsws2 14245 The last symbol of a doubleton word is its second symbol. (Contributed by AV, 8-Feb-2021.)
(𝐵𝑉 → (lastS‘⟨“𝐴𝐵”⟩) = 𝐵)

Theoremlsws3 14246 The last symbol of a 3 letter word is its third symbol. (Contributed by AV, 8-Feb-2021.)
(𝐶𝑉 → (lastS‘⟨“𝐴𝐵𝐶”⟩) = 𝐶)

Theoremlsws4 14247 The last symbol of a 4 letter word is its fourth symbol. (Contributed by AV, 8-Feb-2021.)
(𝐷𝑉 → (lastS‘⟨“𝐴𝐵𝐶𝐷”⟩) = 𝐷)

Theorems2prop 14248 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 14249 Alternate version of s2dm 14231, 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 14250 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 14251 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 14252 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 14253 The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.)
Fun ⟨“𝐴”⟩

Theoremfuncnvs2 14254 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 14255 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 14256 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 14257 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 14258 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 14259 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 14260 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 14261 Mapping a doubleton word by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵”⟩) = ⟨“(𝐹𝐴)(𝐹𝐵)”⟩)

Theorems3co 14262 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 14263 Concatenation of fixed length strings. (This special case of ccatlid 13919 is provided to complete the pattern s0s1 14263, df-s2 14189, df-s3 14190, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
⟨“𝐴”⟩ = (∅ ++ ⟨“𝐴”⟩)

Theorems1s2 14264 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶”⟩)

Theorems1s3 14265 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷”⟩)

Theorems1s4 14266 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸”⟩)

Theorems1s5 14267 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹”⟩)

Theorems1s6 14268 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺”⟩)

Theorems1s7 14269 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩)

Theorems2s2 14270 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷”⟩)

Theorems4s2 14271 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹”⟩)

Theorems4s3 14272 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺”⟩)

Theorems4s4 14273 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺𝐻”⟩)

Theorems3s4 14274 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷𝐸𝐹𝐺”⟩)

Theorems2s5 14275 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷𝐸𝐹𝐺”⟩)

Theorems5s2 14276 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹𝐺”⟩)

Theorems2eq2s1eq 14277 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 14278 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 14279 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 14280 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 14281 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 14282 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 14283 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 14284 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 14285 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 14286 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 14287* A word of length two is a doubleton word. (Contributed by AV, 25-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → ∃𝑠𝑆𝑡𝑆 𝑊 = ⟨“𝑠𝑡”⟩)

Theorempfx2 14288 A prefix of length two. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Theoremwrd3tpop 14289 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)⟩})

Theoremwrdlen3s3 14290 A word of length three as length 3 string. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)(𝑊‘2)”⟩)

Theoremrepsw2 14291 The "repeated symbol word" of length two. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 2) = ⟨“𝑆𝑆”⟩)

Theoremrepsw3 14292 The "repeated symbol word" of length three. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 3) = ⟨“𝑆𝑆𝑆”⟩)

Theoremswrd2lsw 14293 Extract the last two symbols from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉 ∧ 1 < (♯‘𝑊)) → (𝑊 substr ⟨((♯‘𝑊) − 2), (♯‘𝑊)⟩) = ⟨“(𝑊‘((♯‘𝑊) − 2))(lastS‘𝑊)”⟩)

Theorem2swrd2eqwrdeq 14294 Two words of length at least two are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by AV, 12-Oct-2022.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 1 < (♯‘𝑊)) → (𝑊 = 𝑈 ↔ ((♯‘𝑊) = (♯‘𝑈) ∧ ((𝑊 prefix ((♯‘𝑊) − 2)) = (𝑈 prefix ((♯‘𝑊) − 2)) ∧ (𝑊‘((♯‘𝑊) − 2)) = (𝑈‘((♯‘𝑊) − 2)) ∧ (lastS‘𝑊) = (lastS‘𝑈)))))

Theoremccatw2s1ccatws2 14295 The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 29-Jan-2024.)
(𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))

Theoremccatw2s1ccatws2OLD 14296 Obsolete version of ccatw2s1ccatws2 14295 as of 29-Jan-2024. The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))

Theoremccat2s1fvwALT 14297 Alternate proof of ccat2s1fvw 13977 using words of length 2, see df-s2 14189. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 28-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))

Theoremccat2s1fvwALTOLD 14298 Obsolete version of ccat2s1fvwALT 14297 as of 28-Jan-2024. Alternate proof of ccat2s1fvwOLD 13978 using words of length 2, see df-s2 14189. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (♯‘𝑊)) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))

Theoremwwlktovf 14299* Lemma 1 for wrd2f1tovbij 14303. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       𝐹:𝐷𝑅

Theoremwwlktovf1 14300* Lemma 2 for wrd2f1tovbij 14303. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       𝐹:𝐷1-1𝑅

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