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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | s5eqd 14901 | 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 14902 | 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 14903 | 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 14904 | 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 14905 | Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.) |
⊢ (𝐵 = 𝐷 → 〈“𝐴𝐵𝐶”〉 = 〈“𝐴𝐷𝐶”〉) | ||
Theorem | s2cld 14906 | 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 14907 | 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 14908 | 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 14909 | 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 14910 | 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 14911 | 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 14912 | 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 14913 | 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 14914 | 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 14915 | A doubleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵”〉 ∈ Word V | ||
Theorem | s3cli 14916 | A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | ||
Theorem | s4cli 14917 | A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word V | ||
Theorem | s5cli 14918 | A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸”〉 ∈ Word V | ||
Theorem | s6cli 14919 | A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 ∈ Word V | ||
Theorem | s7cli 14920 | A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 ∈ Word V | ||
Theorem | s8cli 14921 | A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 ∈ Word V | ||
Theorem | s2fv0 14922 | 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 14923 | 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 14924 | 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 14925 | The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.) |
⊢ dom 〈“𝐴𝐵”〉 = {0, 1} | ||
Theorem | s3fv0 14926 | Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.) |
⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | ||
Theorem | s3fv1 14927 | Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.) |
⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | ||
Theorem | s3fv2 14928 | Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.) |
⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | ||
Theorem | s3len 14929 | The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (♯‘〈“𝐴𝐵𝐶”〉) = 3 | ||
Theorem | s4fv0 14930 | Extract the first symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘0) = 𝐴) | ||
Theorem | s4fv1 14931 | Extract the second symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.) |
⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘1) = 𝐵) | ||
Theorem | s4fv2 14932 | Extract the third symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.) |
⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘2) = 𝐶) | ||
Theorem | s4fv3 14933 | Extract the fourth symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.) |
⊢ (𝐷 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘3) = 𝐷) | ||
Theorem | s4len 14934 | The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | ||
Theorem | s5len 14935 | The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (♯‘〈“𝐴𝐵𝐶𝐷𝐸”〉) = 5 | ||
Theorem | s6len 14936 | The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (♯‘〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉) = 6 | ||
Theorem | s7len 14937 | The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (♯‘〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉) = 7 | ||
Theorem | s8len 14938 | The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ (♯‘〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉) = 8 | ||
Theorem | lsws2 14939 | The last symbol of a doubleton word is its second symbol. (Contributed by AV, 8-Feb-2021.) |
⊢ (𝐵 ∈ 𝑉 → (lastS‘〈“𝐴𝐵”〉) = 𝐵) | ||
Theorem | lsws3 14940 | 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 14941 | 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 14942 | 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 14943 | Alternate version of s2dm 14925, 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 14944 | 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 14945 | 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 14946 | 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 14947 | The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.) |
⊢ Fun ◡〈“𝐴”〉 | ||
Theorem | funcnvs2 14948 | 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 14949 | 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 14950 | 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 14951 | 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 14952 | 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 14953 | 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 14954 | 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 14955 | 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 14956 | 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 14957 | Concatenation of fixed length strings. (This special case of ccatlid 14620 is provided to complete the pattern s0s1 14957, df-s2 14883, df-s3 14884, ...) (Contributed by Mario Carneiro, 28-Feb-2016.) |
⊢ 〈“𝐴”〉 = (∅ ++ 〈“𝐴”〉) | ||
Theorem | s1s2 14958 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴”〉 ++ 〈“𝐵𝐶”〉) | ||
Theorem | s1s3 14959 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴”〉 ++ 〈“𝐵𝐶𝐷”〉) | ||
Theorem | s1s4 14960 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸”〉 = (〈“𝐴”〉 ++ 〈“𝐵𝐶𝐷𝐸”〉) | ||
Theorem | s1s5 14961 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = (〈“𝐴”〉 ++ 〈“𝐵𝐶𝐷𝐸𝐹”〉) | ||
Theorem | s1s6 14962 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴”〉 ++ 〈“𝐵𝐶𝐷𝐸𝐹𝐺”〉) | ||
Theorem | s1s7 14963 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 = (〈“𝐴”〉 ++ 〈“𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉) | ||
Theorem | s2s2 14964 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶𝐷”〉) | ||
Theorem | s4s2 14965 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹”〉) | ||
Theorem | s4s3 14966 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹𝐺”〉) | ||
Theorem | s4s4 14967 | Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹𝐺𝐻”〉) | ||
Theorem | s3s4 14968 | Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷𝐸𝐹𝐺”〉) | ||
Theorem | s2s5 14969 | Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶𝐷𝐸𝐹𝐺”〉) | ||
Theorem | s5s2 14970 | Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.) |
⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷𝐸”〉 ++ 〈“𝐹𝐺”〉) | ||
Theorem | s2eq2s1eq 14971 | 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.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴𝐵”〉 = 〈“𝐶𝐷”〉 ↔ (〈“𝐴”〉 = 〈“𝐶”〉 ∧ 〈“𝐵”〉 = 〈“𝐷”〉))) | ||
Theorem | s2eq2seq 14972 | Two length 2 words are equal iff the corresponding symbols are equal. (Contributed by AV, 20-Oct-2018.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴𝐵”〉 = 〈“𝐶𝐷”〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | s3eqs2s1eq 14973 | 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.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 = 〈“𝐷𝐸𝐹”〉 ↔ (〈“𝐴𝐵”〉 = 〈“𝐷𝐸”〉 ∧ 〈“𝐶”〉 = 〈“𝐹”〉))) | ||
Theorem | s3eq3seq 14974 | Two length 3 words are equal iff the corresponding symbols are equal. (Contributed by AV, 4-Jan-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 = 〈“𝐷𝐸𝐹”〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | ||
Theorem | swrds2 14975 | Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) | ||
Theorem | swrds2m 14976 | Extract two adjacent symbols from a word in reverse direction. (Contributed by AV, 11-May-2022.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr 〈(𝑁 − 2), 𝑁〉) = 〈“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”〉) | ||
Theorem | wrdlen2i 14977 | Implications of a word of length two. (Contributed by AV, 27-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑉) → (𝑊 = {〈0, 𝑆〉, 〈1, 𝑇〉} → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇)))) | ||
Theorem | wrd2pr2op 14978 | 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)〉}) | ||
Theorem | wrdlen2 14979 | A word of length two. (Contributed by AV, 20-Oct-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑉) → (𝑊 = {〈0, 𝑆〉, 〈1, 𝑇〉} ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇)))) | ||
Theorem | wrdlen2s2 14980 | A word of length two as doubleton word. (Contributed by AV, 20-Oct-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = 〈“(𝑊‘0)(𝑊‘1)”〉) | ||
Theorem | wrdl2exs2 14981* | A word of length two is a doubleton word. (Contributed by AV, 25-Jan-2021.) |
⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → ∃𝑠 ∈ 𝑆 ∃𝑡 ∈ 𝑆 𝑊 = 〈“𝑠𝑡”〉) | ||
Theorem | pfx2 14982 | A prefix of length two. (Contributed by AV, 15-May-2020.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix 2) = 〈“(𝑊‘0)(𝑊‘1)”〉) | ||
Theorem | wrd3tpop 14983 | 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)〉}) | ||
Theorem | wrdlen3s3 14984 | 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)”〉) | ||
Theorem | repsw2 14985 | The "repeated symbol word" of length two. (Contributed by AV, 6-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 2) = 〈“𝑆𝑆”〉) | ||
Theorem | repsw3 14986 | The "repeated symbol word" of length three. (Contributed by AV, 6-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 3) = 〈“𝑆𝑆𝑆”〉) | ||
Theorem | swrd2lsw 14987 | 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)”〉) | ||
Theorem | 2swrd2eqwrdeq 14988 | 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 𝑉 ∧ 𝑈 ∈ 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))))) | ||
Theorem | ccatw2s1ccatws2 14989 | 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”〉)) | ||
Theorem | ccat2s1fvwALT 14990 | Alternate proof of ccat2s1fvw 14672 using words of length 2, see df-s2 14883. 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 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | wwlktovf 14991* | Lemma 1 for wrd2f1tovbij 14995. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ 𝐹:𝐷⟶𝑅 | ||
Theorem | wwlktovf1 14992* | Lemma 2 for wrd2f1tovbij 14995. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ 𝐹:𝐷–1-1→𝑅 | ||
Theorem | wwlktovfo 14993* | Lemma 3 for wrd2f1tovbij 14995. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–onto→𝑅) | ||
Theorem | wwlktovf1o 14994* | Lemma 4 for wrd2f1tovbij 14995. (Contributed by Alexander van der Vekens, 28-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–1-1-onto→𝑅) | ||
Theorem | wrd2f1tovbij 14995* | There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.) |
⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}) | ||
Theorem | eqwrds3 14996 | A word is equal with a length 3 string iff it has length 3 and the same symbol at each position. (Contributed by AV, 12-May-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊 = 〈“𝐴𝐵𝐶”〉 ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = 𝐵 ∧ (𝑊‘2) = 𝐶)))) | ||
Theorem | wrdl3s3 14997* | A word of length 3 is a length 3 string. (Contributed by AV, 18-May-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) | ||
Theorem | s2rn 14998 | Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof shortened by AV, 1-Aug-2025.) |
⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) ⇒ ⊢ (𝜑 → ran 〈“𝐼𝐽”〉 = {𝐼, 𝐽}) | ||
Theorem | s3rn 14999 | Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof shortened by AV, 1-Aug-2025.) |
⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) ⇒ ⊢ (𝜑 → ran 〈“𝐼𝐽𝐾”〉 = {𝐼, 𝐽, 𝐾}) | ||
Theorem | s7rn 15000 | Range of a length 7 string. (Contributed by AV, 30-Jul-2025.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) ⇒ ⊢ (𝜑 → ran 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})) |
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