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Theorem List for Metamath Proof Explorer - 41701-41800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfargshiftf 41701* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)

Theoremfargshiftf1 41702* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)

Theoremfargshiftfo 41703* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)

Theoremfargshiftfva 41704* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))

20.35.6  Words over a set (extension)

20.35.6.1  Last symbol of a word - extension

Theoremlswn0 41705 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( is the last symbol) and invalid cases ( means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (#‘𝑊) ≠ 0) → ( lastS ‘𝑊) ≠ ∅)

20.35.6.2  Prefixes of a word

In https://www.allacronyms.com/prefix/abbreviated, "pfx" is proposed as abbreviation for "prefix". Regarding the meaning of "prefix", it is different in computer science (automata theory/formal languages) compared with linguistics: in linguistics, a prefix has a meaning (see Wikipedia "Prefix" https://en.wikipedia.org/wiki/Prefix), whereas in computer science, a prefix is an arbitrary substring/subword starting at the beginning of a string/word (see Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix or https://math.stackexchange.com/questions/2190559/ is-there-standard-terminology-notation-for-the-prefix-of-a-word ).

Syntaxcpfx 41706 Syntax for the prefix operator.
class prefix

Definitiondf-pfx 41707* Define an operation which extracts prefixes of words, i.e. subwords starting at the beginning of a word. Definition in section 9.1 of [AhoHopUll] p. 318. "pfx" is used as label fragment. (Contributed by AV, 2-May-2020.)
prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))

Theorempfxval 41708 Value of a prefix. (Contributed by AV, 2-May-2020.)
((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))

Theorempfx00 41709 A zero length prefix. (Contributed by AV, 2-May-2020.)
(𝑆 prefix 0) = ∅

Theorempfx0 41710 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
(∅ prefix 𝐿) = ∅

Theorempfxcl 41711 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)

Theorempfxmpt 41712* Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝐿) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆𝑥)))

Theorempfxres 41713 Value of the prefix extractor as restriction. Could replace swrd0val 13466. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝐿) = (𝑆 ↾ (0..^𝐿)))

Theorempfxf 41714 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. Could replace swrd0f 13473. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊))) → (𝑊 prefix 𝐿):(0..^𝐿)⟶𝑉)

Theorempfxfn 41715 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝐿) Fn (0..^𝐿))

Theorempfxlen 41716 Length of a prefix. Could replace swrd0len 13467. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (#‘(𝑆 prefix 𝐿)) = 𝐿)

Theorempfxid 41717 A word is a prefix of itself. (Contributed by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix (#‘𝑆)) = 𝑆)

Theorempfxrn 41718 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ 𝑉)

Theorempfxn0 41719 A prefix consisting of at least one symbol is not empty. Could replace swrdn0 13476. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ ∧ 𝐿 ≤ (#‘𝑊)) → (𝑊 prefix 𝐿) ≠ ∅)

Theorempfxnd 41720 The value of the prefix extractor is the empty set (undefined) if the argument is not within the range of the word. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ0 ∧ (#‘𝑊) < 𝐿) → (𝑊 prefix 𝐿) = ∅)

Theorempfxlen0 41721 Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 13482. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ0 ∧ (#‘𝑊) = (𝐿 + 1)) → (#‘(𝑊 prefix 𝐿)) = 𝐿)

Theoremaddlenrevpfx 41722 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) + (#‘(𝑊 prefix 𝑀))) = (#‘𝑊))

Theoremaddlenpfx 41723 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 prefix 𝑀)) + (#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theorempfxfv 41724 A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 13485. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = (𝑊𝐼))

Theorempfxfv0 41725 The first symbol in a prefix of a word. Could replace swrd0fv0 13486. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(#‘𝑊))) → ((𝑊 prefix 𝐿)‘0) = (𝑊‘0))

Theorempfxtrcfv 41726 A symbol in a word truncated by one symbol. Could replace swrdtrcfv 13487. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅ ∧ 𝐼 ∈ (0..^((#‘𝑊) − 1))) → ((𝑊 prefix ((#‘𝑊) − 1))‘𝐼) = (𝑊𝐼))

Theorempfxtrcfv0 41727 The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 13488. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ((𝑊 prefix ((#‘𝑊) − 1))‘0) = (𝑊‘0))

Theorempfxfvlsw 41728 The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 13489. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 prefix 𝐿)) = (𝑊‘(𝐿 − 1)))

Theorempfxeq 41729* The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 13490. (Contributed by AV, 4-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖))))

Theorempfxtrcfvl 41730 The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 13496. (Contributed by AV, 5-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ( lastS ‘(𝑊 prefix ((#‘𝑊) − 1))) = (𝑊‘((#‘𝑊) − 2)))

Theorempfxsuffeqwrdeq 41731 Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 13499. (Contributed by AV, 5-May-2020.)
((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))

Theorempfxsuff1eqwrdeq 41732 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 13500. (Contributed by AV, 6-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 prefix ((#‘𝑊) − 1)) = (𝑈 prefix ((#‘𝑊) − 1)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Theoremdisjwrdpfx 41733* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 13501. (Contributed by AV, 6-May-2020.)
Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}

Theoremccatpfx 41734 Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))

Theorempfxccat1 41735 Recover the left half of a concatenated word. Could replace swrdccat1 13503. (Contributed by AV, 6-May-2020.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) prefix (#‘𝑆)) = 𝑆)

Theorempfx1 41736 A prefix of length 1. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 prefix 1) = ⟨“(𝑊‘0)”⟩)

Theorempfx2 41737 A prefix of length 2. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Theorempfxswrd 41738 A prefix of a subword. Could replace swrd0swrd 13507. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝐿 ∈ (0...(𝑁𝑀)) → ((𝑊 substr ⟨𝑀, 𝑁⟩) prefix 𝐿) = (𝑊 substr ⟨𝑀, (𝑀 + 𝐿)⟩)))

Theoremswrdpfx 41739 A subword of a prefix. Could replace swrdswrd0 13508. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → ((𝑊 prefix 𝑁) substr ⟨𝐾, 𝐿⟩) = (𝑊 substr ⟨𝐾, 𝐿⟩)))

Theorempfxpfx 41740 A prefix of a prefix. Could replace swrd0swrd0 13509. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝐿 ∈ (0...𝑁)) → ((𝑊 prefix 𝑁) prefix 𝐿) = (𝑊 prefix 𝐿))

Theorempfxpfxid 41741 A prefix of a prefix with the same length is the prefix. Could replace swrd0swrdid 13510. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 𝑁) prefix 𝑁) = (𝑊 prefix 𝑁))

Theorempfxcctswrd 41742 The concatenation of the prefix of a word and the rest of the word yields the word itself. Could replace wrdcctswrd 13511. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 𝑀) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) = 𝑊)

Theoremlenpfxcctswrd 41743 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. Could replace lencctswrd 13512. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 prefix 𝑀) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theoremlenrevpfxcctswrd 41744 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. Could replace lenrevcctswrd 13513. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 substr ⟨𝑀, (#‘𝑊)⟩) ++ (𝑊 prefix 𝑀))) = (#‘𝑊))

Theorempfxlswccat 41745 Reconstruct a nonempty word from its prefix and last symbol. Could replace swrdccatwrd 13514. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ((𝑊 prefix ((#‘𝑊) − 1)) ++ ⟨“( lastS ‘𝑊)”⟩) = 𝑊)

Theoremccats1pfxeq 41746 The last symbol of a word concatenated with the word with the last symbol removed having results in the word itself. Could replace ccats1swrdeq 13515. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) → 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Theoremccats1pfxeqrex 41747* There exists a symbol such that its concatenation with the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. Could replace ccats1swrdeqrex 13524. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩)))

Theorempfxccatin12lem1 41748 Lemma 1 for pfxccatin12 41750. Could replace swrdccatin12lem2b 13532. (Contributed by AV, 9-May-2020.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (𝐾 − (𝐿𝑀)) ∈ (0..^(𝑁𝐿))))

Theorempfxccatin12lem2 41749 Lemma 2 for pfxccatin12 41750. Could replace swrdccatin12lem2 13535. (Contributed by AV, 9-May-2020.)
𝐿 = (#‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐵 prefix (𝑁𝐿))‘(𝐾 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))

Theorempfxccatin12 41750 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 13537. (Contributed by AV, 9-May-2020.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))

Theorempfxccat3 41751 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 13538. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))))

Theorempfxccatpfx1 41752 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))

Theorempfxccatpfx2 41753 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)    &   𝑀 = (#‘𝐵)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))

Theorempfxccat3a 41754 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 13540. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)    &   𝑀 = (#‘𝐵)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿))))))

Theorempfxccatid 41755 A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 13543. (Contributed by AV, 10-May-2020.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) prefix 𝑁) = 𝐴)

Theoremccats1pfxeqbi 41756 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. Could replace ccats1swrdeqbi 13544. (Contributed by AV, 10-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) ↔ 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Theorempfxccatin12d 41757 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 13547. (Contributed by AV, 10-May-2020.)
(𝜑 → (#‘𝐴) = 𝐿)    &   (𝜑 → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))    &   (𝜑𝑀 ∈ (0...𝐿))    &   (𝜑𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))))

Theoremreuccatpfxs1lem 41758* Lemma for reuccatpfxs1 41759. Could replace reuccats1lem 13525. (Contributed by AV, 9-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈𝑋) ∧ ∀𝑠𝑉 ((𝑊 ++ ⟨“𝑠”⟩) ∈ 𝑋𝑆 = 𝑠) ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑊 = (𝑈 prefix (#‘𝑊)) → 𝑈 = (𝑊 ++ ⟨“𝑆”⟩)))

Theoremreuccatpfxs1 41759* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 13526. (Contributed by AV, 10-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑤𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))

Theoremsplvalpfx 41760 Value of the substring replacement operator. (Contributed by AV, 11-May-2020.)
((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))

Theoremrepswpfx 41761 A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
((𝑆𝑉𝑁 ∈ ℕ0𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿))

Theoremcshword2 41762 Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))

Theorempfxco 41763 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝐴𝑁 ∈ (0...(#‘𝑊)) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 prefix 𝑁)) = ((𝐹𝑊) prefix 𝑁))

20.35.7  Number theory (extension)

20.35.7.1  Fermat numbers

At first, the (sequence of) Fermat numbers FermatNo (the 𝑛-th Fermat number is denoted as (FermatNo‘𝑛)) is defined, see df-fmtno 41765, and basic theorems are provided. Afterwards, it is shown that the first five Fermat numbers are prime, the (first) five Fermat primes, see fmtnofz04prm 41814, but that the fifth Fermat number (counting starts at 0!) is not prime, see fmtno5nprm 41820. The fourth Fermat number (i.e., the fifth Fermat prime) (FermatNo‘4) = 65537 is currently the biggest number proven to be prime in set.mm, see 65537prm 41813 (previously, it was 4001, see 4001prm 15899).

Another important result of this section is Goldbach's theorem goldbachth 41784, showing that two different Fermut numbers are coprime. By this, it can be proven that there is an infinite number of primes, see prminf2 41825.

Finally, it is shown that every prime of the form ((2↑𝑘) + 1) must be a Fermat number (i.e., a Fermat prime), see 2pwp1prmfmtno 41829.

Syntaxcfmtno 41764 Extend class notation with the Fermat numbers.
class FermatNo

Definitiondf-fmtno 41765 Define the function that enumerates the Fermat numbers, see definition in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))

Theoremfmtno 41766 The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))

Theoremfmtnoge3 41767 Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ‘3))

Theoremfmtnonn 41768 Each Fermat number is a positive integer. (Contributed by AV, 26-Jul-2021.) (Proof shortened by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ)

Theoremfmtnom1nn 41769 A Fermat number minus one is a power of a power of two. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → ((FermatNo‘𝑁) − 1) = (2↑(2↑𝑁)))

Theoremfmtnoodd 41770 Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.)
(𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁))

Theoremfmtnorn 41771* A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.)
(𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)

Theoremfmtnof1 41772 The enumeration of the Fermat numbers is a one-one function into the positive integers. (Contributed by AV, 3-Aug-2021.)
FermatNo:ℕ01-1→ℕ

Theoremfmtnoinf 41773 The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.)
ran FermatNo ∉ Fin

Theoremfmtnorec1 41774 The first recurrence relation for Fermat numbers, see Wikipedia "Fermat number", https://en.wikipedia.org/wiki/Fermat_number#Basic_properties, 22-Jul-2021. (Contributed by AV, 22-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = ((((FermatNo‘𝑁) − 1)↑2) + 1))

Theoremsqrtpwpw2p 41775 The floor of the square root of 2 to the power of 2 to the power of a positive integer plus a bounded nonnegative integer. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0𝑀 < ((2↑((2↑(𝑁 − 1)) + 1)) + 1)) → (⌊‘(√‘((2↑(2↑𝑁)) + 𝑀))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtnosqrt 41776 The floor of the square root of a Fermat number. (Contributed by AV, 28-Jul-2021.)
(𝑁 ∈ ℕ → (⌊‘(√‘(FermatNo‘𝑁))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtno0 41777 The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) = 3

Theoremfmtno1 41778 The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) = 5

Theoremfmtnorec2lem 41779* Lemma for fmtnorec2 41780 (induction step). (Contributed by AV, 29-Jul-2021.)
(𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))

Theoremfmtnorec2 41780* The second recurrence relation for Fermat numbers, see ProofWiki "Product of Sequence of Fermat Numbers plus 2", 29-Jul-2021, https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 or Wikipedia "Fermat number", 29-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

Theoremfmtnodvds 41781 Any Fermat number divides a greater Fermat number minus 2. Corrolary of fmtnorec2 41780, see ProofWiki "Product of Sequence of Fermat Numbers plus 2/Corollary", 31-Jul-2021. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ) → (FermatNo‘𝑁) ∥ ((FermatNo‘(𝑁 + 𝑀)) − 2))

Theoremgoldbachthlem1 41782 Lemma 1 for goldbachth 41784. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))

Theoremgoldbachthlem2 41783 Lemma 2 for goldbachth 41784. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremgoldbachth 41784 Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑁𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremfmtnorec3 41785* The third recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 2-Aug-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = ((FermatNo‘(𝑁 − 1)) + ((2↑(2↑(𝑁 − 1))) · ∏𝑛 ∈ (0...(𝑁 − 2))(FermatNo‘𝑛))))

Theoremfmtnorec4 41786 The fourth recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 31-Jul-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = (((FermatNo‘(𝑁 − 1))↑2) − (2 · (((FermatNo‘(𝑁 − 2)) − 1)↑2))))

Theoremfmtno2 41787 The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) = 17

Theoremfmtno3 41788 The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘3) = 257

Theoremfmtno4 41789 The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘4) = 65537

Theoremfmtno5lem1 41790 Lemma 1 for fmtno5 41794. (Contributed by AV, 22-Jul-2021.)
(65536 · 6) = 393216

Theoremfmtno5lem2 41791 Lemma 2 for fmtno5 41794. (Contributed by AV, 22-Jul-2021.)
(65536 · 5) = 327680

Theoremfmtno5lem3 41792 Lemma 3 for fmtno5 41794. (Contributed by AV, 22-Jul-2021.)
(65536 · 3) = 196608

Theoremfmtno5lem4 41793 Lemma 4 for fmtno5 41794. (Contributed by AV, 30-Jul-2021.)
(65536↑2) = 4294967296

Theoremfmtno5 41794 The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
(FermatNo‘5) = 4294967297

Theoremfmtno0prm 41795 The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) ∈ ℙ

Theoremfmtno1prm 41796 The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) ∈ ℙ

Theoremfmtno2prm 41797 The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) ∈ ℙ

Theorem257prm 41798 257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
257 ∈ ℙ

Theoremfmtno3prm 41799 The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
(FermatNo‘3) ∈ ℙ

Theoremodz2prm2pw 41800 Any power of two is coprime to any prime not being two. (Contributed by AV, 25-Jul-2021.)
(((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ∧ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))

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