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Type | Label | Description |
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Statement | ||
Axiom | ax-hgprmladder 38901 | There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) |
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Theorem | tgblthelfgott 38902 | The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 38900, ax-hgprmladder 38901 and bgoldbtbnd 38898. (Contributed by AV, 4-Aug-2020.) |
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Theorem | tgoldbachlt 38903* |
The ternary Goldbach conjecture is valid for small odd numbers (i.e. for
all odd numbers less than a fixed big ![]() |
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Axiom | ax-tgoldbachgt 38904* |
The ternary Goldbach conjecture is valid for big odd numbers (i.e. for
all odd numbers greater than a fixed ![]() |
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Theorem | tgoldbach 38905 | The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 38903 and ax-tgoldbachgt 38904. (Contributed by AV, 2-Aug-2020.) |
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Theorem | modexp2m1d 38906 | The square of an integer which is -1 modulo a number greater than 1 is 1 modulo the same modulus. (Contributed by AV, 5-Jul-2020.) |
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Theorem | proththdlem 38907 | Lemma for proththd 38908. (Contributed by AV, 4-Jul-2020.) |
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Theorem | proththd 38908* | Proth's theorem (1878). If P is a Proth number, i.e. a number of the form k2^n+1 with k less than 2^n, and if there exists an integer x for which x^((P-1)/2) is -1 modulo P, then P is prime. Such a prime is called a Proth prime. Like Pocklington's theorem (see pockthg 14843), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.) |
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Theorem | 5tcu2e40 38909 | 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.) |
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Theorem | 3exp4mod41 38910 | 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.) |
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Theorem | 41prothprmlem1 38911 | Lemma 1 for 41prothprm 38913. (Contributed by AV, 4-Jul-2020.) |
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Theorem | 41prothprmlem2 38912 | Lemma 2 for 41prothprm 38913. (Contributed by AV, 5-Jul-2020.) |
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Theorem | 41prothprm 38913 | 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.) |
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Theorem | lswn0 38914 |
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 (![]() ![]() |
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Theorem | ccatw2s1cl 38915 | The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
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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 ). | ||
Syntax | cpfx 38916 | Syntax for the prefix operator. |
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Definition | df-pfx 38917* | 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.) |
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Theorem | pfxval 38918 | Value of a prefix. (Contributed by AV, 2-May-2020.) |
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Theorem | pfx00 38919 | A zero length prefix. (Contributed by AV, 2-May-2020.) |
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Theorem | pfx0 38920 | A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.) |
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Theorem | pfxcl 38921 | Closure of the prefix extractor. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxmpt 38922* | Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxres 38923 | Value of the prefix extractor as restriction. Could replace swrd0val 12772. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxf 38924 | 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 12778. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxfn 38925 | Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxlen 38926 | Length of a prefix. Could replace swrd0len 12773. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxid 38927 | A word is a prefix of itself. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxrn 38928 | 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.) |
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Theorem | pfxn0 38929 | A prefix consisting of at least one symbol is not empty. Could replace swrdn0 12781. (Contributed by AV, 2-May-2020.) |
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Theorem | pfxnd 38930 | 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.) |
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Theorem | pfxlen0 38931 | Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 12787. TODO-AV: Really useful? swrd0len0 12787 is only used in wwlknred 25444. (Contributed by AV, 3-May-2020.) |
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Theorem | addlenrevpfx 38932 | The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by AV, 3-May-2020.) |
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Theorem | addlenpfx 38933 | The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 3-May-2020.) |
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Theorem | pfxfv 38934 | A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 12790. (Contributed by AV, 3-May-2020.) |
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Theorem | pfxfv0 38935 | The first symbol in a prefix of a word. Could replace swrd0fv0 12791. (Contributed by AV, 3-May-2020.) |
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Theorem | pfxtrcfv 38936 | A symbol in a word truncated by one symbol. Could replace swrdtrcfv 12792. (Contributed by AV, 3-May-2020.) |
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Theorem | pfxtrcfv0 38937 | The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 12793. (Contributed by AV, 3-May-2020.) |
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Theorem | pfxfvlsw 38938 | The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 12794. (Contributed by AV, 3-May-2020.) |
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Theorem | pfxeq 38939* | The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 12795. (Contributed by AV, 4-May-2020.) |
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Theorem | pfxtrcfvl 38940 | The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 12801. (Contributed by AV, 5-May-2020.) |
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Theorem | pfxsuffeqwrdeq 38941 | Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 12804. (Contributed by AV, 5-May-2020.) |
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Theorem | pfxsuff1eqwrdeq 38942 | Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 12805. (Contributed by AV, 6-May-2020.) |
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Theorem | disjwrdpfx 38943* | Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 12806. (Contributed by AV, 6-May-2020.) |
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Theorem | ccatpfx 38944 | Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.) |
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Theorem | pfxccat1 38945 | Recover the left half of a concatenated word. Could replace swrdccat1 12808. (Contributed by AV, 6-May-2020.) |
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Theorem | pfx1 38946 | A prefix of length 1. (Contributed by AV, 15-May-2020.) |
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Theorem | pfx2 38947 | A prefix of length 2. (Contributed by AV, 15-May-2020.) |
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Theorem | pfxswrd 38948 | A prefix of a subword. Could replace swrd0swrd 12812. (Contributed by AV, 8-May-2020.) |
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Theorem | swrdpfx 38949 | A subword of a prefix. Could replace swrdswrd0 12813. (Contributed by AV, 8-May-2020.) |
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Theorem | pfxpfx 38950 | A prefix of a prefix. Could replace swrd0swrd0 12814. (Contributed by AV, 8-May-2020.) |
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Theorem | pfxpfxid 38951 | A prefix of a prefix with the same length is the prefix. Could replace swrd0swrdid 12815. (Contributed by AV, 8-May-2020.) |
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Theorem | pfxcctswrd 38952 | The concatenation of the prefix of a word and the rest of the word yields the word itself. Could replace wrdcctswrd 12816. (Contributed by AV, 9-May-2020.) |
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Theorem | lenpfxcctswrd 38953 | 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 12817. (Contributed by AV, 9-May-2020.) |
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Theorem | lenrevpfxcctswrd 38954 | 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 12818. (Contributed by AV, 9-May-2020.) |
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Theorem | pfxlswccat 38955 | Reconstruct a nonempty word from its prefix and last symbol. Could replace wrdeqcats1OLD 12825 resp. swrdccatwrd 12819. (Contributed by AV, 9-May-2020.) |
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Theorem | ccats1pfxeq 38956 | The last symbol of a word concatenated with the word with the last symbol removed having results in the word itself. Could replace ccats1swrdeq 12820. (Contributed by AV, 9-May-2020.) |
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Theorem | ccats1pfxeqrex 38957* | 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 12830. (Contributed by AV, 9-May-2020.) |
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Theorem | pfxccatin12lem1 38958 | Lemma 1 for pfxccatin12 38960. Could replace swrdccatin12lem2b 12837. (Contributed by AV, 9-May-2020.) |
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Theorem | pfxccatin12lem2 38959 | Lemma 2 for pfxccatin12 38960. Could replace swrdccatin12lem2 12840. (Contributed by AV, 9-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pfxccatin12 38960 | The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 12842. (Contributed by AV, 9-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pfxccat3 38961 | 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 12843. (Contributed by AV, 10-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pfxccatpfx1 38962 | A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.) |
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Theorem | pfxccatpfx2 38963 | 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.) |
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Theorem | pfxccat3a 38964 | 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 12845. (Contributed by AV, 10-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pfxccatid 38965 | A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 12848. (Contributed by AV, 10-May-2020.) |
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Theorem | ccats1pfxeqbi 38966 | 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 12849. (Contributed by AV, 10-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pfxccatin12d 38967 | The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 12852. (Contributed by AV, 10-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | reuccatpfxs1lem 38968* | Lemma for reuccatpfxs1 38969. Could replace reuccats1lem 12831. (Contributed by AV, 9-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | reuccatpfxs1 38969* | 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 12832. (Contributed by AV, 10-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | splvalpfx 38970 | Value of the substring replacement operator. (Contributed by AV, 11-May-2020.) |
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Theorem | repswpfx 38971 | A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.) |
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Theorem | cshword2 38972 | Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.) |
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Theorem | pfxco 38973 | Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.) |
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Additional theorems for classical first-order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory. | ||
Theorem | elnelall 38974 | A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
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Theorem | clel5 38975* |
Alternate definition of class membership: a class ![]() ![]() ![]() ![]() |
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Theorem | dfss7 38976* |
Alternate definition of subclass relationship: a class ![]() ![]() ![]() ![]() |
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Theorem | sssseq 38977 | If a class is a subclass of another class, the classes are equal iff the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.) |
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Theorem | prcssprc 38978 | The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
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Theorem | ralnralall 38979* | A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.) |
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Theorem | falseral0 38980* | A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.) |
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Theorem | ralralimp 38981* | Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.) |
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Theorem | n0rex 38982* | There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.) |
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Theorem | ssn0rex 38983* | There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.) |
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Theorem | elpwdifsn 38984 | A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.) |
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Theorem | pr1eqbg 38985 | A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
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Theorem | pr1nebg 38986 | A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
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Theorem | prelpw 38987 | A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.) |
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Theorem | rexdifpr 38988 | Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) |
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Theorem | issn 38989* | A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.) |
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Theorem | n0snor2el 38990* | A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.) |
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Theorem | opidg 38991 |
The ordered pair ![]() ![]() ![]() ![]() ![]() |
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Theorem | snopeqop 38992 | Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.) |
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Theorem | propeqop 38993 | Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | propssopi 38994 | If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ssprss 38995 | A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.) |
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Theorem | ssprsseq 38996 | A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.) |
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Theorem | elpr2elpr 38997* | For an element of an unordered pair which is a subset of a given set, there is another (maybe the same) element of the given set being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.) |
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Theorem | iunxprg 38998* | A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.) |
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Theorem | iunopeqop 38999* | Equivalence for an ordered pair equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | otiunsndisjX 39000* | The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
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