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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwrdeq 11301 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word Word

Theoremwrdexg 11302 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremnfwrd 11303 Hypothesis builder for Word . (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremccatfn 11304 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
concat

Theoremccatfval 11305* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
concat ..^ ..^

Theoremccatcl 11306 The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.)
Word Word concat Word

Theoremccatlen 11307 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word Word concat

Theoremccatval1 11308 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Word Word ..^ concat

Theoremccatval2 11309 Value of a symbol in the right half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Word Word ..^ concat

Theoremccatval3 11310 Value of a symbol in the right half of a concatenated word, using an index relative to the subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word Word ..^ concat

Theoremccatlid 11311 Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word concat

Theoremccatrid 11312 Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word concat

Theoremccatass 11313 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word Word Word concat concat concat concat

Theoremids1 11314 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1val 11315 Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems1eq 11316 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1eqd 11317 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1cl 11318 A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1cld 11319 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1cli 11320 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1len 11321 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems1nz 11322 A singleton is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems1fv 11323 Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theoremeqs1 11324 A word of length 1 is a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word

Theorems111 11325 The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theoremwrdexb 11326 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremswrdval 11327* Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr ..^ ..^

Theoremswrd00 11328 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
substr

Theoremswrdcl 11329 Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word substr Word

Theoremswrdval2 11330* Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word substr ..^

Theoremswrd0val 11331 Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Word substr ..^

Theoremswrd0len 11332 Length of a left-anchored subword. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Word substr

Theoremswrdlen 11333 Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word substr

Theoremswrdfv 11334 A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word ..^ substr

Theoremswrdid 11335 A word is a subword of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word substr

Theoremccatswrd 11336 Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Word substr concat substr substr

Theoremswrdccat1 11337 Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word Word concat substr

Theoremswrdccat2 11338 Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word Word concat substr

Theoremccatopth 11339 An opth 4138-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word Word Word Word concat concat

Theoremccatopth2 11340 An opth 4138-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word Word Word Word concat concat

Theoremccatlcan 11341 Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
Word Word Word concat concat

Theoremccatrcan 11342 Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
Word Word Word concat concat

Theoremsplval 11343 Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
splice substr concat concat substr

Theoremsplcl 11344 Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word Word splice Word

Theoremsplid 11345 Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Word splice substr

Theoremspllen 11346 The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word                      Word        splice

Theoremsplfv1 11347 Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word                      Word        ..^       splice

Theoremsplfv2a 11348 Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word                      Word        ..^       splice

Theoremsplval2 11349 Value of a splice, assuming the input word has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        Word        Word        Word        concat concat                      splice concat concat

Theoremswrds1 11350 Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word ..^ substr

Theoremwrdeqcats1 11351 Decompose a non-empty word by separating off the last symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
Word substr concat

Theoremwrdeqs1cat 11352 Decompose a non-empty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
Word concat substr

Theoremcats1un 11353 Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)
Word concat

Theoremwrdind 11354* Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
concat                      Word        Word

Theoremrevval 11355* Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse ..^

Theoremrevcl 11356 The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word reverse Word

Theoremrevlen 11357 The reverse of a word has the same length as the original. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word reverse

Theoremrevfv 11358 Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word ..^ reverse

Theoremrev0 11359 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse

Theoremrevs1 11360 Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
reverse

Theoremrevccat 11361 Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word Word reverse concat reverse concat reverse

Theoremrevrev 11362 Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word reversereverse

Theoremwrdco 11363 Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word Word

Theoremlenco 11364 Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word

Theorems1co 11365 Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theoremrevco 11366 Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word reverse reverse

Theoremccatco 11367 Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word Word concat concat

5.6.10  Longer string literals

Syntaxcs2 11368 Syntax for the length 2 word constructor.

Syntaxcs3 11369 Syntax for the length 3 word constructor.

Syntaxcs4 11370 Syntax for the length 4 word constructor.

Syntaxcs5 11371 Syntax for the length 5 word constructor.

Syntaxcs6 11372 Syntax for the length 6 word constructor.

Syntaxcs7 11373 Syntax for the length 7 word constructor.

Syntaxcs8 11374 Syntax for the length 8 word constructor.

Definitiondf-s2 11375 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s3 11376 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s4 11377 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s5 11378 Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s6 11379 Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s7 11380 Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s8 11381 Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theoremcats1cld 11382 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word               Word

Theoremcats1co 11383 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word                             concat

Theoremcats1cli 11384 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word        Word

Theoremcats1fvn 11385 The last symbol of a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1fv 11386 A symbol other than the last in a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1len 11387 The length of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1cat 11388 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word        Word        concat        concat        concat

Theorems2eqd 11389 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems3eqd 11390 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems4eqd 11391 Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems5eqd 11392 Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems6eqd 11393 Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems7eqd 11394 Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems8eqd 11395 Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems2cld 11396 A doubleton is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems3cld 11397 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems4cld 11398 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems5cld 11399 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems6cld 11400 A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

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