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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-s5 11501 Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s6 11502 Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s7 11503 Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s8 11504 Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theoremcats1cld 11505 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word               Word

Theoremcats1co 11506 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word                             concat

Theoremcats1cli 11507 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word        Word

Theoremcats1fvn 11508 The last symbol of a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1fv 11509 A symbol other than the last in a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1len 11510 The length of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1cat 11511 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word        Word        concat        concat        concat

Theorems2eqd 11512 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems3eqd 11513 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems4eqd 11514 Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems5eqd 11515 Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems6eqd 11516 Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems7eqd 11517 Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems8eqd 11518 Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems2cld 11519 A doubleton is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems3cld 11520 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems4cld 11521 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems5cld 11522 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems6cld 11523 A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems7cld 11524 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems8cld 11525 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems2cl 11526 A doubleton is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word

Theorems3cl 11527 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems2cli 11528 A doubleton is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems3cli 11529 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems4cli 11530 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems5cli 11531 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems6cli 11532 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems7cli 11533 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems8cli 11534 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems2fv0 11535 Extract the first symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems2fv1 11536 Extract the second symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems2len 11537 The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems3fv0 11538 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3fv1 11539 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3fv2 11540 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3len 11541 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems4len 11542 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems5len 11543 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems6len 11544 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems7len 11545 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems8len 11546 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems2co 11547 Mapping a doubleton by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems3co 11548 Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems0s1 11549 Concatenation of fixed length strings. (This special case of ccatlid 11434 is provided to complete the pattern s0s1 11549, df-s2 11498, df-s3 11499, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
concat

Theorems1s2 11550 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s3 11551 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s4 11552 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s5 11553 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s6 11554 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s7 11555 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems2s2 11556 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s2 11557 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s3 11558 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s4 11559 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theoremswrds2 11560 Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^ substr

5.7  Elementary real and complex functions

5.7.1  The "shift" operation

Syntaxcshi 11561 Extend class notation with function shifter.

Definitiondf-shft 11562* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of ) and produces a new function on . See shftval 11569 for its value. (Contributed by NM, 20-Jul-2005.)

Theoremshftlem 11563* Two ways to write a shifted set . (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremshftuz 11564* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftfval 11565* The value of the sequence shifter operation is a function on . is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftdm 11566* Domain of a relation shifted by . The set on the right is more commonly notated as (meaning add to every element of ). (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremshftfib 11567 Value of a fiber of the relation . (Contributed by Mario Carneiro, 4-Nov-2013.)

Theoremshftfn 11568* Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftval 11569 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremshftval2 11570 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval3 11571 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.)

Theoremshftval4 11572 Value of a sequence shifted by . (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval5 11573 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftf 11574* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theorem2shfti 11575 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt2 11576 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt 11577 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan1 11578 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan2 11579 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremseqshft 11580 Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)

5.7.2  Real and imaginary parts; conjugate

Syntaxccj 11581 Extend class notation to include complex conjugate function.

Syntaxcre 11582 Extend class notation to include real part of a complex number.

Syntaxcim 11583 Extend class notation to include imaginary part of a complex number.

Definitiondf-cj 11584* Define the complex conjugate function. See cjcli 11654 for its closure and cjval 11587 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Definitiondf-re 11585 Define a function whose value is the real part of a complex number. See reval 11591 for its value, recli 11652 for its closure, and replim 11601 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Definitiondf-im 11586 Define a function whose value is the imaginary part of a complex number. See imval 11592 for its value, imcli 11653 for its closure, and replim 11601 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Theoremcjval 11587* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjth 11588 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjf 11589 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjcl 11590 The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremreval 11591 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimval 11592 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimre 11593 The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremreim 11594 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremrecl 11595 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimcl 11596 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremref 11597 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimf 11598 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremcrre 11599 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremcrim 11600 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

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