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Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwrdco 11801 Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word Word

Theoremlenco 11802 Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word

Theorems1co 11803 Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theoremrevco 11804 Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word reverse reverse

Theoremccatco 11805 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 11806 Syntax for the length 2 word constructor.

Syntaxcs3 11807 Syntax for the length 3 word constructor.

Syntaxcs4 11808 Syntax for the length 4 word constructor.

Syntaxcs5 11809 Syntax for the length 5 word constructor.

Syntaxcs6 11810 Syntax for the length 6 word constructor.

Syntaxcs7 11811 Syntax for the length 7 word constructor.

Syntaxcs8 11812 Syntax for the length 8 word constructor.

Definitiondf-s2 11813 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s3 11814 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s4 11815 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s5 11816 Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s6 11817 Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s7 11818 Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Definitiondf-s8 11819 Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theoremcats1cld 11820 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word               Word

Theoremcats1co 11821 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word                             concat

Theoremcats1cli 11822 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word        Word

Theoremcats1fvn 11823 The last symbol of a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1fv 11824 A symbol other than the last in a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1len 11825 The length of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word

Theoremcats1cat 11826 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat        Word        Word        concat        concat        concat

Theorems2eqd 11827 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems3eqd 11828 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems4eqd 11829 Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems5eqd 11830 Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems6eqd 11831 Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems7eqd 11832 Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems8eqd 11833 Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems2cld 11834 A doubleton is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems3cld 11835 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems4cld 11836 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems5cld 11837 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems6cld 11838 A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems7cld 11839 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems8cld 11840 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Word

Theorems2cl 11841 A doubleton is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word

Theorems3cl 11842 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems2cli 11843 A doubleton is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems3cli 11844 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems4cli 11845 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems5cli 11846 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems6cli 11847 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems7cli 11848 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems8cli 11849 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems2fv0 11850 Extract the first symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems2fv1 11851 Extract the second symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems2len 11852 The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems3fv0 11853 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3fv1 11854 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3fv2 11855 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3len 11856 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems4len 11857 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems5len 11858 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems6len 11859 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems7len 11860 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems8len 11861 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems2prop 11862 A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theorems4prop 11863 A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theorems2f1o 11864 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.)

Theoremf1oun2prg 11865 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.)

Theorems4f1o 11866 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.)

Theorems4dom 11867 The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)

Theorems2co 11868 Mapping a doubleton by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems3co 11869 Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems0s1 11870 Concatenation of fixed length strings. (This special case of ccatlid 11749 is provided to complete the pattern s0s1 11870, df-s2 11813, df-s3 11814, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
concat

Theorems1s2 11871 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s3 11872 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s4 11873 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s5 11874 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s6 11875 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s7 11876 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems2s2 11877 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s2 11878 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s3 11879 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s4 11880 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theoremswrds2 11881 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 11882 Extend class notation with function shifter.

Definitiondf-shft 11883* 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 11890 for its value. (Contributed by NM, 20-Jul-2005.)

Theoremshftlem 11884* Two ways to write a shifted set . (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremshftuz 11885* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftfval 11886* 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 11887* 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 11888 Value of a fiber of the relation . (Contributed by Mario Carneiro, 4-Nov-2013.)

Theoremshftfn 11889* Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftval 11890 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremshftval2 11891 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval3 11892 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.)

Theoremshftval4 11893 Value of a sequence shifted by . (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval5 11894 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftf 11895* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theorem2shfti 11896 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt2 11897 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt 11898 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan1 11899 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan2 11900 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

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