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Theorem List for Metamath Proof Explorer - 40901-41000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnpw2blenfzo2 40901 A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.)
 |-  ( N  e.  NN  ->  ( N  =  ( 2 ^ ( (#b `  N )  -  1
 ) )  \/  N  e.  ( ( ( 2 ^ ( (#b `  N )  -  1
 ) )  +  1 )..^ ( 2 ^
 (#b `  N )
 ) ) ) )
 
Theoremnnpw2pmod 40902 Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
 |-  ( N  e.  NN  ->  N  =  ( ( 2 ^ ( (#b `  N )  -  1
 ) )  +  ( N  mod  ( 2 ^
 ( (#b `  N )  -  1 ) ) ) ) )
 
Theoremblen1 40903 The binary length of 1. (Contributed by AV, 21-May-2020.)
 |-  (#b `  1 )  =  1
 
Theoremblen2 40904 The binary length of 2. (Contributed by AV, 21-May-2020.)
 |-  (#b `  2 )  =  2
 
Theoremnnpw2p 40905* Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
 |-  ( N  e.  NN  ->  E. i  e.  NN0  E. r  e.  ( 0..^ ( 2 ^ i ) ) N  =  ( ( 2 ^ i )  +  r ) )
 
Theoremnnpw2pb 40906* A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
 |-  ( N  e.  NN  <->  E. i  e.  NN0  E. r  e.  ( 0..^ ( 2 ^ i
 ) ) N  =  ( ( 2 ^
 i )  +  r
 ) )
 
Theoremblen1b 40907 The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.)
 |-  ( N  e.  NN0  ->  (
 (#b `  N )  =  1  <->  ( N  =  0  \/  N  =  1 ) ) )
 
Theoremblennnt2 40908 The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, 30-May-2010.)
 |-  ( N  e.  NN  ->  (#b
 `  ( 2  x.  N ) )  =  ( (#b `  N )  +  1 )
 )
 
Theoremnnolog2flm1 40909 The floor of the binary logarithm of an odd integer greater than 1 is the floor of the binary logarithm of the integer decreased by 1. (Contributed by AV, 2-Jun-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  ( ( N  +  1 )  /  2
 )  e.  NN )  ->  ( |_ `  (
 2 logb  N ) )  =  ( |_ `  (
 2 logb  ( N  -  1
 ) ) ) )
 
Theoremblennn0em1 40910 The binary length of the half of an even positive integer is the binary length of the integer minus 1. (Contributed by AV, 30-May-2010.)
 |-  (
 ( N  e.  NN  /\  ( N  /  2
 )  e.  NN0 )  ->  (#b `  ( N  /  2 ) )  =  ( (#b `  N )  -  1 ) )
 
Theoremblennngt2o2 40911 The binary length of an odd integer greater than 1 is the binary length of the half of the integer decreased by 1, increased by 1. (Contributed by AV, 3-Jun-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  (#b `  N )  =  ( (#b `  (
 ( N  -  1
 )  /  2 )
 )  +  1 ) )
 
Theoremblengt1fldiv2p1 40912 The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  (#b `  N )  =  (
 (#b `  ( |_ `  ( N  /  2
 ) ) )  +  1 ) )
 
Theoremblennn0e2 40913 The binary length of an even positive integer is the binary length of the half of the integer, increased by 1. (Contributed by AV, 29-May-2020.)
 |-  (
 ( N  e.  NN  /\  ( N  /  2
 )  e.  NN0 )  ->  (#b `  N )  =  ( (#b `  ( N  /  2 ) )  +  1 ) )
 
21.33.16.10  Digits

Generalisation of df-bits 14474. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 40933: 
( ( K (digit 2 ) N ) = 1 <-> K e. ( bits  N ) ).

 
Syntaxcdig 40914 Extend class notation with the class of the digit extraction operation.
 class digit
 
Definitiondf-dig 40915* Definition of an operation to obtain the  k th digit of a nonnegative real number  r in the positional system with base  b.  k  =  - 
1 corresponds to the first digit of the fractional part (for  b  =  10 the first digit after the decimal point),  k  =  0 corresponds to the last digit of the integer part (for  b  =  10 the first digit before the decimal point). See also digit1 12444. Examples (not formal): ( 234.567 ( digit ` 10 ) 0 ) = 4; ( 2.567 ( digit ` 10 ) -2 ) = 6; ( 2345.67 ( digit ` 10 ) 2 ) = 3. (Contributed by AV, 16-May-2020.)
 |- digit  =  ( b  e.  NN  |->  ( k  e.  ZZ ,  r  e.  ( 0 [,) +oo )  |->  ( ( |_ `  ( ( b ^ -u k
 )  x.  r ) )  mod  b ) ) )
 
Theoremdigfval 40916* Operation to obtain the  k th digit of a nonnegative real number  r in the positional system with base  B. (Contributed by AV, 23-May-2020.)
 |-  ( B  e.  NN  ->  (digit `  B )  =  ( k  e.  ZZ ,  r  e.  ( 0 [,) +oo )  |->  ( ( |_ `  ( ( B ^ -u k
 )  x.  r ) )  mod  B ) ) )
 
Theoremdigval 40917 The  K th digit of a nonnegative real number  R in the positional system with base  B. (Contributed by AV, 23-May-2020.)
 |-  (
 ( B  e.  NN  /\  K  e.  ZZ  /\  R  e.  ( 0 [,) +oo ) )  ->  ( K (digit `  B ) R )  =  ( ( |_ `  (
 ( B ^ -u K )  x.  R ) ) 
 mod  B ) )
 
Theoremdigvalnn0 40918 The  K th digit of a nonnegative real number  R in the positional system with base  B is a nonnegative integer. (Contributed by AV, 28-May-2020.)
 |-  (
 ( B  e.  NN  /\  K  e.  ZZ  /\  R  e.  ( 0 [,) +oo ) )  ->  ( K (digit `  B ) R )  e.  NN0 )
 
Theoremnn0digval 40919 The  K th digit of a nonnegative real number  R in the positional system with base  B. (Contributed by AV, 23-May-2020.)
 |-  (
 ( B  e.  NN  /\  K  e.  NN0  /\  R  e.  ( 0 [,) +oo ) )  ->  ( K (digit `  B ) R )  =  (
 ( |_ `  ( R  /  ( B ^ K ) ) ) 
 mod  B ) )
 
Theoremdignn0fr 40920 The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.)
 |-  (
 ( B  e.  NN  /\  K  e.  ( ZZ  \  NN0 )  /\  N  e.  NN0 )  ->  ( K (digit `  B ) N )  =  0
 )
 
Theoremdignn0ldlem 40921 Lemma for dignnld 40922. (Contributed by AV, 25-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN  /\  K  e.  ( ZZ>= `  ( ( |_ `  ( B logb  N ) )  +  1 ) ) )  ->  N  <  ( B ^ K ) )
 
Theoremdignnld 40922 The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN  /\  K  e.  ( ZZ>= `  ( ( |_ `  ( B logb  N ) )  +  1 ) ) )  ->  ( K (digit `  B ) N )  =  0
 )
 
Theoremdig2nn0ld 40923 The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
 |-  (
 ( N  e.  NN  /\  K  e.  ( ZZ>= `  (#b `  N ) ) )  ->  ( K (digit `  2 ) N )  =  0 )
 
Theoremdig2nn1st 40924 The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
 |-  ( N  e.  NN  ->  ( ( (#b `  N )  -  1 ) (digit `  2 ) N )  =  1 )
 
Theoremdig0 40925 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
 |-  (
 ( B  e.  NN  /\  K  e.  ZZ )  ->  ( K (digit `  B ) 0 )  =  0 )
 
Theoremdigexp 40926 The  K th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  K  e.  NN0  /\  N  e.  NN0 )  ->  ( K (digit `  B )
 ( B ^ N ) )  =  if ( K  =  N ,  1 ,  0 ) )
 
Theoremdig1 40927 All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  K  e.  ZZ )  ->  ( K (digit `  B ) 1 )  =  if ( K  =  0 ,  1 ,  0 ) )
 
Theorem0dig1 40928 The  0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.)
 |-  ( B  e.  ( ZZ>= `  2 )  ->  ( 0 (digit `  B )
 1 )  =  1 )
 
Theorem0dig2pr01 40929 The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
 |-  ( N  e.  { 0 ,  1 }  ->  ( 0 (digit `  2
 ) N )  =  N )
 
Theoremdig2nn0 40930 A digit of a nonnegative integer  N in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
 |-  (
 ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( K (digit `  2 ) N )  e.  { 0 ,  1 } )
 
Theorem0dig2nn0e 40931 The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.)
 |-  (
 ( N  e.  NN0  /\  ( N  /  2
 )  e.  NN0 )  ->  ( 0 (digit `  2 ) N )  =  0 )
 
Theorem0dig2nn0o 40932 The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.)
 |-  (
 ( N  e.  NN0  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( 0 (digit `  2 ) N )  =  1 )
 
Theoremdig2bits 40933 The  K th digit of a nonnegative integer  N in a binary system is its  K th bit. (Contributed by AV, 24-May-2020.)
 |-  (
 ( N  e.  NN0  /\  K  e.  NN0 )  ->  ( ( K (digit `  2 ) N )  =  1  <->  K  e.  (bits `  N ) ) )
 
21.33.16.11  Nonnegative integer as sum of its shifted digits
 
Theoremdignn0flhalflem1 40934 Lemma 1 for dignn0flhalf 40937. (Contributed by AV, 7-Jun-2012.)
 |-  (
 ( A  e.  ZZ  /\  ( ( A  -  1 )  /  2
 )  e.  NN  /\  N  e.  NN )  ->  ( |_ `  (
 ( A  /  (
 2 ^ N ) )  -  1 ) )  <  ( |_ `  ( ( A  -  1 )  /  (
 2 ^ N ) ) ) )
 
Theoremdignn0flhalflem2 40935 Lemma 2 for dignn0flhalf 40937. (Contributed by AV, 7-Jun-2012.)
 |-  (
 ( A  e.  ZZ  /\  ( ( A  -  1 )  /  2
 )  e.  NN  /\  N  e.  NN0 )  ->  ( |_ `  ( A 
 /  ( 2 ^
 ( N  +  1 ) ) ) )  =  ( |_ `  (
 ( |_ `  ( A  /  2 ) ) 
 /  ( 2 ^ N ) ) ) )
 
Theoremdignn0ehalf 40936 The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)
 |-  (
 ( ( A  / 
 2 )  e.  NN0  /\  A  e.  NN0  /\  I  e.  NN0 )  ->  (
 ( I  +  1 ) (digit `  2
 ) A )  =  ( I (digit `  2 ) ( A 
 /  2 ) ) )
 
Theoremdignn0flhalf 40937 The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  I  e.  NN0 )  ->  ( ( I  +  1 ) (digit `  2 ) A )  =  ( I (digit `  2 ) ( |_ `  ( A  /  2
 ) ) ) )
 
Theoremnn0sumshdiglemA 40938* Lemma for nn0sumshdig 40942 (induction step, even multiplicant). (Contributed by AV, 3-Jun-2020.)
 |-  (
 ( ( a  e. 
 NN  /\  ( a  /  2 )  e. 
 NN )  /\  y  e.  NN )  ->  ( A. x  e.  NN0  (
 (#b `  x )  =  y  ->  x  = 
 sum_ k  e.  (
 0..^ y ) ( ( k (digit `  2 ) x )  x.  ( 2 ^
 k ) ) ) 
 ->  ( (#b `  a
 )  =  ( y  +  1 )  ->  a  =  sum_ k  e.  ( 0..^ ( y  +  1 ) ) ( ( k (digit `  2 ) a )  x.  ( 2 ^
 k ) ) ) ) )
 
Theoremnn0sumshdiglemB 40939* Lemma for nn0sumshdig 40942 (induction step, odd multiplicant). (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( ( a  e. 
 NN  /\  ( (
 a  -  1 ) 
 /  2 )  e. 
 NN0 )  /\  y  e.  NN )  ->  ( A. x  e.  NN0  (
 (#b `  x )  =  y  ->  x  = 
 sum_ k  e.  (
 0..^ y ) ( ( k (digit `  2 ) x )  x.  ( 2 ^
 k ) ) ) 
 ->  ( (#b `  a
 )  =  ( y  +  1 )  ->  a  =  sum_ k  e.  ( 0..^ ( y  +  1 ) ) ( ( k (digit `  2 ) a )  x.  ( 2 ^
 k ) ) ) ) )
 
Theoremnn0sumshdiglem1 40940* Lemma 1 for nn0sumshdig 40942 (induction step). (Contributed by AV, 7-Jun-2020.)
 |-  (
 y  e.  NN  ->  (
 A. a  e.  NN0  ( (#b `  a )  =  y  ->  a  = 
 sum_ k  e.  (
 0..^ y ) ( ( k (digit `  2 ) a )  x.  ( 2 ^
 k ) ) ) 
 ->  A. a  e.  NN0  ( (#b `  a )  =  ( y  +  1 )  ->  a  =  sum_
 k  e.  ( 0..^ ( y  +  1 ) ) ( ( k (digit `  2
 ) a )  x.  ( 2 ^ k
 ) ) ) ) )
 
Theoremnn0sumshdiglem2 40941* Lemma 2 for nn0sumshdig 40942. (Contributed by AV, 7-Jun-2020.)
 |-  ( L  e.  NN  ->  A. a  e.  NN0  (
 (#b `  a )  =  L  ->  a  = 
 sum_ k  e.  (
 0..^ L ) ( ( k (digit `  2 ) a )  x.  ( 2 ^
 k ) ) ) )
 
Theoremnn0sumshdig 40942* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
 |-  ( A  e.  NN0  ->  A  =  sum_ k  e.  (
 0..^ (#b `  A ) ) ( ( k (digit `  2
 ) A )  x.  ( 2 ^ k
 ) ) )
 
21.33.16.12  Algorithms for the multiplication of nonnegative integers
 
Theoremnn0mulfsum 40943* Trivial algorithm to calculate the product of two nonnegative integers  a and  b by adding up  b  a times. (Contributed by AV, 17-May-2020.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  x.  B )  =  sum_ k  e.  ( 1 ... A ) B )
 
Theoremnn0mullong 40944* Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers  a and  b by multiplying the multiplicand  b by each digit of the multiplier  a and then add up all the properly shifted results. Here, the binary representation of the multiplier  a is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 14546. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  x.  B )  =  sum_ k  e.  ( 0..^ (#b `  A ) ) ( ( ( k (digit `  2 ) A )  x.  ( 2 ^
 k ) )  x.  B ) )
 
21.34  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

 
21.34.1  Natural deduction
 
Theorem19.8ad 40945 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1955. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbidd 40946 An identity theorem for substitution. See sbid 2101. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
 |-  ( ph  ->  [ x  /  x ] ps )   =>    |-  ( ph  ->  ps )
 
Theoremsbidd-misc 40947 An identity theorem for substitution. See sbid 2101. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
 |-  (
 ( ph  ->  [ x  /  x ] ps )  <->  (
 ph  ->  ps ) )
 
21.34.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

 
Syntaxcge-real 40948 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 40950.
 class  >_
 
Syntaxcgt 40949 Extend wff notation to include the 'greater than' relation, see df-gt 40951.
 class  >
 
Definitiondf-gte 40950 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 9699.

We do not write this as  ( x  >_  y  <->  y  <_  x ), and similarly we do not write ` > ` as  ( x  >  y  <->  y  <  x ), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way:  |-  >  =  { <. x ,  y
>.  |  ( (
x  e.  RR*  /\  y  e.  RR* )  /\  y  <  x ) } and  |-  >_  =  { <. x ,  y
>.  |  ( (
x  e.  RR*  /\  y  e.  RR* )  /\  y  <_  x ) } but these are very complicated. This definition of  >_, and the similar one for  > (df-gt 40951), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 40952 for a more conventional expression of the relationship between  < and  >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

 |-  >_  =  `'  <_
 
Definitiondf-gt 40951 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 9570. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 40950 for a discussion on why this approach is used for the definition. See gt-lt 40953 and gt-lth 40955 for more conventional expression of the relationship between  < and  >.

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

 |-  >  =  `'  <
 
Theoremgte-lte 40952 Simple relationship between  <_ and  >_. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >_  B  <->  B 
 <_  A ) )
 
Theoremgt-lt 40953 Simple relationship between  < and  >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >  B  <->  B  <  A ) )
 
Theoremgte-lteh 40954 Relationship between  <_ and  >_ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >_  B  <->  B  <_  A )
 
Theoremgt-lth 40955 Relationship between  < and  > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >  B  <->  B  <  A )
 
Theoremex-gt 40956 Simple example of  >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  -.  0  >  0
 
Theoremex-gte 40957 Simple example of  >_, in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  0  >_  0
 
21.34.3  Hyperbolic trigonometric functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as  ( cos `  ( _i  x.  x ) ). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

 
Syntaxcsinh 40958 Extend class notation to include the hyperbolic sine function, see df-sinh 40961.
 class sinh
 
Syntaxccosh 40959 Extend class notation to include the hyperbolic cosine function. see df-cosh 40962.
 class cosh
 
Syntaxctanh 40960 Extend class notation to include the hyperbolic tangent function, see df-tanh 40963.
 class tanh
 
Definitiondf-sinh 40961 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4454, which requires the form  (
x  e.  A  |->  B ). See sinhval-named 40964 for a simple way to evaluate it. We define this function by dividing by  _i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 40967 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |- sinh  =  ( x  e.  CC  |->  ( ( sin `  ( _i  x.  x ) ) 
 /  _i ) )
 
Definitiondf-cosh 40962 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4454, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- cosh  =  ( x  e.  CC  |->  ( cos `  ( _i  x.  x ) ) )
 
Definitiondf-tanh 40963 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4454, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- tanh  =  ( x  e.  ( `'cosh " ( CC  \  { 0 } )
 )  |->  ( ( tan `  ( _i  x.  x ) )  /  _i ) )
 
Theoremsinhval-named 40964 Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 40961. See sinhval 14285 for a theorem to convert this further. See sinh-conventional 40967 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremcoshval-named 40965 Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 40962. See coshval 14286 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
 
Theoremtanhval-named 40966 Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 40963. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  ( `'cosh " ( CC  \  {
 0 } ) ) 
 ->  (tanh `  A )  =  ( ( tan `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremsinh-conventional 40967 Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  (
 -u _i  x.  ( sin `  ( _i  x.  A ) ) ) )
 
Theoremsinhpcosh 40968 Prove that  (sinh `  A
)  +  (cosh `  A )  =  ( exp `  A ) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.)
 |-  ( A  e.  CC  ->  ( (sinh `  A )  +  (cosh `  A )
 )  =  ( exp `  A ) )
 
21.34.4  Reciprocal trigonometric functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

 
Syntaxcsec 40969 Extend class notation to include the secant function, see df-sec 40972.
 class  sec
 
Syntaxccsc 40970 Extend class notation to include the cosecant function, see df-csc 40973.
 class  csc
 
Syntaxccot 40971 Extend class notation to include the cotangent function, see df-cot 40974.
 class  cot
 
Definitiondf-sec 40972* Define the secant function. We define it this way for cmpt 4454, which requires the form  ( x  e.  A  |->  B ). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  sec  =  ( x  e.  {
 y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1 
 /  ( cos `  x ) ) )
 
Definitiondf-csc 40973* Define the cosecant function. We define it this way for cmpt 4454, which requires the form  ( x  e.  A  |->  B ). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  csc  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1 
 /  ( sin `  x ) ) )
 
Definitiondf-cot 40974* Define the cotangent function. We define it this way for cmpt 4454, which requires the form  ( x  e.  A  |->  B ). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  cot  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
 
Theoremsecval 40975 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  =  ( 1  /  ( cos `  A ) ) )
 
Theoremcscval 40976 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  =  ( 1  /  ( sin `  A ) ) )
 
Theoremcotval 40977 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  =  ( ( cos `  A )  /  ( sin `  A ) ) )
 
Theoremseccl 40978 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  CC )
 
Theoremcsccl 40979 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  CC )
 
Theoremcotcl 40980 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  CC )
 
Theoremreseccl 40981 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  RR )
 
Theoremrecsccl 40982 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  RR )
 
Theoremrecotcl 40983 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  RR )
 
Theoremrecsec 40984 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( cos `  A )  =  ( 1  /  ( sec `  A ) ) )
 
Theoremreccsc 40985 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( sin `  A )  =  ( 1  /  ( csc `  A ) ) )
 
Theoremreccot 40986 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( 1  /  ( cot `  A ) ) )
 
Theoremrectan 40987 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( cot `  A )  =  ( 1  /  ( tan `  A ) ) )
 
Theoremsec0 40988 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 40989 Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( 1  +  (
 ( tan `  A ) ^ 2 ) )  =  ( ( sec `  A ) ^ 2
 ) )
 
Theoremcotsqcscsq 40990 Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( 1  +  (
 ( cot `  A ) ^ 2 ) )  =  ( ( csc `  A ) ^ 2
 ) )
 
21.34.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 40991 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3881 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
21.34.6  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 40995 and df-dp2 40994 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 11075.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 40992 Constant used for decimal fraction constructor. See df-dp2 40994.
 class _ A B
 
Syntaxcdp 40993 Decimal point operator. See df-dp 40995.
 class  period
 
Definitiondf-dp2 40994 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 11075. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 40995* Define the  period (decimal point) operator. For example,  ( 1 period 5 )  =  ( 3  /  2 ), and  -u (; 3 2 period_ 7_ 1 8 )  = 
-u (;;;; 3 2 7 1 8  / ;;; 1 0 0 0 ) Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is  RR, not  QQ; this should simplify some proofs. The LHS is  NN0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression  -u ( A period B ) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

 |-  period  =  ( x  e.  NN0 ,  y  e.  RR  |-> _ x y )
 
Theoremdp2cl 40996 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  -> _ A B  e.  RR )
 
Theoremdpval 40997 Define the value of the decimal point operator. See df-dp 40995. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 40998 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 40995. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 40999 Prove a simple equivalence involving the decimal point. See df-dp 40995 and dpcl 40998. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
21.34.7  Logarithms generalized to arbitrary base using ` logb `

Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base".

 
Theoremlogb2aval 41000 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb  <. B ,  X >. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
 |-  (
 ( B  e.  ( CC  \  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )
 )  ->  ( logb  `  <. B ,  X >. )  =  ( ( log `  X )  /  ( log `  B ) ) )
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