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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiseqfn 8901* The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
 M  ZZ   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S  Fn  ZZ>=
 `  M
 
Theoremiseq1 8902* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
 M  ZZ   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  M  F `  M
 
Theoremiseqcl 8903* Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
 N  ZZ>= `  M   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  N  S
 
Theoremiseqp1 8904* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
 N  ZZ>= `  M   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  N  +  1  seq M 
 .+  ,  F ,  S `  N 
 .+  F `  N  +  1
 
Theoremiseqfveq2 8905* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 K  ZZ>= `  M   &     seq M  .+  ,  F ,  S `  K  G `  K   &     S  V   &     ZZ>= `  M  F `  S   &     ZZ>= `  K  G `  S   &     S  S  .+  S   &     N  ZZ>= `  K   &     k  K  +  1 ... N  F `
  k  G `  k   =>     seq M  .+  ,  F ,  S `  N 
 seq K  .+  ,  G ,  S
 `  N
 
Theoremiseqfeq2 8906* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 K  ZZ>= `  M   &     seq M  .+  ,  F ,  S `  K  G `  K   &     S  V   &     ZZ>= `  M  F `  S   &     ZZ>= `  K  G `  S   &     S  S  .+  S   &     k  ZZ>= `  K  +  1  F `
  k  G `  k   =>     seq M  .+  ,  F ,  S  |`  ZZ>= `  K  seq K 
 .+  ,  G ,  S
 
Theoremiseqfveq 8907* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
 N  ZZ>= `  M   &     k  M ... N  F `
  k  G `  k   &     S  V   &     ZZ>= `  M  F `  S   &     ZZ>= `  M  G `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  N 
 seq M  .+  ,  G ,  S
 `  N
 
3.5.9  Integer powers
 
Syntaxcexp 8908 Extend class notation to include exponentiation of a complex number to an integer power.
 ^
 
Definitiondf-iexp 8909* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8913 and expp1 8916 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that  0 ^ 0  1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case  0 ,  <  0 gives the value  1  0, so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)

 ^  CC ,  ZZ  |->  if  0 ,  1 ,  if 0  <  ,  seq 1  x.  ,  NN  X.  { } ,  CC `  , 
 1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
 
Theoremexpivallem 8910 Lemma for expival 8911. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
 CC #  0  N  NN  seq 1  x.  ,  NN 
 X.  { } ,  CC `  N #  0
 
Theoremexpival 8911 Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
 CC  N  ZZ #  0  0  <_  N  ^ N  if N  0 ,  1 ,  if 0  <  N ,  seq 1  x.  ,  NN  X.  { } ,  CC `  N ,  1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u N
 
Theoremexpinnval 8912 Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC  N  NN  ^ N  seq 1  x.  ,  NN  X.  { } ,  CC `  N
 
Theoremexp0 8913 Value of a complex number raised to the 0th power. Note that under our definition,  0 ^ 0  1, following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
 CC  ^ 0  1
 
Theorem0exp0e1 8914  0 ^
0  1 (common case). This is our convention. It follows the convention used by Gleason; see Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.)
 0 ^ 0  1
 
Theoremexp1 8915 Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
 CC  ^ 1
 
Theoremexpp1 8916 Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
 CC  N  NN0  ^ N  +  1  ^ N  x.
 
Theoremexpnegap0 8917 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC #  0  N  NN0  ^ -u N  1  ^ N
 
Theoremexpineg2 8918 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC #  0  N  CC  -u N  NN0  ^ N  1  ^ -u N
 
Theoremexpn1ap0 8919 A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC #  0  ^ -u 1  1
 
Theoremexpcllem 8920* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
 F  C_  CC   &     F  F  x.  F   &     1  F   =>     F  NN0  ^  F
 
Theoremexpcl2lemap 8921* Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
 F  C_  CC   &     F  F  x.  F   &     1  F   &     F #  0  1  F   =>     F #  0  ZZ  ^  F
 
Theoremnnexpcl 8922 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
 NN  N  NN0  ^ N  NN
 
Theoremnn0expcl 8923 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
 NN0  N  NN0  ^ N  NN0
 
Theoremzexpcl 8924 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
 ZZ  N  NN0  ^ N  ZZ
 
Theoremqexpcl 8925 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
 QQ  N  NN0  ^ N  QQ
 
Theoremreexpcl 8926 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
 RR  N  NN0  ^ N  RR
 
Theoremexpcl 8927 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
 CC  N  NN0  ^ N  CC
 
Theoremrpexpcl 8928 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
 RR+  N  ZZ  ^ N  RR+
 
Theoremreexpclzap 8929 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
 RR #  0  N  ZZ  ^ N  RR
 
Theoremqexpclz 8930 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
 QQ  =/=  0  N  ZZ  ^ N  QQ
 
Theoremm1expcl2 8931 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 N  ZZ  -u 1 ^ N  { -u 1 ,  1 }
 
Theoremm1expcl 8932 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 N  ZZ  -u 1 ^ N  ZZ
 
Theoremexpclzaplem 8933* Closure law for integer exponentiation. Lemma for expclzap 8934 and expap0i 8941. (Contributed by Jim Kingdon, 9-Jun-2020.)
 CC #  0  N  ZZ  ^ N  { 
 CC  | #  0 }
 
Theoremexpclzap 8934 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
 CC #  0  N  ZZ  ^ N  CC
 
Theoremnn0expcli 8935 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     N  NN0   =>     ^ N  NN0
 
Theoremnn0sqcl 8936 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 NN0  ^ 2  NN0
 
Theoremexpm1t 8937 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
 CC  N  NN  ^ N  ^ N  -  1  x.
 
Theorem1exp 8938 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 N  ZZ  1 ^ N  1
 
Theoremexpap0 8939 Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 8940 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC  N  NN 
 ^ N #  0 #  0
 
Theoremexpeq0 8940 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
 CC  N  NN 
 ^ N  0  0
 
Theoremexpap0i 8941 Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC #  0  N  ZZ  ^ N #  0
 
Theoremexpgt0 8942 Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 RR  N  ZZ  0  <  0  < 
 ^ N
 
Theoremexpnegzap 8943 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
 CC #  0  N  ZZ  ^ -u N  1  ^ N
 
Theorem0exp 8944 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
 N  NN  0 ^ N  0
 
Theoremexpge0 8945 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 RR  N  NN0  0  <_  0  <_  ^ N
 
Theoremexpge1 8946 Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 RR  N  NN0  1  <_  1  <_  ^ N
 
Theoremexpgt1 8947 Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 RR  N  NN  1  <  1  < 
 ^ N
 
Theoremmulexp 8948 Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
 CC  CC  N  NN0  x.  ^ N 
 ^ N  x.  ^ N
 
Theoremmulexpzap 8949 Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC #  0  CC #  0  N  ZZ  x. 
 ^ N  ^ N  x.  ^ N
 
Theoremexprecap 8950 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC #  0  N  ZZ  1  ^ N  1  ^ N
 
Theoremexpadd 8951 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
 CC  M  NN0  N  NN0  ^ M  +  N  ^ M  x.  ^ N
 
Theoremexpaddzaplem 8952 Lemma for expaddzap 8953. (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC #  0  M  RR  -u M  NN  N  NN0  ^ M  +  N  ^ M  x.  ^ N
 
Theoremexpaddzap 8953 Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC #  0  M  ZZ  N  ZZ  ^ M  +  N 
 ^ M  x.  ^ N
 
Theoremexpmul 8954 Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
 CC  M  NN0  N  NN0  ^ M  x.  N 
 ^ M ^ N
 
Theoremexpmulzap 8955 Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
 CC #  0  M  ZZ  N  ZZ  ^ M  x.  N 
 ^ M ^ N
 
Theoremexpsubap 8956 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
 CC #  0  M  ZZ  N  ZZ  ^ M  -  N 
 ^ M  ^ N
 
Theoremexpp1zap 8957 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
 CC #  0  N  ZZ  ^ N  +  1  ^ N  x.
 
Theoremexpm1ap 8958 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
 CC #  0  N  ZZ  ^ N  -  1 
 ^ N
 
Theoremexpdivap 8959 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
 CC  CC #  0  N  NN0  ^ N 
 ^ N  ^ N
 
Theoremltexp2a 8960 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 RR  M  ZZ  N  ZZ  1  <  M  <  N  ^ M 
 <  ^ N
 
Theoremleexp2a 8961 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
 RR  1  <_  N  ZZ>= `  M  ^ M  <_  ^ N
 
Theoremleexp2r 8962 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 RR  M  NN0  N  ZZ>= `  M  0  <_  <_  1 
 ^ N  <_  ^ M
 
Theoremleexp1a 8963 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
 RR  RR  N  NN0  0 
 <_  <_  ^ N  <_  ^ N
 
Theoremexple1 8964 Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
 RR  0  <_  <_  1  N  NN0 
 ^ N  <_ 
 1
 
Theoremexpubnd 8965 An upper bound on  ^ N when  2  <_ . (Contributed by NM, 19-Dec-2005.)
 RR  N  NN0  2  <_  ^ N  <_  2 ^ N  x.  -  1 ^ N
 
Theoremsqval 8966 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
 CC  ^ 2  x.
 
Theoremsqneg 8967 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
 CC  -u ^ 2  ^ 2
 
Theoremsqsubswap 8968 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
 CC  CC 
 -  ^
 2  -  ^ 2
 
Theoremsqcl 8969 Closure of square. (Contributed by NM, 10-Aug-1999.)
 CC  ^ 2  CC
 
Theoremsqmul 8970 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
 CC  CC  x.  ^
 2  ^ 2  x.  ^
 2
 
Theoremsqeq0 8971 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
 CC  ^
 2  0  0
 
Theoremsqdivap 8972 Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
 CC  CC #  0 
 ^ 2  ^
 2  ^ 2
 
Theoremsqne0 8973 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
 CC  ^
 2  =/=  0  =/=  0
 
Theoremresqcl 8974 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
 RR  ^ 2  RR
 
Theoremsqgt0ap 8975 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
 RR #  0  0  < 
 ^ 2
 
Theoremnnsqcl 8976 The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 NN  ^ 2  NN
 
Theoremzsqcl 8977 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 ZZ  ^ 2  ZZ
 
Theoremqsqcl 8978 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 QQ  ^ 2  QQ
 
Theoremsq11 8979 The square function is one-to-one for nonnegative reals. (Contributed by NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.)
 RR  0  <_  RR  0  <_  ^ 2  ^ 2
 
Theoremlt2sq 8980 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
 RR  0  <_  RR  0  <_  < 
 ^ 2  <  ^ 2
 
Theoremle2sq 8981 The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
 RR  0  <_  RR  0  <_  <_ 
 ^ 2  <_  ^ 2
 
Theoremle2sq2 8982 The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
 RR  0  <_  RR  <_  ^ 2 
 <_  ^ 2
 
Theoremsqge0 8983 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
 RR  0  <_  ^ 2
 
Theoremzsqcl2 8984 The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)
 ZZ  ^ 2  NN0
 
Theoremsumsqeq0 8985 Two real numbers are equal to 0 iff their Euclidean norm is. (Contributed by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 28-May-2016.)
 RR  RR  0  0  ^ 2  +  ^
 2  0
 
Theoremsqvali 8986 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
 CC   =>     ^ 2  x.
 
Theoremsqcli 8987 Closure of square. (Contributed by NM, 2-Aug-1999.)
 CC   =>     ^ 2  CC
 
Theoremsqeq0i 8988 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
 CC   =>     ^
 2  0  0
 
Theoremsqmuli 8989 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
 CC   &     CC   =>     x.  ^ 2 
 ^ 2  x.  ^ 2
 
Theoremsqdivapi 8990 Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
 CC   &     CC   &    #  0   =>     ^ 2 
 ^ 2  ^ 2
 
Theoremresqcli 8991 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
 RR   =>     ^ 2  RR
 
Theoremsqgt0api 8992 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
 RR   =>    #  0  0  < 
 ^ 2
 
Theoremsqge0i 8993 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
 RR   =>     0  <_  ^ 2
 
Theoremlt2sqi 8994 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
 RR   &     RR   =>     0  <_  0  <_  <  ^ 2 
 <  ^ 2
 
Theoremle2sqi 8995 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
 RR   &     RR   =>     0  <_  0  <_  <_  ^ 2 
 <_  ^ 2
 
Theoremsq11i 8996 The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
 RR   &     RR   =>     0  <_  0  <_ 
 ^ 2  ^ 2
 
Theoremsq0 8997 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
 0 ^ 2  0
 
Theoremsq0i 8998 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
 0  ^ 2  0
 
Theoremsq0id 8999 If a number is zero, its square is zero. Deduction form of sq0i 8998. Converse of sqeq0d 9033. (Contributed by David Moews, 28-Feb-2017.)
 0   =>     ^ 2  0
 
Theoremsq1 9000 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
 1 ^ 2  1
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