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Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremm1expeven 9901 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ (
 2  x.  N ) )  =  1 )
 
Theoremexpsubap 9902 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  -  N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )
 
Theoremexpp1zap 9903 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  +  1 )
 )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpm1ap 9904 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  -  1 ) )  =  ( ( A ^ N )  /  A ) )
 
Theoremexpdivap 9905 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  N  e.  NN0 )  ->  ( ( A 
 /  B ) ^ N )  =  (
 ( A ^ N )  /  ( B ^ N ) ) )
 
Theoremltexp2a 9906 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ M )  <  ( A ^ N ) )
 
Theoremleexp2a 9907 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( A  e.  RR  /\  1  <_  A  /\  N  e.  ( ZZ>= `  M ) )  ->  ( A ^ M ) 
 <_  ( A ^ N ) )
 
Theoremleexp2r 9908 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  NN0  /\  N  e.  ( ZZ>= `  M )
 )  /\  ( 0  <_  A  /\  A  <_  1 ) )  ->  ( A ^ N )  <_  ( A ^ M ) )
 
Theoremleexp1a 9909 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  N  e.  NN0 )  /\  ( 0 
 <_  A  /\  A  <_  B ) )  ->  ( A ^ N )  <_  ( B ^ N ) )
 
Theoremexple1 9910 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.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A  /\  A  <_  1 )  /\  N  e.  NN0 )  ->  ( A ^ N )  <_  1
 )
 
Theoremexpubnd 9911 An upper bound on  A ^ N when  2  <_  A. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  ( A ^ N )  <_  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
 
Theoremsqval 9912 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  =  ( A  x.  A ) )
 
Theoremsqneg 9913 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
 |-  ( A  e.  CC  ->  ( -u A ^ 2
 )  =  ( A ^ 2 ) )
 
Theoremsqsubswap 9914 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^
 2 )  =  ( ( B  -  A ) ^ 2 ) )
 
Theoremsqcl 9915 Closure of square. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  e.  CC )
 
Theoremsqmul 9916 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^
 2 )  =  ( ( A ^ 2
 )  x.  ( B ^ 2 ) ) )
 
Theoremsqeq0 9917 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =  0  <->  A  =  0 )
 )
 
Theoremsqdivap 9918 Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( ( A  /  B ) ^ 2
 )  =  ( ( A ^ 2 ) 
 /  ( B ^
 2 ) ) )
 
Theoremsqne0 9919 A number is nonzero iff its square is nonzero. See also sqap0 9920 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =/=  0  <->  A  =/=  0 ) )
 
Theoremsqap0 9920 A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 ) #  0  <->  A #  0 )
 )
 
Theoremresqcl 9921 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  ( A ^ 2
 )  e.  RR )
 
Theoremsqgt0ap 9922 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  0  <  ( A ^ 2 ) )
 
Theoremnnsqcl 9923 The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  e.  NN  ->  ( A ^ 2
 )  e.  NN )
 
Theoremzsqcl 9924 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  e.  ZZ  ->  ( A ^ 2
 )  e.  ZZ )
 
Theoremqsqcl 9925 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A ^ 2
 )  e.  QQ )
 
Theoremsq11 9926 The square function is one-to-one for nonnegative reals. Also see sq11ap 10017 which would easily follow from this given excluded middle, but which for us is proved another way. (Contributed by NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A ^
 2 )  =  ( B ^ 2 )  <->  A  =  B )
 )
 
Theoremlt2sq 9927 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( A  <  B  <->  ( A ^
 2 )  <  ( B ^ 2 ) ) )
 
Theoremle2sq 9928 The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( A  <_  B  <->  ( A ^
 2 )  <_  ( B ^ 2 ) ) )
 
Theoremle2sq2 9929 The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  A  <_  B ) )  ->  ( A ^ 2 ) 
 <_  ( B ^ 2
 ) )
 
Theoremsqge0 9930 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  0  <_  ( A ^ 2 ) )
 
Theoremzsqcl2 9931 The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ  ->  ( A ^ 2
 )  e.  NN0 )
 
Theoremsumsqeq0 9932 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  0 ) )
 
Theoremsqvali 9933 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  =  ( A  x.  A )
 
Theoremsqcli 9934 Closure of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  e.  CC
 
Theoremsqeq0i 9935 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( A ^ 2 )  =  0  <->  A  =  0
 )
 
Theoremsqmuli 9936 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  x.  B ) ^ 2
 )  =  ( ( A ^ 2 )  x.  ( B ^
 2 ) )
 
Theoremsqdivapi 9937 Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( ( A 
 /  B ) ^
 2 )  =  ( ( A ^ 2
 )  /  ( B ^ 2 ) )
 
Theoremresqcli 9938 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( A ^
 2 )  e.  RR
 
Theoremsqgt0api 9939 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  0  <  ( A ^ 2 ) )
 
Theoremsqge0i 9940 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  0  <_  ( A ^ 2 )
 
Theoremlt2sqi 9941 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <  B  <-> 
 ( A ^ 2
 )  <  ( B ^ 2 ) ) )
 
Theoremle2sqi 9942 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <_  B  <-> 
 ( A ^ 2
 )  <_  ( B ^ 2 ) ) )
 
Theoremsq11i 9943 The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( ( A ^ 2 )  =  ( B ^ 2
 ) 
 <->  A  =  B ) )
 
Theoremsq0 9944 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
 |-  ( 0 ^ 2
 )  =  0
 
Theoremsq0i 9945 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
 |-  ( A  =  0 
 ->  ( A ^ 2
 )  =  0 )
 
Theoremsq0id 9946 If a number is zero, its square is zero. Deduction form of sq0i 9945. Converse of sqeq0d 9982. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( A ^ 2 )  =  0 )
 
Theoremsq1 9947 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
 |-  ( 1 ^ 2
 )  =  1
 
Theoremneg1sqe1 9948  -u 1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( -u 1 ^ 2
 )  =  1
 
Theoremsq2 9949 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
 |-  ( 2 ^ 2
 )  =  4
 
Theoremsq3 9950 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
 |-  ( 3 ^ 2
 )  =  9
 
Theoremcu2 9951 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 2 ^ 3
 )  =  8
 
Theoremirec 9952 The reciprocal of  _i. (Contributed by NM, 11-Oct-1999.)
 |-  ( 1  /  _i )  =  -u _i
 
Theoremi2 9953  _i squared. (Contributed by NM, 6-May-1999.)
 |-  ( _i ^ 2
 )  =  -u 1
 
Theoremi3 9954  _i cubed. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 3
 )  =  -u _i
 
Theoremi4 9955  _i to the fourth power. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 4
 )  =  1
 
Theoremnnlesq 9956 A positive integer is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  ( N  e.  NN  ->  N  <_  ( N ^ 2 ) )
 
Theoremiexpcyc 9957 Taking  _i to the  K-th power is the same as using the  K  mod  4 -th power instead, by i4 9955. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( K  e.  ZZ  ->  ( _i ^ ( K  mod  4 ) )  =  ( _i ^ K ) )
 
Theoremexpnass 9958 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
 |-  ( ( 3 ^
 3 ) ^ 3
 )  <  ( 3 ^ ( 3 ^
 3 ) )
 
Theoremsubsq 9959 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B ) ) )
 
Theoremsubsq2 9960 Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  ( ( ( A  -  B ) ^ 2 )  +  ( ( 2  x.  B )  x.  ( A  -  B ) ) ) )
 
Theorembinom2i 9961 The square of a binomial. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  +  B ) ^ 2
 )  =  ( ( ( A ^ 2
 )  +  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
 2 ) )
 
Theoremsubsqi 9962 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A ^
 2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B ) )
 
Theorembinom2 9963 The square of a binomial. (Contributed by FL, 10-Dec-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^
 2 )  =  ( ( ( A ^
 2 )  +  (
 2  x.  ( A  x.  B ) ) )  +  ( B ^ 2 ) ) )
 
Theorembinom21 9964 Special case of binom2 9963 where  B  =  1. (Contributed by Scott Fenton, 11-May-2014.)
 |-  ( A  e.  CC  ->  ( ( A  +  1 ) ^ 2
 )  =  ( ( ( A ^ 2
 )  +  ( 2  x.  A ) )  +  1 ) )
 
Theorembinom2sub 9965 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^
 2 )  =  ( ( ( A ^
 2 )  -  (
 2  x.  ( A  x.  B ) ) )  +  ( B ^ 2 ) ) )
 
Theorembinom2subi 9966 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  -  B ) ^ 2
 )  =  ( ( ( A ^ 2
 )  -  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
 2 ) )
 
Theoremmulbinom2 9967 The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( C  x.  A )  +  B ) ^ 2
 )  =  ( ( ( ( C  x.  A ) ^ 2
 )  +  ( ( 2  x.  C )  x.  ( A  x.  B ) ) )  +  ( B ^
 2 ) ) )
 
Theorembinom3 9968 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^
 3 )  =  ( ( ( A ^
 3 )  +  (
 3  x.  ( ( A ^ 2 )  x.  B ) ) )  +  ( ( 3  x.  ( A  x.  ( B ^
 2 ) ) )  +  ( B ^
 3 ) ) ) )
 
Theoremzesq 9969 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  ( N  e.  ZZ  ->  ( ( N  / 
 2 )  e.  ZZ  <->  (
 ( N ^ 2
 )  /  2 )  e.  ZZ ) )
 
Theoremnnesq 9970 A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  ( N  e.  NN  ->  ( ( N  / 
 2 )  e.  NN  <->  (
 ( N ^ 2
 )  /  2 )  e.  NN ) )
 
Theorembernneq 9971 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  -u 1  <_  A ) 
 ->  ( 1  +  ( A  x.  N ) ) 
 <_  ( ( 1  +  A ) ^ N ) )
 
Theorembernneq2 9972 Variation of Bernoulli's inequality bernneq 9971. (Contributed by NM, 18-Oct-2007.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  ( ( ( A  -  1 )  x.  N )  +  1 )  <_  ( A ^ N ) )
 
Theorembernneq3 9973 A corollary of bernneq 9971. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( P  e.  ( ZZ>= `  2 )  /\  N  e.  NN0 )  ->  N  <  ( P ^ N ) )
 
Theoremexpnbnd 9974* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B ) 
 ->  E. k  e.  NN  A  <  ( B ^
 k ) )
 
Theoremexpnlbnd 9975* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  ( 1  /  ( B ^ k ) )  <  A )
 
Theoremexpnlbnd2 9976* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( 1 
 /  ( B ^
 k ) )  <  A )
 
Theoremexp0d 9977 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 0 )  =  1 )
 
Theoremexp1d 9978 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 1 )  =  A )
 
Theoremexpeq0d 9979 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( A ^ N )  =  0
 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremsqvald 9980 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
 
Theoremsqcld 9981 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 CC )
 
Theoremsqeq0d 9982 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( A ^ 2 )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremexpcld 9983 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^ N )  e. 
 CC )
 
Theoremexpp1d 9984 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 Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpaddd 9985 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  NN0 )   =>    |-  ( ph  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpmuld 9986 Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  NN0 )   =>    |-  ( ph  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N ) )
 
Theoremsqrecapd 9987 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 ( 1  /  A ) ^ 2 )  =  ( 1  /  ( A ^ 2 ) ) )
 
Theoremexpclzapd 9988 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ N )  e.  CC )
 
Theoremexpap0d 9989 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ N ) #  0 )
 
Theoremexpnegapd 9990 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexprecapd 9991 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( ( 1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpp1zapd 9992 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpm1apd 9993 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ ( N  -  1 ) )  =  ( ( A ^ N )  /  A ) )
 
Theoremexpsubapd 9994 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ ( M  -  N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )
 
Theoremsqmuld 9995 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  x.  B ) ^ 2 )  =  ( ( A ^
 2 )  x.  ( B ^ 2 ) ) )
 
Theoremsqdivapd 9996 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A 
 /  B ) ^
 2 )  =  ( ( A ^ 2
 )  /  ( B ^ 2 ) ) )
 
Theoremexpdivapd 9997 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A  /  B ) ^ N )  =  ( ( A ^ N )  /  ( B ^ N ) ) )
 
Theoremmulexpd 9998 Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theorem0expd 9999 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  (
 0 ^ N )  =  0 )
 
Theoremreexpcld 10000 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^ N )  e. 
 RR )
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