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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqne0 11101 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =/=  0  <->  A  =/=  0 ) )
 
Theoremresqcl 11102 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  ( A ^ 2
 )  e.  RR )
 
Theoremsqgt0 11103 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  0  <  ( A ^ 2 ) )
 
Theoremnnsqcl 11104 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 11105 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 11106 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A ^ 2
 )  e.  QQ )
 
Theoremsq11 11107 The square function is one-to-one for nonnegative reals. (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 11108 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 11109 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 11110 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 11111 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  0  <_  ( A ^ 2 ) )
 
Theoremzsqcl2 11112 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 11113 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 11114 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  =  ( A  x.  A )
 
Theoremsqcli 11115 Closure of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  e.  CC
 
Theoremsqeq0i 11116 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( A ^ 2 )  =  0  <->  A  =  0
 )
 
Theoremsqrecii 11117 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( ( 1  /  A ) ^ 2
 )  =  ( 1 
 /  ( A ^
 2 ) )
 
Theoremsqmuli 11118 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 ) )
 
Theoremsqdivi 11119 Distribution of square over division. (Contributed by NM, 20-Aug-2001.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   =>    |-  (
 ( A  /  B ) ^ 2 )  =  ( ( A ^
 2 )  /  ( B ^ 2 ) )
 
Theoremresqcli 11120 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( A ^
 2 )  e.  RR
 
Theoremsqgt0i 11121 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  0  <  ( A ^ 2 ) )
 
Theoremsqge0i 11122 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  0  <_  ( A ^ 2 )
 
Theoremlt2sqi 11123 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 11124 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 11125 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 11126 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
 |-  ( 0 ^ 2
 )  =  0
 
Theoremsq0i 11127 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
 |-  ( A  =  0 
 ->  ( A ^ 2
 )  =  0 )
 
Theoremsq0id 11128 If a number is zero, its square is zero. Deduction form of sq0i 11127. Converse of sqeq0d 11175. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( A ^ 2 )  =  0 )
 
Theoremsq1 11129 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
 |-  ( 1 ^ 2
 )  =  1
 
Theoremsq2 11130 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
 |-  ( 2 ^ 2
 )  =  4
 
Theoremsq3 11131 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
 |-  ( 3 ^ 2
 )  =  9
 
Theoremcu2 11132 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 2 ^ 3
 )  =  8
 
Theoremirec 11133 The reciprocal of  _i. (Contributed by NM, 11-Oct-1999.)
 |-  ( 1  /  _i )  =  -u _i
 
Theoremi2 11134  _i squared. (Contributed by NM, 6-May-1999.)
 |-  ( _i ^ 2
 )  =  -u 1
 
Theoremi3 11135  _i cubed. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 3
 )  =  -u _i
 
Theoremi4 11136  _i to the fourth power. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 4
 )  =  1
 
Theoremnnlesq 11137 A natural number 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 11138 Taking  _i to the  K-th power is the same as using the  K  mod  4 -th power instead, by i4 11136. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( K  e.  ZZ  ->  ( _i ^ ( K  mod  4 ) )  =  ( _i ^ K ) )
 
Theoremexpnass 11139 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
 |-  ( ( 3 ^
 3 ) ^ 3
 )  <  ( 3 ^ ( 3 ^
 3 ) )
 
Theoremsqlecan 11140 Cancel one factor of a square in a 
<_ comparison. Unlike lemul1 9541, the common factor  A may be zero. (Contributed by NM, 17-Jan-2008.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A ^
 2 )  <_  ( B  x.  A )  <->  A  <_  B ) )
 
Theoremsubsq 11141 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 11142 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 11143 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 ) )
 
Theorembinom2aiOLD 11144 Product of sum and difference. (Contributed by NM, 7-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  +  B )  x.  ( A  -  B ) )  =  ( ( A ^ 2 )  -  ( B ^ 2 ) )
 
Theoremsubsqi 11145 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 ) )
 
Theoremsqeqori 11146 The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by NM, 15-Jan-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A ^
 2 )  =  ( B ^ 2 )  <-> 
 ( A  =  B  \/  A  =  -u B ) )
 
Theoremsubsq0i 11147 The two solutions to the difference of squares set equal to zero. (Contributed by NM, 25-Apr-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( ( A ^ 2 )  -  ( B ^ 2 ) )  =  0  <->  ( A  =  B  \/  A  =  -u B ) )
 
Theoremsqeqor 11148 The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^ 2 )  =  ( B ^ 2
 ) 
 <->  ( A  =  B  \/  A  =  -u B ) ) )
 
Theorembinom2 11149 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 11150 Special case of binom2 11149 where  B  =  1. (Contributed by Scott Fenton, 11-May-2014.)
 |-  ( A  e.  CC  ->  ( ( A  +  1 ) ^ 2
 )  =  ( ( ( A ^ 2
 )  +  ( 2  x.  A ) )  +  1 ) )
 
Theorembinom2sub 11151 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 11152 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 ) )
 
Theorembinom3 11153 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 ) ) ) )
 
Theoremsq01 11154 If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =  A  <->  ( A  =  0  \/  A  =  1 ) ) )
 
Theoremzesq 11155 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 11156 A natural number 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 ) )
 
Theoremcrreczi 11157 Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Jeff Hankins, 16-Dec-2013.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( A  =/=  0  \/  B  =/=  0
 )  ->  ( 1  /  ( A  +  ( _i  x.  B ) ) )  =  ( ( A  -  ( _i 
 x.  B ) ) 
 /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) ) )
 
Theorembernneq 11158 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 11159 Variation of Bernoulli's inequality bernneq 11158. (Contributed by NM, 18-Oct-2007.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  ( ( ( A  -  1 )  x.  N )  +  1 )  <_  ( A ^ N ) )
 
Theorembernneq3 11160 A corollary of bernneq 11158. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( P  e.  ( ZZ>= `  2 )  /\  N  e.  NN0 )  ->  N  <  ( P ^ N ) )
 
Theoremexpnbnd 11161* 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 11162* 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 11163* 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 )
 
Theoremexpmulnbnd 11164* Exponentiation with a mantissa greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B ) 
 ->  E. j  e.  NN0  A. k  e.  ( ZZ>= `  j ) ( A  x.  k )  < 
 ( B ^ k
 ) )
 
Theoremdigit2 11165 Two ways to express the  K th digit in the decimal (when base  B  =  10) expansion of a number  A.  K  =  1 corresponds to the first digit after the decimal point. (Contributed by NM, 25-Dec-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( ( |_ `  (
 ( B ^ K )  x.  A ) ) 
 mod  B )  =  ( ( |_ `  (
 ( B ^ K )  x.  A ) )  -  ( B  x.  ( |_ `  ( ( B ^ ( K  -  1 ) )  x.  A ) ) ) ) )
 
Theoremdigit1 11166 Two ways to express the  K th digit in the decimal expansion of a number  A (when base  B  =  10). 
K  =  1 corresponds to the first digit after the decimal point. (Contributed by NM, 3-Jan-2009.)
 |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( ( |_ `  (
 ( B ^ K )  x.  A ) ) 
 mod  B )  =  ( ( ( |_ `  (
 ( B ^ K )  x.  A ) ) 
 mod  ( B ^ K ) )  -  ( ( B  x.  ( |_ `  ( ( B ^ ( K  -  1 ) )  x.  A ) ) )  mod  ( B ^ K ) ) ) )
 
Theoremmodexp 11167 Exponentiation property of the modulo operation. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D ) )  ->  ( ( A ^ C ) 
 mod  D )  =  ( ( B ^ C )  mod  D ) )
 
Theoremdiscr1 11168* A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  RR )  ->  0  <_  ( ( ( A  x.  ( x ^
 2 ) )  +  ( B  x.  x ) )  +  C ) )   &    |-  X  =  if ( 1  <_  (
 ( ( B  +  if ( 0  <_  C ,  C ,  0 ) )  +  1 ) 
 /  -u A ) ,  ( ( ( B  +  if ( 0 
 <_  C ,  C , 
 0 ) )  +  1 )  /  -u A ) ,  1 )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremdiscr 11169* If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is non-positive. (Contributed by NM, 10-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  RR )  ->  0  <_  ( ( ( A  x.  ( x ^
 2 ) )  +  ( B  x.  x ) )  +  C ) )   =>    |-  ( ph  ->  (
 ( B ^ 2
 )  -  ( 4  x.  ( A  x.  C ) ) ) 
 <_  0 )
 
Theoremexp0d 11170 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 11171 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 11172 Natural number 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 11173 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
 
Theoremsqcld 11174 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 CC )
 
Theoremsqeq0d 11175 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 11176 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 11177 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 11178 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 11179 Product of exponents law for natural number 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 ) )
 
Theoremsqrecd 11180 Square of reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( ( 1  /  A ) ^ 2 )  =  ( 1  /  ( A ^ 2 ) ) )
 
Theoremexpclzd 11181 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ N )  e. 
 CC )
 
Theoremexpne0d 11182 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ N )  =/=  0 )
 
Theoremexpnegd 11183 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexprecd 11184 Nonnegative integer exponentiation of a reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  (
 ( 1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpp1zd 11185 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpm1d 11186 Value of a complex number raised to an integer power minus one. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ ( N  -  1 ) )  =  ( ( A ^ N )  /  A ) )
 
Theoremexpsubd 11187 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ ( M  -  N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )
 
Theoremsqmuld 11188 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 ) ) )
 
Theoremsqdivd 11189 Distribution of square over division. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B ) ^ 2 )  =  ( ( A ^
 2 )  /  ( B ^ 2 ) ) )
 
Theoremexpdivd 11190 Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A  /  B ) ^ N )  =  ( ( A ^ N )  /  ( B ^ N ) ) )
 
Theoremmulexpd 11191 Natural number 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 11192 Value of zero raised to a natural number power. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  (
 0 ^ N )  =  0 )
 
Theoremreexpcld 11193 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 )
 
Theoremexpge0d 11194 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( A ^ N ) )
 
Theoremexpge1d 11195 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  1 
 <_  A )   =>    |-  ( ph  ->  1  <_  ( A ^ N ) )
 
Theoremnnsqcld 11196 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 NN )
 
Theoremnnexpcld 11197 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^ N )  e. 
 NN )
 
Theoremnn0expcld 11198 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^ N )  e. 
 NN0 )
 
Theoremrpexpcld 11199 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ N )  e.  RR+ )
 
Theoremltexp2rd 11200 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  <  1 )   =>    |-  ( ph  ->  ( M  <  N  <->  ( A ^ N )  <  ( A ^ M ) ) )
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