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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpne0i 11101 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0 )
 
Theoremexpgt0 11102 Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  0  <  A ) 
 ->  0  <  ( A ^ N ) )
 
Theoremexpnegz 11103 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
 
Theorem0exp 11104 Value of zero raised to a natural number power. (Contributed by NM, 19-Aug-2004.)
 |-  ( N  e.  NN  ->  ( 0 ^ N )  =  0 )
 
Theoremexpge0 11105 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  0  <_  ( A ^ N ) )
 
Theoremexpge1 11106 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.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  1  <_  A )  ->  1  <_  ( A ^ N ) )
 
Theoremexpgt1 11107 Natural number exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A ) 
 ->  1  <  ( A ^ N ) )
 
Theoremmulexp 11108 Natural number exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theoremmulexpz 11109 Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  ZZ )  ->  (
 ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theoremexprec 11110 Nonnegative integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( ( 1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpadd 11111 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
 |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N )
 )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpaddzlem 11112 Lemma for expaddz 11113. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpaddz 11113 Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpmul 11114 Product of exponents law for natural number exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
 |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N ) )
 
Theoremexpmulz 11115 Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N ) )
 
Theoremexpsub 11116 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  -  N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )
 
Theoremexpp1z 11117 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  +  1 )
 )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpm1 11118 Value of a complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  -  1 ) )  =  ( ( A ^ N )  /  A ) )
 
Theoremexpdiv 11119 Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  N  e.  NN0 )  ->  ( ( A  /  B ) ^ N )  =  (
 ( A ^ N )  /  ( B ^ N ) ) )
 
Theoremltexp2a 11120 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 ) )
 
Theoremexpcan 11121 Cancellation law 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 )  ->  (
 ( A ^ M )  =  ( A ^ N )  <->  M  =  N ) )
 
Theoremltexp2 11122 Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) )
 
Theoremleexp2 11123 Ordering law for exponentiation. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  <_  N  <->  ( A ^ M )  <_  ( A ^ N ) ) )
 
Theoremleexp2a 11124 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 ) )
 
Theoremltexp2r 11125 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
 )  ->  ( M  <  N  <->  ( A ^ N )  <  ( A ^ M ) ) )
 
Theoremleexp2r 11126 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 11127 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 11128 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 11129 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 11130 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  =  ( A  x.  A ) )
 
Theoremsqneg 11131 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
 |-  ( A  e.  CC  ->  ( -u A ^ 2
 )  =  ( A ^ 2 ) )
 
Theoremsqsubswap 11132 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 11133 Closure of square. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  e.  CC )
 
Theoremsqmul 11134 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 11135 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =  0  <->  A  =  0 )
 )
 
Theoremsqdiv 11136 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( A  /  B ) ^ 2
 )  =  ( ( A ^ 2 ) 
 /  ( B ^
 2 ) ) )
 
Theoremsqne0 11137 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =/=  0  <->  A  =/=  0 ) )
 
Theoremresqcl 11138 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  ( A ^ 2
 )  e.  RR )
 
Theoremsqgt0 11139 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  0  <  ( A ^ 2 ) )
 
Theoremnnsqcl 11140 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 11141 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 11142 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A ^ 2
 )  e.  QQ )
 
Theoremsq11 11143 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 11144 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 11145 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 11146 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 11147 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  0  <_  ( A ^ 2 ) )
 
Theoremzsqcl2 11148 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 11149 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 11150 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  =  ( A  x.  A )
 
Theoremsqcli 11151 Closure of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  e.  CC
 
Theoremsqeq0i 11152 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( A ^ 2 )  =  0  <->  A  =  0
 )
 
Theoremsqrecii 11153 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( ( 1  /  A ) ^ 2
 )  =  ( 1 
 /  ( A ^
 2 ) )
 
Theoremsqmuli 11154 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 11155 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 11156 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( A ^
 2 )  e.  RR
 
Theoremsqgt0i 11157 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  0  <  ( A ^ 2 ) )
 
Theoremsqge0i 11158 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  0  <_  ( A ^ 2 )
 
Theoremlt2sqi 11159 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 11160 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 11161 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 11162 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
 |-  ( 0 ^ 2
 )  =  0
 
Theoremsq0i 11163 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
 |-  ( A  =  0 
 ->  ( A ^ 2
 )  =  0 )
 
Theoremsq0id 11164 If a number is zero, its square is zero. Deduction form of sq0i 11163. Converse of sqeq0d 11211. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( A ^ 2 )  =  0 )
 
Theoremsq1 11165 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
 |-  ( 1 ^ 2
 )  =  1
 
Theoremsq2 11166 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
 |-  ( 2 ^ 2
 )  =  4
 
Theoremsq3 11167 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
 |-  ( 3 ^ 2
 )  =  9
 
Theoremcu2 11168 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 2 ^ 3
 )  =  8
 
Theoremirec 11169 The reciprocal of  _i. (Contributed by NM, 11-Oct-1999.)
 |-  ( 1  /  _i )  =  -u _i
 
Theoremi2 11170  _i squared. (Contributed by NM, 6-May-1999.)
 |-  ( _i ^ 2
 )  =  -u 1
 
Theoremi3 11171  _i cubed. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 3
 )  =  -u _i
 
Theoremi4 11172  _i to the fourth power. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 4
 )  =  1
 
Theoremnnlesq 11173 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 11174 Taking  _i to the  K-th power is the same as using the  K  mod  4 -th power instead, by i4 11172. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( K  e.  ZZ  ->  ( _i ^ ( K  mod  4 ) )  =  ( _i ^ K ) )
 
Theoremexpnass 11175 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 11176 Cancel one factor of a square in a 
<_ comparison. Unlike lemul1 9576, 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 11177 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 11178 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 11179 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 11180 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 11181 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 11182 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 11183 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 11184 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 11185 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 11186 Special case of binom2 11185 where  B  =  1. (Contributed by Scott Fenton, 11-May-2014.)
 |-  ( A  e.  CC  ->  ( ( A  +  1 ) ^ 2
 )  =  ( ( ( A ^ 2
 )  +  ( 2  x.  A ) )  +  1 ) )
 
Theorembinom2sub 11187 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 11188 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 11189 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 11190 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 11191 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 11192 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 11193 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 11194 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 11195 Variation of Bernoulli's inequality bernneq 11194. (Contributed by NM, 18-Oct-2007.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  ( ( ( A  -  1 )  x.  N )  +  1 )  <_  ( A ^ N ) )
 
Theorembernneq3 11196 A corollary of bernneq 11194. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( P  e.  ( ZZ>= `  2 )  /\  N  e.  NN0 )  ->  N  <  ( P ^ N ) )
 
Theoremexpnbnd 11197* 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 11198* 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 11199* 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 11200* 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
 ) )
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