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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremserle 11101* Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  F ) `  N )  <_  (  seq  M (  +  ,  G ) `  N ) )
 
Theoremser1const 11102 Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1
 (  +  ,  ( NN  X.  { A }
 ) ) `  N )  =  ( N  x.  A ) )
 
Theoremseqof 11103* Distribute function operation through a sequence. Note that  G
( z ) is an implicit function on  z. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  x  e.  ( M
 ... N ) ) 
 ->  ( F `  x )  =  ( z  e.  A  |->  ( G `  x ) ) )   =>    |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N )  =  ( z  e.  A  |->  (  seq  M (  .+  ,  G ) `
  N ) ) )
 
5.6.4  Integer powers
 
Syntaxcexp 11104 Extend class notation to include exponentiation of a complex number to an integer power.
 class  ^
 
Definitiondf-exp 11105* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 11108 and expp1 11110 provide a 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  x  =  0 ,  y  <  0 gives the value  ( 1  /  0 ), so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)
 |- 
 ^  =  ( x  e.  CC ,  y  e.  ZZ  |->  if ( y  =  0 ,  1 ,  if ( 0  < 
 y ,  (  seq  1 (  x.  ,  ( NN  X.  { x }
 ) ) `  y
 ) ,  ( 1 
 /  (  seq  1
 (  x.  ,  ( NN  X.  { x }
 ) ) `  -u y
 ) ) ) ) )
 
Theoremexpval 11106 Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A ^ N )  =  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq  1 (  x.  ,  ( NN 
 X.  { A } )
 ) `  N ) ,  ( 1  /  (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) ) ) )
 
Theoremexpnnval 11107 Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N )  =  (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `  N ) )
 
Theoremexp0 11108 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.)
 |-  ( A  e.  CC  ->  ( A ^ 0
 )  =  1 )
 
Theoremexp1 11109 Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
 |-  ( A  e.  CC  ->  ( A ^ 1
 )  =  A )
 
Theoremexpp1 11110 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.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^
 ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpneg 11111 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpneg2 11112 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
 
Theoremexpn1 11113 A number to the negative one power is the reciprocal. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( A  e.  CC  ->  ( A ^ -u 1
 )  =  ( 1 
 /  A ) )
 
Theoremexpcllem 11114* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
 |-  F  C_  CC   &    |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y )  e.  F )   &    |-  1  e.  F   =>    |-  (
 ( A  e.  F  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  F )
 
Theoremexpcl2lem 11115* Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
 |-  F  C_  CC   &    |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y )  e.  F )   &    |-  1  e.  F   &    |-  (
 ( x  e.  F  /\  x  =/=  0
 )  ->  ( 1  /  x )  e.  F )   =>    |-  ( ( A  e.  F  /\  A  =/=  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
 
Theoremnnexpcl 11116 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN )
 
Theoremnn0expcl 11117 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN0 )
 
Theoremzexpcl 11118 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  ZZ )
 
Theoremqexpcl 11119 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  QQ )
 
Theoremreexpcl 11120 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  RR )
 
Theoremexpcl 11121 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
 
Theoremrpexpcl 11122 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR+ )
 
Theoremreexpclz 11123 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( A  e.  RR  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR )
 
Theoremqexpclz 11124 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  QQ )
 
Theoremm1expcl2 11125 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
 
Theoremm1expcl 11126 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  ZZ )
 
Theoremexpclzlem 11127 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  ( CC  \  { 0 } )
 )
 
Theoremexpclz 11128 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
 
Theoremnn0expcli 11129 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( A ^ N )  e.  NN0
 
Theoremexpm1t 11130 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N )  =  (
 ( A ^ ( N  -  1 ) )  x.  A ) )
 
Theorem1exp 11131 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( N  e.  ZZ  ->  ( 1 ^ N )  =  1 )
 
Theoremexpeq0 11132 Natural number exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <->  A  =  0
 ) )
 
Theoremexpne0 11133 Natural number exponentiation is nonzero iff its mantissa is nonzero. (Contributed by NM, 6-May-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =/=  0  <->  A  =/=  0
 ) )
 
Theoremexpne0i 11134 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 11135 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 11136 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 11137 Value of zero raised to a natural number power. (Contributed by NM, 19-Aug-2004.)
 |-  ( N  e.  NN  ->  ( 0 ^ N )  =  0 )
 
Theoremexpge0 11138 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 11139 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 11140 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 11141 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 11142 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 11143 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 11144 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 11145 Lemma for expaddz 11146. (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 11146 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 11147 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 11148 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 11149 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 11150 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 11151 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 11152 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 11153 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 11154 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 11155 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 11156 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 11157 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 11158 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 11159 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 11160 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 11161 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 11162 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 11163 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  =  ( A  x.  A ) )
 
Theoremsqneg 11164 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
 |-  ( A  e.  CC  ->  ( -u A ^ 2
 )  =  ( A ^ 2 ) )
 
Theoremsqsubswap 11165 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 11166 Closure of square. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  e.  CC )
 
Theoremsqmul 11167 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 11168 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =  0  <->  A  =  0 )
 )
 
Theoremsqdiv 11169 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 11170 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =/=  0  <->  A  =/=  0 ) )
 
Theoremresqcl 11171 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  ( A ^ 2
 )  e.  RR )
 
Theoremsqgt0 11172 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  0  <  ( A ^ 2 ) )
 
Theoremnnsqcl 11173 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 11174 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 11175 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A ^ 2
 )  e.  QQ )
 
Theoremsq11 11176 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 11177 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 11178 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 11179 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 11180 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  0  <_  ( A ^ 2 ) )
 
Theoremzsqcl2 11181 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 11182 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 11183 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  =  ( A  x.  A )
 
Theoremsqcli 11184 Closure of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  e.  CC
 
Theoremsqeq0i 11185 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( A ^ 2 )  =  0  <->  A  =  0
 )
 
Theoremsqrecii 11186 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( ( 1  /  A ) ^ 2
 )  =  ( 1 
 /  ( A ^
 2 ) )
 
Theoremsqmuli 11187 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 11188 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 11189 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( A ^
 2 )  e.  RR
 
Theoremsqgt0i 11190 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  0  <  ( A ^ 2 ) )
 
Theoremsqge0i 11191 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  0  <_  ( A ^ 2 )
 
Theoremlt2sqi 11192 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 11193 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 11194 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 11195 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
 |-  ( 0 ^ 2
 )  =  0
 
Theoremsq0i 11196 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
 |-  ( A  =  0 
 ->  ( A ^ 2
 )  =  0 )
 
Theoremsq0id 11197 If a number is zero, its square is zero. Deduction form of sq0i 11196. Converse of sqeq0d 11244. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( A ^ 2 )  =  0 )
 
Theoremsq1 11198 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
 |-  ( 1 ^ 2
 )  =  1
 
Theoremsq2 11199 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
 |-  ( 2 ^ 2
 )  =  4
 
Theoremsq3 11200 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
 |-  ( 3 ^ 2
 )  =  9
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