HomeHome Intuitionistic Logic Explorer
Theorem List (p. 106 of 141)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpgt1 10501 A real greater than 1 raised to a positive integer 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 10502 Nonnegative 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theoremmulexpzap 10503 Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  ZZ )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theoremexprecap 10504 Integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( ( 1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpadd 10505 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 ) ) )
 
Theoremexpaddzaplem 10506 Lemma for expaddzap 10507. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpaddzap 10507 Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpmul 10508 Product of exponents law for nonnegative integer 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 ) )
 
Theoremexpmulzap 10509 Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N ) )
 
Theoremm1expeven 10510 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 10511 Exponent subtraction law for 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 10512 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 10513 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 10514 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 10515 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 10516 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 10517 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 10518 Weak base 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 10519 A real between 0 and 1 inclusive raised to a nonnegative integer 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 10520 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 10521 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  =  ( A  x.  A ) )
 
Theoremsqneg 10522 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
 |-  ( A  e.  CC  ->  ( -u A ^ 2
 )  =  ( A ^ 2 ) )
 
Theoremsqsubswap 10523 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 10524 Closure of square. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  e.  CC )
 
Theoremsqmul 10525 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 10526 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =  0  <->  A  =  0 )
 )
 
Theoremsqdivap 10527 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 10528 A number is nonzero iff its square is nonzero. See also sqap0 10529 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 10529 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 10530 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  ( A ^ 2
 )  e.  RR )
 
Theoremsqgt0ap 10531 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  0  <  ( A ^ 2 ) )
 
Theoremnnsqcl 10532 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 10533 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 10534 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A ^ 2
 )  e.  QQ )
 
Theoremsq11 10535 The square function is one-to-one for nonnegative reals. Also see sq11ap 10630 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 10536 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 10537 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 10538 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 10539 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
 |-  ( A  e.  RR  ->  0  <_  ( A ^ 2 ) )
 
Theoremzsqcl2 10540 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 10541 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 10542 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  =  ( A  x.  A )
 
Theoremsqcli 10543 Closure of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A ^
 2 )  e.  CC
 
Theoremsqeq0i 10544 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( A ^ 2 )  =  0  <->  A  =  0
 )
 
Theoremsqmuli 10545 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 10546 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 10547 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( A ^
 2 )  e.  RR
 
Theoremsqgt0api 10548 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  0  <  ( A ^ 2 ) )
 
Theoremsqge0i 10549 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  0  <_  ( A ^ 2 )
 
Theoremlt2sqi 10550 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 10551 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 10552 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 10553 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
 |-  ( 0 ^ 2
 )  =  0
 
Theoremsq0i 10554 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
 |-  ( A  =  0 
 ->  ( A ^ 2
 )  =  0 )
 
Theoremsq0id 10555 If a number is zero, its square is zero. Deduction form of sq0i 10554. Converse of sqeq0d 10595. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( A ^ 2 )  =  0 )
 
Theoremsq1 10556 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
 |-  ( 1 ^ 2
 )  =  1
 
Theoremneg1sqe1 10557  -u 1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( -u 1 ^ 2
 )  =  1
 
Theoremsq2 10558 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
 |-  ( 2 ^ 2
 )  =  4
 
Theoremsq3 10559 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
 |-  ( 3 ^ 2
 )  =  9
 
Theoremsq4e2t8 10560 The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
 |-  ( 4 ^ 2
 )  =  ( 2  x.  8 )
 
Theoremcu2 10561 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 2 ^ 3
 )  =  8
 
Theoremirec 10562 The reciprocal of  _i. (Contributed by NM, 11-Oct-1999.)
 |-  ( 1  /  _i )  =  -u _i
 
Theoremi2 10563  _i squared. (Contributed by NM, 6-May-1999.)
 |-  ( _i ^ 2
 )  =  -u 1
 
Theoremi3 10564  _i cubed. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 3
 )  =  -u _i
 
Theoremi4 10565  _i to the fourth power. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 4
 )  =  1
 
Theoremnnlesq 10566 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 10567 Taking  _i to the  K-th power is the same as using the  K  mod  4 -th power instead, by i4 10565. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( K  e.  ZZ  ->  ( _i ^ ( K  mod  4 ) )  =  ( _i ^ K ) )
 
Theoremexpnass 10568 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 10569 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 10570 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 10571 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 10572 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 ) )
 
Theoremqsqeqor 10573 The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( ( A ^ 2 )  =  ( B ^ 2
 ) 
 <->  ( A  =  B  \/  A  =  -u B ) ) )
 
Theorembinom2 10574 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 10575 Special case of binom2 10574 where  B  =  1. (Contributed by Scott Fenton, 11-May-2014.)
 |-  ( A  e.  CC  ->  ( ( A  +  1 ) ^ 2
 )  =  ( ( ( A ^ 2
 )  +  ( 2  x.  A ) )  +  1 ) )
 
Theorembinom2sub 10576 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 ) ) )
 
Theorembinom2sub1 10577 Special case of binom2sub 10576 where  B  =  1. (Contributed by AV, 2-Aug-2021.)
 |-  ( A  e.  CC  ->  ( ( A  -  1 ) ^ 2
 )  =  ( ( ( A ^ 2
 )  -  ( 2  x.  A ) )  +  1 ) )
 
Theorembinom2subi 10578 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 10579 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 10580 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 10581 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 10582 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 10583 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 10584 Variation of Bernoulli's inequality bernneq 10583. (Contributed by NM, 18-Oct-2007.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  ( ( ( A  -  1 )  x.  N )  +  1 )  <_  ( A ^ N ) )
 
Theorembernneq3 10585 A corollary of bernneq 10583. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( P  e.  ( ZZ>= `  2 )  /\  N  e.  NN0 )  ->  N  <  ( P ^ N ) )
 
Theoremexpnbnd 10586* Exponentiation with a base 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 10587* The reciprocal of exponentiation with a base greater than 1 has no positive 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 10588* The reciprocal of exponentiation with a base greater than 1 has no positive 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 )
 
Theoremmodqexp 10589 Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  0  <  D )   &    |-  ( ph  ->  ( A  mod  D )  =  ( B 
 mod  D ) )   =>    |-  ( ph  ->  ( ( A ^ C )  mod  D )  =  ( ( B ^ C )  mod  D ) )
 
Theoremexp0d 10590 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 10591 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 10592 Positive integer exponentiation is 0 iff its base 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 10593 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
 
Theoremsqcld 10594 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 CC )
 
Theoremsqeq0d 10595 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 10596 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 10597 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 10598 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 10599 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 10600 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 ( 1  /  A ) ^ 2 )  =  ( 1  /  ( A ^ 2 ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14073
  Copyright terms: Public domain < Previous  Next >