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Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqeq0i 9501 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( A ^ 2 )  =  0  <->  A  =  0
 )
 
Theoremsqmuli 9502 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 9503 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 9504 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( A ^
 2 )  e.  RR
 
Theoremsqgt0api 9505 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  0  <  ( A ^ 2 ) )
 
Theoremsqge0i 9506 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  0  <_  ( A ^ 2 )
 
Theoremlt2sqi 9507 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 9508 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 9509 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 9510 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
 |-  ( 0 ^ 2
 )  =  0
 
Theoremsq0i 9511 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
 |-  ( A  =  0 
 ->  ( A ^ 2
 )  =  0 )
 
Theoremsq0id 9512 If a number is zero, its square is zero. Deduction form of sq0i 9511. Converse of sqeq0d 9548. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( A ^ 2 )  =  0 )
 
Theoremsq1 9513 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
 |-  ( 1 ^ 2
 )  =  1
 
Theoremneg1sqe1 9514  -u 1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( -u 1 ^ 2
 )  =  1
 
Theoremsq2 9515 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
 |-  ( 2 ^ 2
 )  =  4
 
Theoremsq3 9516 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
 |-  ( 3 ^ 2
 )  =  9
 
Theoremcu2 9517 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 2 ^ 3
 )  =  8
 
Theoremirec 9518 The reciprocal of  _i. (Contributed by NM, 11-Oct-1999.)
 |-  ( 1  /  _i )  =  -u _i
 
Theoremi2 9519  _i squared. (Contributed by NM, 6-May-1999.)
 |-  ( _i ^ 2
 )  =  -u 1
 
Theoremi3 9520  _i cubed. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 3
 )  =  -u _i
 
Theoremi4 9521  _i to the fourth power. (Contributed by NM, 31-Jan-2007.)
 |-  ( _i ^ 4
 )  =  1
 
Theoremnnlesq 9522 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 9523 Taking  _i to the  K-th power is the same as using the  K  mod  4 -th power instead, by i4 9521. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( K  e.  ZZ  ->  ( _i ^ ( K  mod  4 ) )  =  ( _i ^ K ) )
 
Theoremexpnass 9524 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 9525 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 9526 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 9527 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 9528 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 9529 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 9530 Special case of binom2 9529 where  B  =  1. (Contributed by Scott Fenton, 11-May-2014.)
 |-  ( A  e.  CC  ->  ( ( A  +  1 ) ^ 2
 )  =  ( ( ( A ^ 2
 )  +  ( 2  x.  A ) )  +  1 ) )
 
Theorembinom2sub 9531 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 9532 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 9533 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 9534 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 9535 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 9536 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 9537 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 9538 Variation of Bernoulli's inequality bernneq 9537. (Contributed by NM, 18-Oct-2007.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  ( ( ( A  -  1 )  x.  N )  +  1 )  <_  ( A ^ N ) )
 
Theorembernneq3 9539 A corollary of bernneq 9537. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( P  e.  ( ZZ>= `  2 )  /\  N  e.  NN0 )  ->  N  <  ( P ^ N ) )
 
Theoremexpnbnd 9540* 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 9541* 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 9542* 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 9543 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 9544 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 9545 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 9546 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
 
Theoremsqcld 9547 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 CC )
 
Theoremsqeq0d 9548 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 9549 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 9550 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 9551 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 9552 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 9553 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 9554 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 9555 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 9556 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 9557 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 9558 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 9559 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 9560 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 9561 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 9562 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 9563 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 9564 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 9565 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 9566 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 9567 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 9568 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 ) )
 
Theoremsqoddm1div8 9569 A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.)
 |-  ( ( N  e.  ZZ  /\  M  =  ( ( 2  x.  N )  +  1 )
 )  ->  ( (
 ( M ^ 2
 )  -  1 ) 
 /  8 )  =  ( ( N  x.  ( N  +  1
 ) )  /  2
 ) )
 
Theoremnnsqcld 9570 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 NN )
 
Theoremnnexpcld 9571 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 9572 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 9573 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+ )
 
Theoremreexpclzapd 9574 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ N )  e.  RR )
 
Theoremresqcld 9575 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 RR )
 
Theoremsqge0d 9576 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A ^ 2
 ) )
 
Theoremsqgt0apd 9577 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  0  <  ( A ^ 2
 ) )
 
Theoremleexp2ad 9578 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ph  ->  ( A ^ M )  <_  ( A ^ N ) )
 
Theoremleexp2rd 9579 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A 
 <_  1 )   =>    |-  ( ph  ->  ( A ^ N )  <_  ( A ^ M ) )
 
Theoremlt2sqd 9580 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A ^
 2 )  <  ( B ^ 2 ) ) )
 
Theoremle2sqd 9581 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A ^
 2 )  <_  ( B ^ 2 ) ) )
 
Theoremsq11d 9582 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  ( A ^ 2 )  =  ( B ^
 2 ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsq11ap 9583 Analogue to sq11 9492 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A ^
 2 ) #  ( B ^ 2 )  <->  A #  B )
 )
 
Theoremsq10 9584 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
 |-  (; 1 0 ^ 2
 )  = ;; 1 0 0
 
Theoremsq10e99m1 9585 The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
 |-  (; 1 0 ^ 2
 )  =  (; 9 9  +  1 )
 
Theorem3dec 9586 A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |- ;; A B C  =  ( ( ( (; 1
 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B ) )  +  C )
 
3.6.6  Ordered pair theorem for nonnegative integers
 
Theoremnn0le2msqd 9587 The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  x.  A )  <_  ( B  x.  B ) ) )
 
Theoremnn0opthlem1d 9588 A rather pretty lemma for nn0opth2 9592. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   =>    |-  ( ph  ->  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  <  ( C  x.  C ) ) )
 
Theoremnn0opthlem2d 9589 Lemma for nn0opth2 9592. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A  +  B )  <  C  ->  (
 ( C  x.  C )  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) ) )
 
Theoremnn0opthd 9590 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers  A and  B by  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B ). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 3412 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( (
 ( C  +  D )  x.  ( C  +  D ) )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremnn0opth2d 9591 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 9590. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( ( A  +  B ) ^
 2 )  +  B )  =  ( (
 ( C  +  D ) ^ 2 )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremnn0opth2 9592 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 9590. (Contributed by NM, 22-Jul-2004.)
 |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  ->  (
 ( ( ( A  +  B ) ^
 2 )  +  B )  =  ( (
 ( C  +  D ) ^ 2 )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
3.6.7  Factorial function
 
Syntaxcfa 9593 Extend class notation to include the factorial of nonnegative integers.
 class  !
 
Definitiondf-fac 9594 Define the factorial function on nonnegative integers. For example,  ( ! `  5 )  =  1 2 0 because  1  x.  2  x.  3  x.  4  x.  5  =  1 2 0 (ex-fac 10281). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)
 |-  !  =  ( { <. 0 ,  1 >. }  u.  seq 1 (  x.  ,  _I  ,  CC ) )
 
Theoremfacnn 9595 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( N  e.  NN  ->  ( ! `  N )  =  (  seq 1 (  x.  ,  _I  ,  CC ) `  N ) )
 
Theoremfac0 9596 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  0
 )  =  1
 
Theoremfac1 9597 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  1
 )  =  1
 
Theoremfacp1 9598 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( N  e.  NN0  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
  N )  x.  ( N  +  1 ) ) )
 
Theoremfac2 9599 The factorial of 2. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  2
 )  =  2
 
Theoremfac3 9600 The factorial of 3. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  3
 )  =  6
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