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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsq01 11101 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 11102 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 11103 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 11104 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 11105 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 11106 Variation of Bernoulli's inequality bernneq 11105. (Contributed by NM, 18-Oct-2007.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  ( ( ( A  -  1 )  x.  N )  +  1 )  <_  ( A ^ N ) )
 
Theorembernneq3 11107 A corollary of bernneq 11105. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( P  e.  ( ZZ>= `  2 )  /\  N  e.  NN0 )  ->  N  <  ( P ^ N ) )
 
Theoremexpnbnd 11108* 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 11109* 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 11110* 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 11111* 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 11112 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 11113 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 11114 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 11115* 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 11116* 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 11117 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 11118 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 11119 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 11120 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
 
Theoremsqcld 11121 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 CC )
 
Theoremsqeq0d 11122 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 11123 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 11124 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 11125 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 11126 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 11127 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 11128 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 11129 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 11130 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 11131 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 11132 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 11133 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 11134 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 11135 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 11136 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 11137 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 11138 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 11139 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 11140 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 11141 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 11142 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 11143 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 NN )
 
Theoremnnexpcld 11144 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 11145 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 11146 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 11147 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 ) ) )
 
Theoremreexpclzd 11148 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( A ^ N )  e. 
 RR )
 
Theoremresqcld 11149 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A ^ 2 )  e. 
 RR )
 
Theoremsqge0d 11150 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A ^ 2
 ) )
 
Theoremsqgt0d 11151 The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  0  <  ( A ^
 2 ) )
 
Theoremltexp2d 11152 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  1  <  A )   =>    |-  ( ph  ->  ( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) )
 
Theoremleexp2d 11153 Ordering law for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  1  <  A )   =>    |-  ( ph  ->  ( M  <_  N  <->  ( A ^ M )  <_  ( A ^ N ) ) )
 
Theoremexpcand 11154 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  ( A ^ M )  =  ( A ^ N ) )   =>    |-  ( ph  ->  M  =  N )
 
Theoremleexp2ad 11155 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 11156 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 11157 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 11158 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 11159 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 )
 
5.6.5  Ordered pair theorem for nonnegative integers
 
Theoremnn0le2msqi 11160 The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |-  ( A  <_  B  <->  ( A  x.  A ) 
 <_  ( B  x.  B ) )
 
Theoremnn0opthlem1 11161 A rather pretty lemma for nn0opthi 11163. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  A  e.  NN0   &    |-  C  e.  NN0   =>    |-  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  < 
 ( C  x.  C ) )
 
Theoremnn0opthlem2 11162 Lemma for nn0opthi 11163. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   =>    |-  ( ( A  +  B )  <  C  ->  ( ( C  x.  C )  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
 
Theoremnn0opthi 11163 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 3553 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   =>    |-  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  (
 ( ( C  +  D )  x.  ( C  +  D )
 )  +  D )  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremnn0opth2i 11164 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 11163. (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 ) )
 
Theoremnn0opth2 11165 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 11163. (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 ) ) )
 
5.6.6  Factorial function
 
Syntaxcfa 11166 Extend class notation to include the factorial of nonnegative integers.
 class  !
 
Definitiondf-fac 11167 Define the factorial function on nonnegative integers. For example,  ( ! `  4 )  = ; 2
4 because  ( 4  x.  (
3  x.  ( 2  x.  1 ) ) )  = ; 2 4 (fac4 11174). 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  ) )
 
Theoremfacnn 11168 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  ) `  N ) )
 
Theoremfac0 11169 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  0
 )  =  1
 
Theoremfac1 11170 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  1
 )  =  1
 
Theoremfacp1 11171 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 11172 The factorial of 2. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  2
 )  =  2
 
Theoremfac3 11173 The factorial of 3. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  3
 )  =  6
 
Theoremfac4 11174 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ! `  4
 )  = ; 2 4
 
Theoremfacnn2 11175 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN  ->  ( ! `  N )  =  ( ( ! `  ( N  -  1 ) )  x.  N ) )
 
Theoremfaccl 11176 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  e.  NN )
 
Theoremfacne0 11177 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  =/=  0 )
 
Theoremfacdiv 11178 A natural number divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN  /\  N  <_  M )  ->  ( ( ! `  M )  /  N )  e.  NN )
 
Theoremfacndiv 11179 No natural number (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  <  N  /\  N  <_  M ) )  ->  -.  ( ( ( ! `
  M )  +  1 )  /  N )  e.  ZZ )
 
Theoremfacwordi 11180 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( ! `  M )  <_  ( ! `  N ) )
 
Theoremfaclbnd 11181 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ ( N  +  1 )
 )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd2 11182 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( ( 2 ^ N )  /  2
 )  <_  ( ! `  N ) )
 
Theoremfaclbnd3 11183 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ N )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4lem1 11184 Lemma for faclbnd4 11188. Prepare the induction step. (Contributed by NM, 20-Dec-2005.)
 |-  N  e.  NN   &    |-  K  e.  NN0   &    |-  M  e.  NN0   =>    |-  ( ( ( ( N  -  1 ) ^ K )  x.  ( M ^ ( N  -  1 ) ) )  <_  ( (
 ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
 ( M  +  K ) ) )  x.  ( ! `  ( N  -  1 ) ) )  ->  ( ( N ^ ( K  +  1 ) )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ (
 ( K  +  1 ) ^ 2 ) )  x.  ( M ^ ( M  +  ( K  +  1
 ) ) ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4lem2 11185 Lemma for faclbnd4 11188. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 11184 to antecedents. (Contributed by NM, 23-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  K  e.  NN0  /\  N  e.  NN )  ->  (
 ( ( ( N  -  1 ) ^ K )  x.  ( M ^ ( N  -  1 ) ) ) 
 <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  ( N  -  1 ) ) ) 
 ->  ( ( N ^
 ( K  +  1 ) )  x.  ( M ^ N ) ) 
 <_  ( ( ( 2 ^ ( ( K  +  1 ) ^
 2 ) )  x.  ( M ^ ( M  +  ( K  +  1 ) ) ) )  x.  ( ! `  N ) ) ) )
 
Theoremfaclbnd4lem3 11186 Lemma for faclbnd4 11188. The  N  =  0 case. (Contributed by NM, 23-Dec-2005.)
 |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0 )  ->  ( ( N ^ K )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4lem4 11187 Lemma for faclbnd4 11188. Prove the  0  <  N case by induction on  K. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( N  e.  NN  /\  K  e.  NN0  /\  M  e.  NN0 )  ->  ( ( N ^ K )  x.  ( M ^ N ) ) 
 <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4 11188 Variant of faclbnd5 11189 providing a non-strict lower bound. (Contributed by NM, 23-Dec-2005.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN0  /\  M  e.  NN0 )  ->  (
 ( N ^ K )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd5 11189 The factorial function grows faster than powers and exponentiations. If we consider  K and  M to be constants, the right-hand side of the inequality is a constant times 
N-factorial. (Contributed by NM, 24-Dec-2005.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN0  /\  M  e.  NN )  ->  (
 ( N ^ K )  x.  ( M ^ N ) )  < 
 ( ( 2  x.  ( ( 2 ^
 ( K ^ 2
 ) )  x.  ( M ^ ( M  +  K ) ) ) )  x.  ( ! `
  N ) ) )
 
Theoremfaclbnd6 11190 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
 |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( ! `  N )  x.  (
 ( N  +  1 ) ^ M ) )  <_  ( ! `  ( N  +  M ) ) )
 
Theoremfacubnd 11191 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( N  e.  NN0  ->  ( ! `  N ) 
 <_  ( N ^ N ) )
 
Theoremfacavg 11192 The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ! `  ( |_ `  ( ( M  +  N )  / 
 2 ) ) ) 
 <_  ( ( ! `  M )  x.  ( ! `  N ) ) )
 
5.6.7  The binomial coefficient operation
 
Syntaxcbc 11193 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
 class  _C
 
Definitiondf-bc 11194* Define the binomial coefficient operation. In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C".  ( N  _C  K ) is read " N choose  K." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  k  <_  n does not hold. (Contributed by NM, 10-Jul-2005.)
 |- 
 _C  =  ( n  e.  NN0 ,  k  e. 
 ZZ  |->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `  n )  /  ( ( ! `  ( n  -  k
 ) )  x.  ( ! `  k ) ) ) ,  0 ) )
 
Theorembcval 11195 Value of the binomial coefficient, 
N choose  K. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  K  <_  N does not hold. See bcval2 11196 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  if ( K  e.  ( 0 ... N ) ,  (
 ( ! `  N )  /  ( ( ! `
  ( N  -  K ) )  x.  ( ! `  K ) ) ) ,  0 ) )
 
Theorembcval2 11196 Value of the binomial coefficient, 
N choose  K, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `
  N )  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) ) )
 
Theorembcval3 11197 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0
 ... N ) ) 
 ->  ( N  _C  K )  =  0 )
 
Theorembcval4 11198 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  ( K  <  0  \/  N  <  K ) )  ->  ( N  _C  K )  =  0 )
 
Theorembcrpcl 11199 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 11213.) (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
 
Theorembccmpl 11200 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
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