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Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmsq11 8301 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A  x.  A )  =  ( B  x.  B )  <->  A  =  B ) )
 
Theoremledivp1 8302 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A  /  ( B  +  1
 ) )  x.  B )  <_  A )
 
Theoremsqueeze0 8303* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
 |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  ( 0  <  x  ->  A  <  x ) )  ->  A  =  0 )
 
Theoremltp1i 8304 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  A  e.  RR   =>    |-  A  <  ( A  +  1 )
 
Theoremrecgt0i 8305 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <  A  ->  0  <  (
 1  /  A )
 )
 
Theoremrecgt0ii 8306 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  0  <  ( 1 
 /  A )
 
Theoremprodgt0i 8307 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <  ( A  x.  B ) ) 
 ->  0  <  B )
 
Theoremprodge0i 8308 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <_  ( A  x.  B ) ) 
 ->  0  <_  B )
 
Theoremdivgt0i 8309 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  0  <  ( A  /  B ) )
 
Theoremdivge0i 8310 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <  B )  ->  0  <_  ( A  /  B ) )
 
Theoremltreci 8311 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  ( A  <  B  <-> 
 ( 1  /  B )  <  ( 1  /  A ) ) )
 
Theoremlereci 8312 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  ( A  <_  B  <-> 
 ( 1  /  B )  <_  ( 1  /  A ) ) )
 
Theoremlt2msqi 8313 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <  B  <-> 
 ( A  x.  A )  <  ( B  x.  B ) ) )
 
Theoremle2msqi 8314 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <_  B  <-> 
 ( A  x.  A )  <_  ( B  x.  B ) ) )
 
Theoremmsq11i 8315 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( ( A  x.  A )  =  ( B  x.  B ) 
 <->  A  =  B ) )
 
Theoremdivgt0i2i 8316 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  B   =>    |-  (
 0  <  A  ->  0  <  ( A  /  B ) )
 
Theoremltrecii 8317 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  A   &    |-  0  <  B   =>    |-  ( A  <  B  <->  ( 1  /  B )  <  ( 1  /  A ) )
 
Theoremdivgt0ii 8318 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  A   &    |-  0  <  B   =>    |-  0  <  ( A 
 /  B )
 
Theoremltmul1i 8319 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
Theoremltdiv1i 8320 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <  B  <->  ( A  /  C )  <  ( B 
 /  C ) ) )
 
Theoremltmuldivi 8321 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( ( A  x.  C )  <  B  <->  A  <  ( B 
 /  C ) ) )
 
Theoremltmul2i 8322 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
 
Theoremlemul1i 8323 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremlemul2i 8324 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <_  B  <->  ( C  x.  A )  <_  ( C  x.  B ) ) )
 
Theoremltdiv23i 8325 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( 0  <  B  /\  0  <  C ) 
 ->  ( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )
 
Theoremltdiv23ii 8326 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  0  <  B   &    |-  0  <  C   =>    |-  (
 ( A  /  B )  <  C  <->  ( A  /  C )  <  B )
 
Theoremltmul1ii 8327 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  0  <  C   =>    |-  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) )
 
Theoremltdiv1ii 8328 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  0  <  C   =>    |-  ( A  <  B  <->  ( A  /  C )  <  ( B  /  C ) )
 
Theoremltp1d 8329 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <  ( A  +  1 ) )
 
Theoremlep1d 8330 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1d 8331 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <  A )
 
Theoremlem1d 8332 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <_  A )
 
Theoremrecgt0d 8333 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  0  <  ( 1  /  A ) )
 
Theoremdivgt0d 8334 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  /  B ) )
 
Theoremmulgt1d 8335 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  1  <  B )   =>    |-  ( ph  ->  1  <  ( A  x.  B ) )
 
Theoremlemulge11d 8336 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  1  <_  B )   =>    |-  ( ph  ->  A  <_  ( A  x.  B ) )
 
Theoremlemulge12d 8337 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  1  <_  B )   =>    |-  ( ph  ->  A  <_  ( B  x.  A ) )
 
Theoremlemul1ad 8338 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  C ) )
 
Theoremlemul2ad 8339 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( C  x.  A )  <_  ( C  x.  B ) )
 
Theoremltmul12ad 8340 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  C  <  D )   =>    |-  ( ph  ->  ( A  x.  C )  < 
 ( B  x.  D ) )
 
Theoremlemul12ad 8341 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  0 
 <_  C )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  D ) )
 
Theoremlemul12bd 8342 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  0 
 <_  D )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  D ) )
 
Theoremmulle0r 8343 Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  0  /\  0  <_  B ) )  ->  ( A  x.  B )  <_  0 )
 
3.3.10  Suprema
 
Theoremlbreu 8344* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  E! x  e.  S  A. y  e.  S  x  <_  y
 )
 
Theoremlbcl 8345* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  ( iota_ x  e.  S  A. y  e.  S  x  <_  y
 )  e.  S )
 
Theoremlble 8346* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y  /\  A  e.  S ) 
 ->  ( iota_ x  e.  S  A. y  e.  S  x  <_  y )  <_  A )
 
Theoremlbinf 8347* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  -> inf ( S ,  RR ,  <  )  =  ( iota_ x  e.  S  A. y  e.  S  x  <_  y ) )
 
Theoremlbinfcl 8348* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  -> inf ( S ,  RR ,  <  )  e.  S )
 
Theoremlbinfle 8349* If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y  /\  A  e.  S ) 
 -> inf ( S ,  RR ,  <  )  <_  A )
 
Theoremsuprubex 8350* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  B  <_  sup ( A ,  RR ,  <  ) )
 
Theoremsuprlubex 8351* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
 
Theoremsuprnubex 8352* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( -.  B  <  sup ( A ,  RR ,  <  )  <->  A. z  e.  A  -.  B  <  z ) )
 
Theoremsuprleubex 8353* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  <_  B  <->  A. z  e.  A  z 
 <_  B ) )
 
Theoremnegiso 8354 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  F  =  ( x  e.  RR  |->  -u x )   =>    |-  ( F  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
 
Theoremdfinfre 8355* The infimum of a set of reals  A. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( A  C_  RR  -> inf ( A ,  RR ,  <  )  =  U. { x  e.  RR  |  ( A. y  e.  A  x  <_  y  /\  A. y  e.  RR  ( x  <  y  ->  E. z  e.  A  z  <  y
 ) ) } )
 
3.3.11  Imaginary and complex number properties
 
Theoremcrap0 8356 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A #  0  \/  B #  0
 ) 
 <->  ( A  +  ( _i  x.  B ) ) #  0 ) )
 
Theoremcreur 8357* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremcreui 8358* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremcju 8359* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  E! x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) )
 
3.4  Integer sets
 
3.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 8360 Extend class notation to include the class of positive integers.
 class  NN
 
Definitiondf-inn 8361* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8362 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
 |- 
 NN  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theoremdfnn2 8362* Definition of the set of positive integers. Another name for df-inn 8361. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
 |- 
 NN  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theorempeano5nni 8363* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( 1  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  A )
 
Theoremnnssre 8364 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |- 
 NN  C_  RR
 
Theoremnnsscn 8365 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 NN  C_  CC
 
Theoremnnex 8366 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 NN  e.  _V
 
Theoremnnre 8367 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  RR )
 
Theoremnncn 8368 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  CC )
 
Theoremnnrei 8369 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  RR
 
Theoremnncni 8370 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  CC
 
Theorem1nn 8371 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
 |-  1  e.  NN
 
Theorempeano2nn 8372 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
 
Theoremnnred 8373 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnncnd 8374 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  CC )
 
Theorempeano2nnd 8375 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A  +  1 )  e.  NN )
 
3.4.2  Principle of mathematical induction
 
Theoremnnind 8376* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 8380 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  NN  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  NN  ->  ta )
 
TheoremnnindALT 8377* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 8376 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

 |-  ( y  e.  NN  ->  ( ch  ->  th )
 )   &    |- 
 ps   &    |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   =>    |-  ( A  e.  NN  ->  ta )
 
Theoremnn1m1nn 8378 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1 )  e.  NN ) )
 
Theoremnn1suc 8379* If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  (
 ph 
 <-> 
 th ) )   &    |-  ps   &    |-  (
 y  e.  NN  ->  ch )   =>    |-  ( A  e.  NN  ->  th )
 
Theoremnnaddcl 8380 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B )  e.  NN )
 
Theoremnnmulcl 8381 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B )  e.  NN )
 
Theoremnnmulcli 8382 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  x.  B )  e.  NN
 
Theoremnnge1 8383 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  1  <_  A )
 
Theoremnnle1eq1 8384 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
 |-  ( A  e.  NN  ->  ( A  <_  1  <->  A  =  1 ) )
 
Theoremnngt0 8385 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
 |-  ( A  e.  NN  ->  0  <  A )
 
Theoremnnnlt1 8386 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  NN  ->  -.  A  <  1
 )
 
Theorem0nnn 8387 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
 |- 
 -.  0  e.  NN
 
Theoremnnne0 8388 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  e.  NN  ->  A  =/=  0 )
 
Theoremnnap0 8389 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( A  e.  NN  ->  A #  0 )
 
Theoremnngt0i 8390 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  NN   =>    |-  0  <  A
 
Theoremnnne0i 8391 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  =/=  0
 
Theoremnn2ge 8392* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  E. x  e.  NN  ( A  <_  x  /\  B  <_  x ) )
 
Theoremnn1gt1 8393 A positive integer is either one or greater than one. This is for  NN; 0elnn 4407 is a similar theorem for  om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  1  <  A ) )
 
Theoremnngt1ne1 8394 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
 |-  ( A  e.  NN  ->  ( 1  <  A  <->  A  =/=  1 ) )
 
Theoremnndivre 8395 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
 |-  ( ( A  e.  RR  /\  N  e.  NN )  ->  ( A  /  N )  e.  RR )
 
Theoremnnrecre 8396 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
 |-  ( N  e.  NN  ->  ( 1  /  N )  e.  RR )
 
Theoremnnrecgt0 8397 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  0  <  ( 1 
 /  A ) )
 
Theoremnnsub 8398 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  <-> 
 ( B  -  A )  e.  NN )
 )
 
Theoremnnsubi 8399 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  <  B  <->  ( B  -  A )  e.  NN )
 
Theoremnndiv 8400* Two ways to express " A divides  B " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. x  e.  NN  ( A  x.  x )  =  B  <->  ( B  /  A )  e.  NN ) )
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