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Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlereci 9601 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 9602 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 9603 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 9604 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 9605 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 9606 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 9607 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 9608 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 9609 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 9610 '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 9611 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 9612 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 9613 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 9614 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 ) )
 
Theoremledivp1i 9615 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1
 ) ) )  ->  ( A  x.  C )  <_  B )
 
Theoremltdivp1i 9616 Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1
 ) ) )  ->  ( A  x.  C )  <  B )
 
Theoremltdiv23ii 9617 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 9618 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 9619 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 9620 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <  ( A  +  1 ) )
 
Theoremlep1d 9621 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 9622 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <  A )
 
Theoremlem1d 9623 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 9624 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 9625 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 9626 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 9627 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 9628 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 9629 The square of a nonnegative number is a one-to-one function. (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 9630 The square of a nonnegative number is a one-to-one function. (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 9631 The square of a nonnegative number is a one-to-one function. (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 9632 The square of a nonnegative number is a one-to-one function. (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 9633 The square of a nonnegative number is a one-to-one function. (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 ) )
 
5.3.8  Completeness Axiom and Suprema
 
Theoremfimaxre 9634* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  C_  RR  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  y 
 <_  x )
 
Theoremfimaxre2 9635* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
 |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. x  e.  RR  A. y  e.  A  y 
 <_  x )
 
Theoremfimaxre3 9636* A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.)
 |-  ( ( A  e.  Fin  /\  A. y  e.  A  B  e.  RR )  ->  E. x  e.  RR  A. y  e.  A  B  <_  x )
 
Theoremlbreu 9637* 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 9638* 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 9639* 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 )
 
Theoremlbinfm 9640* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  sup ( S ,  RR ,  `'  <  )  =  ( iota_ x  e.  S A. y  e.  S  x  <_  y
 ) )
 
Theoremlbinfmcl 9641* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  sup ( S ,  RR ,  `'  <  )  e.  S )
 
Theoremlbinfmle 9642* If a set of reals contains a lower bound, its infmimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y  /\  A  e.  S ) 
 ->  sup ( S ,  RR ,  `'  <  )  <_  A )
 
Theoremsup2 9643* A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  ( y  < 
 x  \/  y  =  x ) )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
 y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoremsup3 9644* A version of the completeness axiom for reals. (Contributed by NM, 12-Oct-2004.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
 y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoreminfm3lem 9645* Lemma for infm3 9646. (Contributed by NM, 14-Jun-2005.)
 |-  ( x  e.  RR  ->  E. y  e.  RR  x  =  -u y )
 
Theoreminfm3 9646* The completeness axiom for reals in terms of infimum: a non-empty, bounded-below set of reals has a infimum. (This theorem is the dual of sup3 9644.) (Contributed by NM, 14-Jun-2005.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
 )  ->  E. x  e.  RR  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e. 
 RR  ( x  < 
 y  ->  E. z  e.  A  z  <  y
 ) ) )
 
Theoremsuprcl 9647* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Oct-2004.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR ,  <  )  e. 
 RR )
 
Theoremsuprub 9648* A member of a non-empty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.)
 |-  ( ( ( A 
 C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y 
 <_  x )  /\  B  e.  A )  ->  B  <_  sup ( A ,  RR ,  <  ) )
 
Theoremsuprlub 9649* The supremum of a non-empty bounded set of reals is the least upper bound. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( ( ( A 
 C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y 
 <_  x )  /\  B  e.  RR )  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
 
Theoremsuprnub 9650* An upper bound is not less than the supremum of a non-empty bounded set of reals. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( ( ( A 
 C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y 
 <_  x )  /\  B  e.  RR )  ->  ( -.  B  <  sup ( A ,  RR ,  <  )  <->  A. z  e.  A  -.  B  <  z ) )
 
Theoremsuprleub 9651* The supremum of a non-empty 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.)
 |-  ( ( ( A 
 C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y 
 <_  x )  /\  B  e.  RR )  ->  ( sup ( A ,  RR ,  <  )  <_  B  <->  A. z  e.  A  z 
 <_  B ) )
 
Theoremsupmul1 9652* The supremum function distributes over multiplication, in the sense that  A  x.  ( sup B )  =  sup ( A  x.  B
), where  A  x.  B is shorthand for  { A  x.  b  |  b  e.  B } and is defined as  C below. This is the simple version, with only one set argument; see supmul 9655 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)
 |-  C  =  { z  |  E. v  e.  B  z  =  ( A  x.  v ) }   &    |-  ( ph 
 <->  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  B  0  <_  x )  /\  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) ) )   =>    |-  ( ph  ->  ( A  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
 
Theoremsupmullem1 9653* Lemma for supmul 9655. (Contributed by Mario Carneiro, 5-Jul-2013.)
 |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }   &    |-  ( ph 
 <->  ( ( A. x  e.  A  0  <_  x  /\  A. x  e.  B  0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) ) )   =>    |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
 
Theoremsupmullem2 9654* Lemma for supmul 9655. (Contributed by Mario Carneiro, 5-Jul-2013.)
 |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }   &    |-  ( ph 
 <->  ( ( A. x  e.  A  0  <_  x  /\  A. x  e.  B  0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) ) )   =>    |-  ( ph  ->  ( C  C_  RR  /\  C  =/= 
 (/)  /\  E. x  e. 
 RR  A. w  e.  C  w  <_  x ) )
 
Theoremsupmul 9655* The supremum function distributes over multiplication, in the sense that  ( sup A
)  x.  ( sup B )  =  sup ( A  x.  B
), where  A  x.  B is shorthand for  { a  x.  b  |  a  e.  A ,  b  e.  B } and is defined as  C below. We made use of this in our definition of multiplication in the Dedekind cut construction of the reals (see df-mp 8541). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }   &    |-  ( ph 
 <->  ( ( A. x  e.  A  0  <_  x  /\  A. x  e.  B  0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) ) )   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x. 
 sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
 
Theoremsup3ii 9656* A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-1999.)
 |-  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |- 
 E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
 y  <  x  ->  E. z  e.  A  y  <  z ) )
 
Theoremsuprclii 9657* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Sep-1999.)
 |-  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |- 
 sup ( A ,  RR ,  <  )  e. 
 RR
 
Theoremsuprubii 9658* A member of a non-empty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-1999.)
 |-  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |-  ( B  e.  A  ->  B  <_  sup ( A ,  RR ,  <  ) )
 
Theoremsuprlubii 9659* The supremum of a non-empty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |-  ( B  e.  RR  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
 
Theoremsuprnubii 9660* An upper bound is not less than the supremum of a non-empty bounded set of reals. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |-  ( B  e.  RR  ->  ( -.  B  <  sup ( A ,  RR ,  <  )  <->  A. z  e.  A  -.  B  <  z ) )
 
Theoremsuprleubii 9661* The supremum of a non-empty 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.)
 |-  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |-  ( B  e.  RR  ->  ( sup ( A ,  RR ,  <  ) 
 <_  B  <->  A. z  e.  A  z  <_  B ) )
 
Theoremriotaneg 9662* The negative of the unique real such that  ph. (Contributed by NM, 13-Jun-2005.)
 |-  ( x  =  -u y  ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  RR  ph  ->  ( iota_ x  e.  RR ph )  =  -u ( iota_ y  e. 
 RR ps ) )
 
Theoremnegiso 9663 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 )
 
Theoremdfinfmr 9664* The infimum (expressed as supremum with converse 'less-than') of a set of reals  A. (Contributed by NM, 9-Oct-2005.)
 |-  ( A  C_  RR  ->  sup ( A ,  RR ,  `'  <  )  =  U. { x  e. 
 RR  |  ( A. y  e.  A  x  <_  y  /\  A. y  e.  RR  ( x  < 
 y  ->  E. z  e.  A  z  <  y
 ) ) } )
 
Theoreminfmsup 9665* The infimum (expressed as supremum with converse 'less-than') of a set of reals  A is the negative of the supremum of the negatives of its elements. The antecedent ensures that  A is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
 )  ->  sup ( A ,  RR ,  `'  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  ) )
 
Theoreminfmrcl 9666* Closure of infimum of a non-empty bounded set of reals. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
 )  ->  sup ( A ,  RR ,  `'  <  )  e.  RR )
 
Theoreminfmrgelb 9667* Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( ( ( A 
 C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y )  /\  B  e.  RR )  ->  ( B  <_  sup ( A ,  RR ,  `'  <  )  <->  A. z  e.  A  B  <_  z ) )
 
Theoreminfmrlb 9668* If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.)
 |-  ( ( B  C_  RR  /\  E. x  e. 
 RR  A. y  e.  B  x  <_  y  /\  A  e.  B )  ->  sup ( B ,  RR ,  `'  <  )  <_  A )
 
5.3.9  Imaginary and complex number properties
 
Theoreminelr 9669 The imaginary unit  _i is not a real number. (Contributed by NM, 6-May-1999.)
 |- 
 -.  _i  e.  RR
 
Theoremrimul 9670 A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  ->  A  =  0 )
 
Theoremcru 9671 The representation of complex numbers in terms of real and imaginary parts 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.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  +  ( _i  x.  B ) )  =  ( C  +  ( _i  x.  D ) )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremcrne0 9672 The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =/=  0  \/  B  =/=  0 )  <->  ( A  +  ( _i  x.  B ) )  =/=  0
 ) )
 
Theoremcreur 9673* 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 9674* 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 9675* 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 ) )
 
5.3.10  Function operation analogue theorems
 
Theoremofsubeq0 9676 Function analog of subeq0 9006. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G )  =  ( A  X.  { 0 } )  <->  F  =  G ) )
 
Theoremofnegsub 9677 Function analog of negsub 9028. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  o F  +  (
 ( A  X.  { -u 1 } )  o F  x.  G ) )  =  ( F  o F  -  G ) )
 
Theoremofsubge0 9678 Function analog of subge0 9220. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( ( A  X.  { 0 } )  o R  <_  ( F  o F  -  G )  <->  G  o R  <_  F ) )
 
5.4  Integer sets
 
5.4.1  Natural numbers (as a subset of complex numbers)
 
Syntaxcn 9679 Extend class notation to include the class of positive integers.
 class  NN
 
Definitiondf-n 9680 The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set  om, df-om 4594, which start at zero). This is the convention used by most analysis books, and it is often convenient in proofs because we don't have to worry about division by zero. See nnind 9697 for the principle of mathematical induction. See dfnn2 9692 for a slight variant. See df-n0 9898 for the set of nonnegative integers  NN0 starting at zero. See dfn2 9910 for  NN defined in terms of  NN0.

This is a technical definition that helps us avoid the Axiom of Infinity in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing 
1 as well as the successor of every member") see dfnn3 9693. (Contributed by NM, 10-Jan-1997.)

 |- 
 NN  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  1 ) " om )
 
TheoremnnexALT 9681 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
 |- 
 NN  e.  _V
 
Theorempeano5nni 9682* 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 9683 The natural numbers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |- 
 NN  C_  RR
 
Theoremnnsscn 9684 The natural numbers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 NN  C_  CC
 
Theoremnnex 9685 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 NN  e.  _V
 
Theoremnnre 9686 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  RR )
 
Theoremnncn 9687 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  CC )
 
Theoremnnrei 9688 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  RR
 
Theoremnncni 9689 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  CC
 
Theorem1nn 9690 Peano postulate: 1 is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  1  e.  NN
 
Theorempeano2nn 9691 Peano postulate: a successor of a natural number is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
 
Theoremdfnn2 9692* Alternate definition of the set of natural numbers. This was our original definition, before the current df-n 9680 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (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 ) }
 
Theoremdfnn3 9693* Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)
 |- 
 NN  =  |^| { x  |  ( x  C_  RR  /\  1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
 
Theoremnnred 9694 A natural number is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnncnd 9695 A natural number is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  CC )
 
Theorempeano2nnd 9696 Peano postulate: a successor of a natural number is a natural number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A  +  1 )  e.  NN )
 
5.4.2  Principle of mathematical induction
 
Theoremnnind 9697* 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 hypothesis. See nnaddcl 9701 for an example of its use. See nn0ind 10040 for induction on nonnegative integers and uzind 10035, uzind4 10208 for induction on an arbitrary set of upper integers. See indstr 10219 for strong induction. (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 9698* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis. (This ALT version of nnind 9697 is easier to use with the Proof Assistant since 'assign last' will be applied to the substitution instances first. We may switch to it as the official version.) (Contributed by NM, 7-Dec-2005.)
 |-  ( 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 9699 Every natural number is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1 )  e.  NN ) )
 
Theoremnn1suc 9700* If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. 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 )
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