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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltsubrp 10401 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  -  B )  <  A )
 
Theoremltaddrp 10402 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  <  ( A  +  B )
 )
 
Theoremdifrp 10403 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( B  -  A )  e.  RR+ ) )
 
Theoremelrpd 10404 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremnnrpd 10405 A natural number is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremrpred 10406 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremrpxrd 10407 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrpcnd 10408 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremrpgt0d 10409 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <  A )
 
Theoremrpge0d 10410 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremrpne0d 10411 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremrpregt0d 10412 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremrprege0d 10413 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A ) )
 
Theoremrprene0d 10414 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpcnne0d 10415 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  CC  /\  A  =/=  0 ) )
 
Theoremrpreccld 10416 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR+ )
 
Theoremrprecred 10417 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremrphalfcld 10418 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  2 )  e.  RR+ )
 
Theoremreclt1d 10419 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  1  <->  1  <  (
 1  /  A )
 ) )
 
Theoremrecgt1d 10420 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  ( 1  /  A )  <  1 ) )
 
Theoremrpaddcld 10421 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  +  B )  e.  RR+ )
 
Theoremrpmulcld 10422 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  x.  B )  e.  RR+ )
 
Theoremrpdivcld 10423 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR+ )
 
Theoremltrecd 10424 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( 1  /  B )  <  ( 1 
 /  A ) ) )
 
Theoremlerecd 10425 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( 1  /  B )  <_  ( 1 
 /  A ) ) )
 
Theoremltrec1d 10426 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( 1  /  A )  <  B )   =>    |-  ( ph  ->  ( 1  /  B )  <  A )
 
Theoremlerec2d 10427 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A 
 <_  ( 1  /  B ) )   =>    |-  ( ph  ->  B  <_  ( 1  /  A ) )
 
Theoremlediv2ad 10428 Division of both sides of 'less than or equal to' into 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  /  B )  <_  ( C  /  A ) )
 
Theoremltdiv2d 10429 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  /  B )  <  ( C 
 /  A ) ) )
 
Theoremlediv2d 10430 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  /  B )  <_  ( C 
 /  A ) ) )
 
Theoremledivdivd 10431 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B ) 
 <_  ( C  /  D ) )   =>    |-  ( ph  ->  ( D  /  C )  <_  ( B  /  A ) )
 
Theoremge0p1rpd 10432 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
 
Theoremrerpdivcld 10433 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  B )  e. 
 RR )
 
Theoremltsubrpd 10434 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  -  B )  <  A )
 
Theoremltaddrpd 10435 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  A  <  ( A  +  B ) )
 
Theoremltaddrp2d 10436 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  A  <  ( B  +  A ) )
 
Theoremltmulgt11d 10437 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  B  <  ( B  x.  A ) ) )
 
Theoremltmulgt12d 10438 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  B  <  ( A  x.  B ) ) )
 
Theoremgt0divd 10439 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 0  <  A  <->  0  <  ( A  /  B ) ) )
 
Theoremge0divd 10440 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 0  <_  A  <->  0  <_  ( A  /  B ) ) )
 
Theoremrpgecld 10441 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B 
 <_  A )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremdivge0d 10442 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( A  /  B ) )
 
Theoremltmul1d 10443 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
Theoremltmul2d 10444 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
 
Theoremlemul1d 10445 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremlemul2d 10446 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  x.  A )  <_  ( C  x.  B ) ) )
 
Theoremltdiv1d 10447 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  /  C )  <  ( B 
 /  C ) ) )
 
Theoremlediv1d 10448 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  /  C )  <_  ( B 
 /  C ) ) )
 
Theoremltmuldivd 10449 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  C )  <  B  <->  A  <  ( B 
 /  C ) ) )
 
Theoremltmuldiv2d 10450 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( C  x.  A )  <  B  <->  A  <  ( B 
 /  C ) ) )
 
Theoremlemuldivd 10451 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  C )  <_  B  <->  A  <_  ( B 
 /  C ) ) )
 
Theoremlemuldiv2d 10452 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( C  x.  A )  <_  B  <->  A  <_  ( B 
 /  C ) ) )
 
Theoremltdivmuld 10453 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <  B  <->  A  <  ( C  x.  B ) ) )
 
Theoremltdivmul2d 10454 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <  B  <->  A  <  ( B  x.  C ) ) )
 
Theoremledivmuld 10455 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <_  B  <->  A  <_  ( C  x.  B ) ) )
 
Theoremledivmul2d 10456 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <_  B  <->  A  <_  ( B  x.  C ) ) )
 
Theoremltmul1dd 10457 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  x.  C )  < 
 ( B  x.  C ) )
 
Theoremltmul2dd 10458 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( C  x.  A )  < 
 ( C  x.  B ) )
 
Theoremltdiv1dd 10459 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  /  C )  < 
 ( B  /  C ) )
 
Theoremlediv1dd 10460 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  /  C )  <_  ( B  /  C ) )
 
Theoremlediv12ad 10461 Comparison of ratio 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  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  /  D )  <_  ( B  /  C ) )
 
Theoremltdiv23d 10462 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B )  <  C )   =>    |-  ( ph  ->  ( A  /  C )  <  B )
 
Theoremlediv23d 10463 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B ) 
 <_  C )   =>    |-  ( ph  ->  ( A  /  C )  <_  B )
 
Theoremlt2mul2divd 10464 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  B )  <  ( C  x.  D )  <->  ( A  /  D )  <  ( C 
 /  B ) ) )
 
5.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 10465 Extend class notation to include the negative of an extended real.
 class  - e A
 
Syntaxcxad 10466 Extend class notation to include addition of extended reals.
 class  + e
 
Syntaxcxmu 10467 Extend class notation to include multiplication of extended reals.
 class  x e
 
Definitiondf-xneg 10468 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
 |-  - e A  =  if ( A  =  +oo , 
 -oo ,  if ( A  =  -oo ,  +oo ,  -u A ) )
 
Definitiondf-xadd 10469* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |- 
 + e  =  ( x  e.  RR* ,  y  e.  RR*  |->  if ( x  = 
 +oo ,  if (
 y  =  -oo , 
 0 ,  +oo ) ,  if ( x  = 
 -oo ,  if (
 y  =  +oo , 
 0 ,  -oo ) ,  if ( y  = 
 +oo ,  +oo ,  if ( y  =  -oo , 
 -oo ,  ( x  +  y ) ) ) ) ) )
 
Definitiondf-xmul 10470* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  x e  =  ( x  e.  RR* ,  y  e.  RR*  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  if (
 ( ( ( 0  <  y  /\  x  =  +oo )  \/  (
 y  <  0  /\  x  =  -oo ) )  \/  ( ( 0  <  x  /\  y  =  +oo )  \/  ( x  <  0  /\  y  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  y  /\  x  =  -oo )  \/  ( y  <  0  /\  x  =  +oo ) )  \/  (
 ( 0  <  x  /\  y  =  -oo )  \/  ( x  < 
 0  /\  y  =  +oo ) ) ) , 
 -oo ,  ( x  x.  y ) ) ) ) )
 
Theorempnfxr 10471 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
 |- 
 +oo  e.  RR*
 
Theoremmnfxr 10472 Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |- 
 -oo  e.  RR*
 
Theoremltxr 10473 The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  =  -oo  /\  B  =  +oo ) )  \/  ( ( A  e.  RR  /\  B  =  +oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) )
 
Theoremelxr 10474 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo ) )
 
Theorempnfnemnf 10475 Plus and minus infinity are distinguished elements of  RR*. (Contributed by NM, 14-Oct-2005.)
 |- 
 +oo  =/=  -oo
 
Theoremxrnemnf 10476 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  <->  ( A  e.  RR  \/  A  =  +oo ) )
 
Theoremxrnepnf 10477 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  <->  ( A  e.  RR  \/  A  =  -oo ) )
 
Theoremxrltnr 10478 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  A )
 
Theoremltpnf 10479 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  <  +oo )
 
Theoremmnflt 10480 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  -oo  <  A )
 
Theoremmnfltpnf 10481 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |- 
 -oo  <  +oo
 
Theoremmnfltxr 10482 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
 |-  ( ( A  e.  RR  \/  A  =  +oo )  ->  -oo  <  A )
 
Theorempnfnlt 10483 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  +oo  <  A )
 
Theoremnltmnf 10484 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  -oo )
 
Theorempnfge 10485 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  A  <_  +oo )
 
Theoremmnfle 10486 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  e.  RR*  ->  -oo  <_  A )
 
Theoremxrltnsym 10487 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
 
Theoremxrltnsym2 10488 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  ( A  <  B 
 /\  B  <  A ) )
 
Theoremxrlttri 10489 Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 8827 or axlttri 8910. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
 
Theoremxrlttr 10490 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <  B  /\  B  <  C ) 
 ->  A  <  C ) )
 
Theoremxrltso 10491 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
 |- 
 <  Or  RR*
 
Theoremxrlttri2 10492 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) ) )
 
Theoremxrlttri3 10493 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremxrleloe 10494 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
 ) )
 
Theoremxrleltne 10495 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  <  B  <->  B  =/=  A ) )
 
Theoremxrltlen 10496 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) ) )
 
Theoremdfle2 10497 Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |- 
 <_  =  (  <  u.  (  _I  |`  RR* )
 )
 
Theoremdflt2 10498 Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |- 
 <  =  (  <_  \  _I  )
 
Theoremxrltle 10499 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B )
 )
 
Theoremxrleid 10500 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( A  e.  RR*  ->  A  <_  A )
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