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Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulneg1i 7401 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  x.  B )  =  -u ( A  x.  B )
 
Theoremmulneg2i 7402 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  -u B )  =  -u ( A  x.  B )
 
Theoremmul2negi 7403 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  x.  -u B )  =  ( A  x.  B )
 
Theoremsubdii 7404 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C ) )
 
Theoremsubdiri 7405 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  x.  C )  =  ( ( A  x.  C )  -  ( B  x.  C ) )
 
Theoremmuladdi 7406 Product of two sums. (Contributed by NM, 17-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  x.  ( C  +  D )
 )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  +  ( ( A  x.  D )  +  ( C  x.  B ) ) )
 
Theoremmulm1d 7407 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
 
Theoremmulneg1d 7408 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  x.  B )  =  -u ( A  x.  B ) )
 
Theoremmulneg2d 7409 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  x.  -u B )  =  -u ( A  x.  B ) )
 
Theoremmul2negd 7410 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  x.  -u B )  =  ( A  x.  B ) )
 
Theoremsubdid 7411 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C ) ) )
 
Theoremsubdird 7412 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  x.  C )  =  ( ( A  x.  C )  -  ( B  x.  C ) ) )
 
Theoremmuladdd 7413 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  x.  ( C  +  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  +  (
 ( A  x.  D )  +  ( C  x.  B ) ) ) )
 
Theoremmulsubd 7414 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  x.  ( C  -  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  -  (
 ( A  x.  D )  +  ( C  x.  B ) ) ) )
 
Theoremmulsubfacd 7415 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  x.  B )  -  B )  =  ( ( A  -  1 )  x.  B ) )
 
3.3.4  Ordering on reals (cont.)
 
Theoremltadd2 7416 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) ) )
 
Theoremltadd2i 7417 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremltadd2d 7418 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) ) )
 
Theoremltadd2dd 7419 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremltletrd 7420 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremlelttrdi 7421 If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
 |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR ) )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  ( A  <  B  ->  A  <  C ) )
 
Theoremgt0ne0 7422 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0
 )
 
Theoremlt0ne0 7423 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  RR  /\  A  <  0
 )  ->  A  =/=  0 )
 
Theoremltadd1 7424 Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C ) ) )
 
Theoremleadd1 7425 Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A  +  C ) 
 <_  ( B  +  C ) ) )
 
Theoremleadd2 7426 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  +  A ) 
 <_  ( C  +  B ) ) )
 
Theoremltsubadd 7427 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  -  B )  <  C  <->  A  <  ( C  +  B ) ) )
 
Theoremltsubadd2 7428 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  -  B )  <  C  <->  A  <  ( B  +  C ) ) )
 
Theoremlesubadd 7429 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  -  B )  <_  C  <->  A  <_  ( C  +  B ) ) )
 
Theoremlesubadd2 7430 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  -  B )  <_  C  <->  A  <_  ( B  +  C ) ) )
 
Theoremltaddsub 7431 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  +  B )  <  C  <->  A  <  ( C  -  B ) ) )
 
Theoremltaddsub2 7432 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  +  B )  <  C  <->  B  <  ( C  -  A ) ) )
 
Theoremleaddsub 7433 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  +  B )  <_  C  <->  A  <_  ( C  -  B ) ) )
 
Theoremleaddsub2 7434 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  +  B )  <_  C  <->  B  <_  ( C  -  A ) ) )
 
Theoremsuble 7435 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  -  B )  <_  C  <->  ( A  -  C )  <_  B ) )
 
Theoremlesub 7436 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  ( B  -  C )  <->  C  <_  ( B  -  A ) ) )
 
Theoremltsub23 7437 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  -  B )  <  C  <->  ( A  -  C )  <  B ) )
 
Theoremltsub13 7438 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  ( B  -  C )  <->  C  <  ( B  -  A ) ) )
 
Theoremle2add 7439 Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <_  C 
 /\  B  <_  D )  ->  ( A  +  B )  <_  ( C  +  D ) ) )
 
Theoremlt2add 7440 Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <  C 
 /\  B  <  D )  ->  ( A  +  B )  <  ( C  +  D ) ) )
 
Theoremltleadd 7441 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <  C 
 /\  B  <_  D )  ->  ( A  +  B )  <  ( C  +  D ) ) )
 
Theoremleltadd 7442 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <_  C 
 /\  B  <  D )  ->  ( A  +  B )  <  ( C  +  D ) ) )
 
Theoremaddgt0 7443 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <  B ) ) 
 ->  0  <  ( A  +  B ) )
 
Theoremaddgegt0 7444 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  0  <  B ) ) 
 ->  0  <  ( A  +  B ) )
 
Theoremaddgtge0 7445 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <_  B ) ) 
 ->  0  <  ( A  +  B ) )
 
Theoremaddge0 7446 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  0  <_  B ) ) 
 ->  0  <_  ( A  +  B ) )
 
Theoremltaddpos 7447 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A 
 <->  B  <  ( B  +  A ) ) )
 
Theoremltaddpos2 7448 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A 
 <->  B  <  ( A  +  B ) ) )
 
Theoremltsubpos 7449 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A 
 <->  ( B  -  A )  <  B ) )
 
Theoremposdif 7450 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 0  <  ( B  -  A ) ) )
 
Theoremlesub1 7451 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A  -  C ) 
 <_  ( B  -  C ) ) )
 
Theoremlesub2 7452 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  -  B ) 
 <_  ( C  -  A ) ) )
 
Theoremltsub1 7453 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A  -  C )  <  ( B  -  C ) ) )
 
Theoremltsub2 7454 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  -  B )  <  ( C  -  A ) ) )
 
Theoremlt2sub 7455 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <  C 
 /\  D  <  B )  ->  ( A  -  B )  <  ( C  -  D ) ) )
 
Theoremle2sub 7456 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <_  C 
 /\  D  <_  B )  ->  ( A  -  B )  <_  ( C  -  D ) ) )
 
Theoremltneg 7457 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A ) )
 
Theoremltnegcon1 7458 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  <  B  <->  -u B  <  A ) )
 
Theoremltnegcon2 7459 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  -u B  <->  B  <  -u A ) )
 
Theoremleneg 7460 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -u B  <_  -u A ) )
 
Theoremlenegcon1 7461 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  <_  B  <->  -u B  <_  A ) )
 
Theoremlenegcon2 7462 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  -u B 
 <->  B  <_  -u A ) )
 
Theoremlt0neg1 7463 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
 |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
 
Theoremlt0neg2 7464 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  -u A  <  0 ) )
 
Theoremle0neg1 7465 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 |-  ( A  e.  RR  ->  ( A  <_  0  <->  0 
 <_  -u A ) )
 
Theoremle0neg2 7466 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  -u A  <_  0 ) )
 
Theoremaddge01 7467 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B 
 <->  A  <_  ( A  +  B ) ) )
 
Theoremaddge02 7468 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B 
 <->  A  <_  ( B  +  A ) ) )
 
Theoremadd20 7469 Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 )
 ) )
 
Theoremsubge0 7470 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  ( A  -  B ) 
 <->  B  <_  A )
 )
 
Theoremsuble0 7471 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  -  B )  <_ 
 0 
 <->  A  <_  B )
 )
 
Theoremleaddle0 7472 The sum of a real number and a second real number is less then the real number iff the second real number is negative. (Contributed by Alexander van der Vekens, 30-May-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  <_  A 
 <->  B  <_  0 )
 )
 
Theoremsubge02 7473 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B 
 <->  ( A  -  B )  <_  A ) )
 
Theoremlesub0 7474 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0 
 <_  A  /\  B  <_  ( B  -  A ) )  <->  A  =  0
 ) )
 
Theoremmullt0 7475 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
 |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
 0  <  ( A  x.  B ) )
 
Theorem0le1 7476 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  0  <_  1
 
Theoremleidi 7477 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  A  <_  A
 
Theoremgt0ne0i 7478 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <  A  ->  A  =/=  0
 )
 
Theoremgt0ne0ii 7479 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  =/=  0
 
Theoremaddgt0i 7480 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  0  <  ( A  +  B )
 )
 
Theoremaddge0i 7481 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  0  <_  ( A  +  B )
 )
 
Theoremaddgegt0i 7482 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <  B )  ->  0  <  ( A  +  B )
 )
 
Theoremaddgt0ii 7483 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  A   &    |-  0  <  B   =>    |-  0  <  ( A  +  B )
 
Theoremadd20i 7484 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
 
Theoremltnegi 7485 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  <->  -u B  <  -u A )
 
Theoremlenegi 7486 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <_  B  <->  -u B  <_  -u A )
 
Theoremltnegcon2i 7487 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  -u B  <->  B  <  -u A )
 
Theoremlesub0i 7488 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  B  <_  ( B  -  A ) )  <->  A  =  0 )
 
Theoremltaddposi 7489 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( 0  <  A  <->  B  <  ( B  +  A ) )
 
Theoremposdifi 7490 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  <->  0  <  ( B  -  A ) )
 
Theoremltnegcon1i 7491 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( -u A  <  B  <->  -u B  <  A )
 
Theoremlenegcon1i 7492 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( -u A  <_  B  <->  -u B  <_  A )
 
Theoremsubge0i 7493 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( 0  <_  ( A  -  B )  <->  B  <_  A )
 
Theoremltadd1i 7494 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C ) )
 
Theoremleadd1i 7495 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <_  B  <->  ( A  +  C )  <_  ( B  +  C ) )
 
Theoremleadd2i 7496 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <_  B  <->  ( C  +  A )  <_  ( C  +  B ) )
 
Theoremltsubaddi 7497 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <  C  <->  A  <  ( C  +  B ) )
 
Theoremlesubaddi 7498 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <_  C  <->  A  <_  ( C  +  B ) )
 
Theoremltsubadd2i 7499 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <  C  <->  A  <  ( B  +  C ) )
 
Theoremlesubadd2i 7500 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <_  C  <->  A  <_  ( B  +  C ) )
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