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Theorem List for Metamath Proof Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwloglei 9301* Form of wlogle 9302 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  ( ( z  =  x  /\  w  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( z  =  y 
 /\  w  =  x )  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  S 
 C_  RR )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  th )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  ch )   =>    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ch )
 
Theoremwlogle 9302* If the predicate  ch ( x ,  y ) is symmetric under interchange of  x ,  y, then "without loss of generality" we can assume that  x  <_  y. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)
 |-  ( ( z  =  x  /\  w  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( z  =  y 
 /\  w  =  x )  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  S 
 C_  RR )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( ch  <->  th ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  ch )   =>    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
 
Theoremleidi 9303 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  A  <_  A
 
Theoremgt0ne0i 9304 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <  A  ->  A  =/=  0
 )
 
Theoremgt0ne0ii 9305 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  =/=  0
 
Theoremmsqgt0i 9306 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  0  <  ( A  x.  A ) )
 
Theoremmsqge0i 9307 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   =>    |-  0  <_  ( A  x.  A )
 
Theoremaddgt0i 9308 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 9309 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 9310 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 9311 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 9312 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 9313 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 9314 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 9315 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  -u B  <->  B  <  -u A )
 
Theoremmulge0i 9316 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  0  <_  ( A  x.  B ) )
 
Theoremlesub0i 9317 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 9318 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 9319 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 9320 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 9321 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 9322 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( 0  <_  ( A  -  B )  <->  B  <_  A )
 
Theoremltadd1i 9323 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 9324 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 9325 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 9326 '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 9327 '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 9328 '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 9329 '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 ) )
 
Theoremltaddsubi 9330 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  +  B )  <  C  <->  A  <  ( C  -  B ) )
 
Theoremlt2addi 9331 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   =>    |-  ( ( A  <  C 
 /\  B  <  D )  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremle2addi 9332 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   =>    |-  ( ( A  <_  C 
 /\  B  <_  D )  ->  ( A  +  B )  <_  ( C  +  D ) )
 
Theoremgt0ne0d 9333 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremlt0ne0d 9334 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  <  0 )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremleidd 9335 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  A )
 
Theoremmsqgt0d 9336 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  0  <  ( A  x.  A ) )
 
Theoremmsqge0d 9337 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A  x.  A ) )
 
Theoremlt0neg1d 9338 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  <  0  <->  0  <  -u A ) )
 
Theoremlt0neg2d 9339 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 0  <  A  <->  -u A  <  0
 ) )
 
Theoremle0neg1d 9340 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  <_  0  <->  0  <_  -u A ) )
 
Theoremle0neg2d 9341 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 0  <_  A  <->  -u A  <_  0
 ) )
 
Theoremaddgegt0d 9342 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  +  B ) )
 
Theoremaddgt0d 9343 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  +  B ) )
 
Theoremaddge0d 9344 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  ( A  +  B ) )
 
Theoremmulge0d 9345 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  ( A  x.  B ) )
 
Theoremltnegd 9346 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  -u B  <  -u A ) )
 
Theoremlenegd 9347 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  -u B  <_  -u A ) )
 
Theoremltnegcon1d 9348 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  -u A  <  B )   =>    |-  ( ph  ->  -u B  <  A )
 
Theoremltnegcon2d 9349 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  -u B )   =>    |-  ( ph  ->  B  < 
 -u A )
 
Theoremlenegcon1d 9350 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  -u A  <_  B )   =>    |-  ( ph  ->  -u B  <_  A )
 
Theoremlenegcon2d 9351 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  -u B )   =>    |-  ( ph  ->  B  <_ 
 -u A )
 
Theoremltaddposd 9352 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  B  <  ( B  +  A ) ) )
 
Theoremltaddpos2d 9353 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  B  <  ( A  +  B ) ) )
 
Theoremltsubposd 9354 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  ( B  -  A )  <  B ) )
 
Theoremposdifd 9355 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
 
Theoremaddge01d 9356 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( A  +  B ) ) )
 
Theoremaddge02d 9357 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( B  +  A ) ) )
 
Theoremsubge0d 9358 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  ( A  -  B )  <->  B  <_  A ) )
 
Theoremsuble0d 9359 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( A  -  B )  <_  0  <->  A  <_  B ) )
 
Theoremsubge02d 9360 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  ( A  -  B )  <_  A ) )
 
Theoremltadd1d 9361 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C ) ) )
 
Theoremleadd1d 9362 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  +  C )  <_  ( B  +  C ) ) )
 
Theoremleadd2d 9363 Addition to both sides of 'less than or equal to'. (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 ) ) )
 
Theoremltsubaddd 9364 'Less than' relationship between subtraction and addition. (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 ) ) )
 
Theoremlesubaddd 9365 'Less than or equal to' relationship between subtraction and addition. (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 ) ) )
 
Theoremltsubadd2d 9366 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <  C  <->  A  <  ( B  +  C ) ) )
 
Theoremlesubadd2d 9367 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <_  C  <->  A  <_  ( B  +  C ) ) )
 
Theoremltaddsubd 9368 'Less than' relationship between subtraction and addition. (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 ) ) )
 
Theoremltaddsub2d 9369 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <  C  <->  B  <  ( C  -  A ) ) )
 
Theoremleaddsub2d 9370 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <_  C  <->  B  <_  ( C  -  A ) ) )
 
Theoremsubled 9371 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  ( A  -  B )  <_  C )   =>    |-  ( ph  ->  ( A  -  C )  <_  B )
 
Theoremlesubd 9372 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  ( B  -  C ) )   =>    |-  ( ph  ->  C  <_  ( B  -  A ) )
 
Theoremltsub23d 9373 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  ( A  -  B )  <  C )   =>    |-  ( ph  ->  ( A  -  C )  <  B )
 
Theoremltsub13d 9374 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  ( B  -  C ) )   =>    |-  ( ph  ->  C  <  ( B  -  A ) )
 
Theoremlesub1d 9375 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  -  C )  <_  ( B  -  C ) ) )
 
Theoremlesub2d 9376 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  -  B )  <_  ( C  -  A ) ) )
 
Theoremltsub1d 9377 Subtraction from 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  <->  ( A  -  C )  <  ( B  -  C ) ) )
 
Theoremltsub2d 9378 Subtraction of 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  -  B )  <  ( C  -  A ) ) )
 
Theoremltadd1dd 9379 Addition to both sides of 'less than'. Theorem I.18 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  ->  ( A  +  C )  <  ( B  +  C ) )
 
Theoremltsub1dd 9380 Subtraction from 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  ->  ( A  -  C )  < 
 ( B  -  C ) )
 
Theoremltsub2dd 9381 Subtraction of 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  -  B )  < 
 ( C  -  A ) )
 
Theoremleadd1dd 9382 Addition to both sides of 'less than or equal to'. (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 ) )
 
Theoremleadd2dd 9383 Addition to both sides of 'less than or equal to'. (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 ) )
 
Theoremlesub1dd 9384 Subtraction from both sides of 'less than or equal to'. (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 ) )
 
Theoremlesub2dd 9385 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( C  -  B )  <_  ( C  -  A ) )
 
Theoremle2addd 9386 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B 
 <_  D )   =>    |-  ( ph  ->  ( A  +  B )  <_  ( C  +  D ) )
 
Theoremle2subd 9387 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B 
 <_  D )   =>    |-  ( ph  ->  ( A  -  D )  <_  ( C  -  B ) )
 
Theoremltleaddd 9388 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <_  D )   =>    |-  ( ph  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremleltaddd 9389 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B  <  D )   =>    |-  ( ph  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremlt2addd 9390 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <  D )   =>    |-  ( ph  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremlt2subd 9391 Adding both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <  D )   =>    |-  ( ph  ->  ( A  -  D )  < 
 ( C  -  B ) )
 
Theorem1le1 9392  1  <_  1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  1  <_  1
 
5.3.5  Reciprocals
 
Theoremixi 9393  _i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( _i  x.  _i )  =  -u 1
 
Theoremrecextlem1 9394 Lemma for recex 9396. (Contributed by Eric Schmidt, 23-May-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  +  ( B  x.  B ) ) )
 
Theoremrecextlem2 9395 Lemma for recex 9396. (Contributed by Eric Schmidt, 23-May-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) )  =/=  0 )  ->  ( ( A  x.  A )  +  ( B  x.  B ) )  =/=  0 )
 
Theoremrecex 9396* Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  E. x  e.  CC  ( A  x.  x )  =  1
 )
 
Theoremmulcand 9397 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( C  x.  A )  =  ( C  x.  B )  <->  A  =  B ) )
 
Theoremmulcan2d 9398 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  x.  C )  =  ( B  x.  C )  <->  A  =  B ) )
 
Theoremmulcanad 9399 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 9397. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   &    |-  ( ph  ->  ( C  x.  A )  =  ( C  x.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremmulcan2ad 9400 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 9398. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   &    |-  ( ph  ->  ( A  x.  C )  =  ( B  x.  C ) )   =>    |-  ( ph  ->  A  =  B )
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