HomeHome Metamath Proof Explorer
Theorem List (p. 95 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-22421)
  Hilbert Space Explorer  Hilbert Space Explorer
(22422-23944)
  Users' Mathboxes  Users' Mathboxes
(23945-32762)
 

Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnegidd 9401 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u A )  =  0 )
 
Theoremnegnegd 9402 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u -u A  =  A )
 
Theoremnegeq0d 9403 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  0  <->  -u A  =  0 ) )
 
Theoremnegne0bd 9404 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =/=  0  <->  -u A  =/=  0
 ) )
 
Theoremnegcon1d 9405 Contraposition law for unary minus. Deduction form of negcon1 9353. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  B  <->  -u B  =  A )
 )
 
Theoremnegcon1ad 9406 Contraposition law for unary minus. One-way deduction form of negcon1 9353. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  =  B )   =>    |-  ( ph  ->  -u B  =  A )
 
Theoremneg11ad 9407 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 9352. Generalization of neg11d 9423. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  -u B 
 <->  A  =  B ) )
 
Theoremnegned 9408 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 9423. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -u A  =/=  -u B )
 
Theoremnegne0d 9409 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  -u A  =/=  0 )
 
Theoremnegrebd 9410 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremsubcld 9411 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  B )  e.  CC )
 
Theorempncand 9412 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  B )  =  A )
 
Theorempncan2d 9413 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  A )  =  B )
 
Theorempncan3d 9414 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( B  -  A ) )  =  B )
 
Theoremnpcand 9415 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  +  B )  =  A )
 
Theoremnncand 9416 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( A  -  B ) )  =  B )
 
Theoremnegsubd 9417 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u B )  =  ( A  -  B ) )
 
Theoremsubnegd 9418 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  -u B )  =  ( A  +  B ) )
 
Theoremsubeq0d 9419 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  0
 )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsubne0d 9420 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  -  B )  =/=  0 )
 
Theoremsubeq0ad 9421 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 9327. Generalization of subeq0d 9419. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  =  0  <->  A  =  B ) )
 
Theoremsubne0ad 9422 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 9420. Contrapositive of subeq0bd 9463. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =/=  0
 )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremneg11d 9423 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  -u A  =  -u B )   =>    |-  ( ph  ->  A  =  B )
 
Theoremnegdid 9424 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  +  B )  =  ( -u A  +  -u B ) )
 
Theoremnegdi2d 9425 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdid 9426 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  -  B )  =  ( -u A  +  B ) )
 
Theoremnegsubdi2d 9427 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2subd 9428 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  -  -u B )  =  ( B  -  A ) )
 
Theoremsubaddd 9429 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  C  <->  ( B  +  C )  =  A ) )
 
Theoremsubadd2d 9430 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  C  <->  ( C  +  B )  =  A ) )
 
Theoremaddsubassd 9431 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) ) )
 
Theoremaddsubd 9432 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B ) )
 
Theoremsubadd23d 9433 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  C )  =  ( A  +  ( C  -  B ) ) )
 
Theoremaddsub12d 9434 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( B  -  C ) )  =  ( B  +  ( A  -  C ) ) )
 
Theoremnpncand 9435 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( B  -  C ) )  =  ( A  -  C ) )
 
Theoremnppcand 9436 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( ( A  -  B )  +  C )  +  B )  =  ( A  +  C ) )
 
Theoremnppcan2d 9437 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  ( B  +  C )
 )  +  C )  =  ( A  -  B ) )
 
Theoremnppcan3d 9438 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( C  +  B ) )  =  ( A  +  C ) )
 
Theoremsubsubd 9439 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( ( A  -  B )  +  C ) )
 
Theoremsubsub2d 9440 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) ) )
 
Theoremsubsub3d 9441 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( ( A  +  C )  -  B ) )
 
Theoremsubsub4d 9442 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  C )  =  ( A  -  ( B  +  C )
 ) )
 
Theoremsub32d 9443 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  C )  =  ( ( A  -  C )  -  B ) )
 
Theoremnnncand 9444 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  ( B  -  C ) )  -  C )  =  ( A  -  B ) )
 
Theoremnnncan1d 9445 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  ( A  -  C ) )  =  ( C  -  B ) )
 
Theoremnnncan2d 9446 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  C )  -  ( B  -  C ) )  =  ( A  -  B ) )
 
Theoremnpncan3d 9447 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( C  -  A ) )  =  ( C  -  B ) )
 
Theorempnpcand 9448 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( A  +  C ) )  =  ( B  -  C ) )
 
Theorempnpcan2d 9449 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  C )  -  ( B  +  C ) )  =  ( A  -  B ) )
 
Theorempnncand 9450 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C ) )
 
Theoremppncand 9451 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  ( C  -  B ) )  =  ( A  +  C ) )
 
Theoremsubcand 9452 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  ( A  -  C ) )   =>    |-  ( ph  ->  B  =  C )
 
Theoremsubcan2d 9453 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  -  C )  =  ( B  -  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsubcanad 9454 Cancellation law for subtraction. Deduction form of subcan 9356. Generalization of subcand 9452. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  ( A  -  C )  <->  B  =  C ) )
 
Theoremsubneintrd 9455 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcand 9452. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  ( A  -  B )  =/=  ( A  -  C ) )
 
Theoremsubcan2ad 9456 Cancellation law for subtraction. Deduction form of subcan2 9326. Generalization of subcan2d 9453. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  C )  =  ( B  -  C )  <->  A  =  B ) )
 
Theoremsubneintr2d 9457 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcan2d 9453. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  -  C )  =/=  ( B  -  C ) )
 
Theoremaddsub4d 9458 Rearrangement of 4 terms in a mixed addition and subtraction. (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 )  -  ( C  +  D ) )  =  ( ( A  -  C )  +  ( B  -  D ) ) )
 
Theoremsubadd4d 9459 Rearrangement of 4 terms in a mixed addition and subtraction. (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 )  -  ( C  -  D ) )  =  ( ( A  +  D )  -  ( B  +  C )
 ) )
 
Theoremsub4d 9460 Rearrangement of 4 terms in a subtraction. (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 )  -  ( C  -  D ) )  =  ( ( A  -  C )  -  ( B  -  D ) ) )
 
Theorem2addsubd 9461 Law for subtraction and addition. (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 )  +  C )  -  D )  =  ( ( ( A  +  C )  -  D )  +  B ) )
 
Theoremaddsubeq4d 9462 Relation between sums and 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 )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theoremsubeq0bd 9463 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 9421. Converse of subeq0d 9419. Contrapositive of subne0ad 9422. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  -  B )  =  0 )
 
Theoremrenegcld 9464 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -u A  e.  RR )
 
Theoremresubcld 9465 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  -  B )  e.  RR )
 
5.3.3  Multiplication
 
Theoremmuladd 9466 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( 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 ) ) ) )
 
Theoremsubdi 9467 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C ) ) )
 
Theoremsubdir 9468 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  x.  C )  =  ( ( A  x.  C )  -  ( B  x.  C ) ) )
 
Theoremine0 9469 The imaginary unit  _i is not zero. (Contributed by NM, 6-May-1999.)
 |-  _i  =/=  0
 
Theoremmulneg1 9470 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  B )  =  -u ( A  x.  B ) )
 
Theoremmulneg2 9471 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  -u B )  =  -u ( A  x.  B ) )
 
Theoremmulneg12 9472 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  B )  =  ( A  x.  -u B ) )
 
Theoremmul2neg 9473 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  -u B )  =  ( A  x.  B ) )
 
Theoremsubmul2 9474 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  x.  C ) )  =  ( A  +  ( B  x.  -u C ) ) )
 
Theoremmulm1 9475 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
 |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
 
Theoremmulsub 9476 Product of two differences. (Contributed by NM, 14-Jan-2006.)
 |-  ( ( ( 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 ) ) ) )
 
Theoremmulsub2 9477 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  x.  ( C  -  D ) )  =  ( ( B  -  A )  x.  ( D  -  C ) ) )
 
Theoremmulm1i 9478 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
 |-  A  e.  CC   =>    |-  ( -u 1  x.  A )  =  -u A
 
Theoremmulneg1i 9479 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 9480 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 9481 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 9482 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 9483 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 9484 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 9485 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 9486 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 9487 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 9488 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 9489 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 9490 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 9491 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 9492 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 ) ) ) )
 
5.3.4  Ordering on reals (cont.)
 
Theoremgt0ne0 9493 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 9494 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 9495 Addition to both sides of 'less than'. (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 9496 Addition to both sides of 'less than or equal to'. (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 9497 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 9498 '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 9499 '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 9500 '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 ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32762
  Copyright terms: Public domain < Previous  Next >