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Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnegsubdi2d 7401 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 7402 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 7403 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 7404 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 7405 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 7406 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 7407 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 7408 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 7409 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 7410 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 7411 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 7412 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 7413 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 7414 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 7415 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 7416 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 7417 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 7418 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 7419 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 7420 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 7421 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 7422 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 7423 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 7424 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 7425 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 7426 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 7427 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 7428 Cancellation law for subtraction. Deduction form of subcan 7329. Generalization of subcand 7426. (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 7429 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7426. (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 7430 Cancellation law for subtraction. Deduction form of subcan2 7299. Generalization of subcan2d 7427. (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 7431 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7427. (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 7432 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 7433 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 7434 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 7435 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 7436 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 ) ) )
 
Theoremmvlraddd 7437 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  C )   =>    |-  ( ph  ->  A  =  ( C  -  B ) )
 
Theoremmvrraddd 7438 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  ( A  -  C )  =  B )
 
Theoremsubaddeqd 7439 Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  ( C  +  D ) )   =>    |-  ( ph  ->  ( A  -  D )  =  ( C  -  B ) )
 
Theoremaddlsub 7440 Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  C  <->  A  =  ( C  -  B ) ) )
 
Theoremaddrsub 7441 Right-subtraction: Subtraction of the right summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  C  <->  B  =  ( C  -  A ) ) )
 
Theoremsubexsub 7442 A subtraction law: Exchanging the subtrahend and the result of the subtraction. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  =  ( C  -  B )  <->  B  =  ( C  -  A ) ) )
 
Theoremaddid0 7443 If adding a number to a another number yields the other number, the added number must be  0. This shows that  0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
 |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  +  Y )  =  X  <->  Y  =  0
 ) )
 
Theoremaddn0nid 7444 Adding a nonzero number to a complex number does not yield the complex number. (Contributed by AV, 17-Jan-2021.)
 |-  ( ( X  e.  CC  /\  Y  e.  CC  /\  Y  =/=  0 ) 
 ->  ( X  +  Y )  =/=  X )
 
Theorempnpncand 7445 Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  ( B  -  C ) )  +  ( C  -  B ) )  =  A )
 
Theoremsubeqrev 7446 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  =  ( C  -  D )  <->  ( B  -  A )  =  ( D  -  C ) ) )
 
Theorempncan1 7447 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( A  e.  CC  ->  ( ( A  +  1 )  -  1
 )  =  A )
 
Theoremnpcan1 7448 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
 |-  ( A  e.  CC  ->  ( ( A  -  1 )  +  1
 )  =  A )
 
Theoremsubeq0bd 7449 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 7395. Converse of subeq0d 7393. Contrapositive of subne0ad 7396. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  -  B )  =  0 )
 
Theoremrenegcld 7450 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -u A  e.  RR )
 
Theoremresubcld 7451 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 )
 
3.3.3  Multiplication
 
Theoremkcnktkm1cn 7452 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
 
Theoremmuladd 7453 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 7454 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 7455 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 ) ) )
 
Theoremmul02 7456 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( 0  x.  A )  =  0 )
 
Theoremmul02lem2 7457 Zero times a real is zero. Although we prove it as a corollary of mul02 7456, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 7456. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  RR  ->  ( 0  x.  A )  =  0 )
 
Theoremmul01 7458 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( A  x.  0
 )  =  0 )
 
Theoremmul02i 7459 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( 0  x.  A )  =  0
 
Theoremmul01i 7460 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( A  x.  0 )  =  0
 
Theoremmul02d 7461 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 0  x.  A )  =  0 )
 
Theoremmul01d 7462 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  0 )  =  0 )
 
Theoremine0 7463 The imaginary unit  _i is not zero. (Contributed by NM, 6-May-1999.)
 |-  _i  =/=  0
 
Theoremmulneg1 7464 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 7465 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 7466 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 7467 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 7468 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 7469 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
 |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
 
Theoremmulsub 7470 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 7471 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 7472 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
 |-  A  e.  CC   =>    |-  ( -u 1  x.  A )  =  -u A
 
Theoremmulneg1i 7473 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 7474 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 7475 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 7476 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 7477 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 7478 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 7479 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 7480 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 7481 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 7482 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 7483 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 7484 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 7485 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 7486 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 7487 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 7488 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 7489 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 7490 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 7491 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 7492 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 )
 
Theoremltaddneg 7493 Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  0  <-> 
 ( B  +  A )  <  B ) )
 
Theoremltaddnegr 7494 Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  0  <-> 
 ( A  +  B )  <  B ) )
 
Theoremlelttrdi 7495 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 7496 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 7497 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 7498 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 7499 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 7500 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 ) ) )
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