HomeHome Intuitionistic Logic Explorer
Theorem List (p. 74 of 110)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlelttrd 7301 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremlttrd 7302 Transitive law deduction for 'less than'. (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 )
 
Theorem0lt1 7303 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
 |-  0  <  1
 
3.2.5  Initial properties of the complex numbers
 
Theoremmul12 7304 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) ) )
 
Theoremmul32 7305 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B ) )
 
Theoremmul31 7306 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( C  x.  B )  x.  A ) )
 
Theoremmul4 7307 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
 
Theoremmuladd11 7308 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  ( 1  +  B ) )  =  (
 ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) ) )
 
Theorem1p1times 7309 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( ( 1  +  1 )  x.  A )  =  ( A  +  A ) )
 
Theorempeano2cn 7310 A theorem for complex numbers analogous the second Peano postulate peano2 4344. (Contributed by NM, 17-Aug-2005.)
 |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
 
Theorempeano2re 7311 A theorem for reals analogous the second Peano postulate peano2 4344. (Contributed by NM, 5-Jul-2005.)
 |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
 
Theoremaddcom 7312 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
 
Theoremaddid1 7313  0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Theoremaddid2 7314  0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( 0  +  A )  =  A )
 
Theoremreaddcan 7315 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( C  +  A )  =  ( C  +  B )  <->  A  =  B ) )
 
Theorem00id 7316  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( 0  +  0 )  =  0
 
Theoremaddid1i 7317  0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( A  +  0 )  =  A
 
Theoremaddid2i 7318  0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( 0  +  A )  =  A
 
Theoremaddcomi 7319 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  B )  =  ( B  +  A )
 
Theoremaddcomli 7320 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  ( B  +  A )  =  C
 
Theoremmul12i 7321 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) )
 
Theoremmul32i 7322 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B )
 
Theoremmul4i 7323 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) )
 
Theoremaddid1d 7324  0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  0 )  =  A )
 
Theoremaddid2d 7325  0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 0  +  A )  =  A )
 
Theoremaddcomd 7326 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  B )  =  ( B  +  A ) )
 
Theoremmul12d 7327 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) ) )
 
Theoremmul32d 7328 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B ) )
 
Theoremmul31d 7329 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( ( C  x.  B )  x.  A ) )
 
Theoremmul4d 7330 Rearrangement of 4 factors. (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  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
 
Theoremmuladd11r 7331 A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  1 )  x.  ( B  +  1 ) )  =  ( ( ( A  x.  B )  +  ( A  +  B )
 )  +  1 ) )
 
Theoremcomraddd 7332 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  A  =  ( C  +  B ) )
 
3.3  Real and complex numbers - basic operations
 
3.3.1  Addition
 
Theoremadd12 7333 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( B  +  ( A  +  C ) ) )
 
Theoremadd32 7334 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B ) )
 
Theoremadd32r 7335 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( ( A  +  C )  +  B ) )
 
Theoremadd4 7336 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( B  +  D ) ) )
 
Theoremadd42 7337 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( D  +  B ) ) )
 
Theoremadd12i 7338 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  +  ( B  +  C ) )  =  ( B  +  ( A  +  C )
 )
 
Theoremadd32i 7339 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B )
 
Theoremadd4i 7340 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( B  +  D ) )
 
Theoremadd42i 7341 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( D  +  B ) )
 
Theoremadd12d 7342 Commutative/associative law that swaps the first two terms in a triple sum. (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 )
 ) )
 
Theoremadd32d 7343 Commutative/associative law that swaps the last two terms in a triple sum. (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 ) )
 
Theoremadd4d 7344 Rearrangement of 4 terms in a sum. (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 )
 ) )
 
Theoremadd42d 7345 Rearrangement of 4 terms in a sum. (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 )
 ) )
 
3.3.2  Subtraction
 
Syntaxcmin 7346 Extend class notation to include subtraction.
 class  -
 
Syntaxcneg 7347 Extend class notation to include unary minus. The symbol  -u is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus ( -u) and subtraction cmin 7346 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 5543, if we used the same symbol then " (  -  A  -  B ) " could mean either " -  A " minus " B", or it could represent the (meaningless) operation of classes " - " and " -  B " connected with "operation" " A". On the other hand, " ( -u A  -  B ) " is unambiguous.
 class  -u A
 
Definitiondf-sub 7348* Define subtraction. Theorem subval 7367 shows its value (and describes how this definition works), theorem subaddi 7462 relates it to addition, and theorems subcli 7451 and resubcli 7438 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
 |- 
 -  =  ( x  e.  CC ,  y  e.  CC  |->  ( iota_ z  e. 
 CC  ( y  +  z )  =  x ) )
 
Definitiondf-neg 7349 Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 7347 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
 |-  -u A  =  (
 0  -  A )
 
Theoremcnegexlem1 7350 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7353. (Contributed by Eric Schmidt, 22-May-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremcnegexlem2 7351 Existence of a real number which produces a real number when multiplied by  _i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 7353. (Contributed by Eric Schmidt, 22-May-2007.)
 |- 
 E. y  e.  RR  ( _i  x.  y
 )  e.  RR
 
Theoremcnegexlem3 7352* Existence of real number difference. Lemma for cnegex 7353. (Contributed by Eric Schmidt, 22-May-2007.)
 |-  ( ( b  e. 
 RR  /\  y  e.  RR )  ->  E. c  e.  RR  ( b  +  c )  =  y
 )
 
Theoremcnegex 7353* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
 
Theoremcnegex2 7354* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
 
Theoremaddcan 7355 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremaddcan2 7356 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
 
Theoremaddcani 7357 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  =  ( A  +  C )  <->  B  =  C )
 
Theoremaddcan2i 7358 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  C )  =  ( B  +  C )  <->  A  =  B )
 
Theoremaddcand 7359 Cancellation law for addition. Theorem I.1 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 )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremaddcan2d 7360 Cancellation law for addition. Theorem I.1 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  +  C )  =  ( B  +  C )  <->  A  =  B ) )
 
Theoremaddcanad 7361 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7359. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  ( A  +  C ) )   =>    |-  ( ph  ->  B  =  C )
 
Theoremaddcan2ad 7362 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7360. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  +  C )  =  ( B  +  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremaddneintrd 7363 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7361. Consequence of addcand 7359. (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 ) )
 
Theoremaddneintr2d 7364 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7362. Consequence of addcan2d 7360. (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 ) )
 
Theorem0cnALT 7365 Alternate proof of 0cn 7173. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  0  e.  CC
 
Theoremnegeu 7366* Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E! x  e. 
 CC  ( A  +  x )  =  B )
 
Theoremsubval 7367* Value of subtraction, which is the (unique) element  x such that  B  +  x  =  A. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B )  =  ( iota_ x  e.  CC  ( B  +  x )  =  A ) )
 
Theoremnegeq 7368 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
 |-  ( A  =  B  -> 
 -u A  =  -u B )
 
Theoremnegeqi 7369 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
 |-  A  =  B   =>    |-  -u A  =  -u B
 
Theoremnegeqd 7370 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -u A  =  -u B )
 
Theoremnfnegd 7371 Deduction version of nfneg 7372. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x -u A )
 
Theoremnfneg 7372 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x -u A
 
Theoremcsbnegg 7373 Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
 
Theoremsubcl 7374 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B )  e.  CC )
 
Theoremnegcl 7375 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
 |-  ( A  e.  CC  -> 
 -u A  e.  CC )
 
Theoremnegicn 7376  -u _i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  -u _i  e.  CC
 
Theoremsubf 7377 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 -  : ( CC 
 X.  CC ) --> CC
 
Theoremsubadd 7378 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  C  <->  ( B  +  C )  =  A ) )
 
Theoremsubadd2 7379 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  C  <->  ( C  +  B )  =  A ) )
 
Theoremsubsub23 7380 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  C  <->  ( A  -  C )  =  B ) )
 
Theorempncan 7381 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  B )  =  A )
 
Theorempncan2 7382 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  A )  =  B )
 
Theorempncan3 7383 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  -  A ) )  =  B )
 
Theoremnpcan 7384 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  B )  =  A )
 
Theoremaddsubass 7385 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) ) )
 
Theoremaddsub 7386 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B ) )
 
Theoremsubadd23 7387 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  C )  =  ( A  +  ( C  -  B ) ) )
 
Theoremaddsub12 7388 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  -  C ) )  =  ( B  +  ( A  -  C ) ) )
 
Theorem2addsub 7389 Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( ( A  +  B )  +  C )  -  D )  =  ( (
 ( A  +  C )  -  D )  +  B ) )
 
Theoremaddsubeq4 7390 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theorempncan3oi 7391 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 7452 and pncan 7381, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 7487. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  +  B )  -  B )  =  A
 
Theoremmvrraddi 7392 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  B  e.  CC   &    |-  C  e.  CC   &    |-  A  =  ( B  +  C )   =>    |-  ( A  -  C )  =  B
 
Theoremmvlladdi 7393 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  B  =  ( C  -  A )
 
Theoremsubid 7394 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  A )  =  0 )
 
Theoremsubid1 7395 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  0
 )  =  A )
 
Theoremnpncan 7396 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( B  -  C ) )  =  ( A  -  C ) )
 
Theoremnppcan 7397 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  +  C )  +  B )  =  ( A  +  C ) )
 
Theoremnnpcan 7398 Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  -  C )  +  B )  =  ( A  -  C ) )
 
Theoremnppcan3 7399 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( C  +  B )
 )  =  ( A  +  C ) )
 
Theoremsubcan2 7400 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  ( B  -  C )  <->  A  =  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 10901-10953
  Copyright terms: Public domain < Previous  Next >