HomeHome Intuitionistic Logic Explorer
Theorem List (p. 70 of 94)
< 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 - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmul31 6901 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
 CC  CC  C  CC  x.  x.  C  C  x.  x.
 
Theoremmul4 6902 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
 CC  CC  C  CC  D  CC  x.  x.  C  x.  D  x.  C  x.  x.  D
 
Theoremmuladd11 6903 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
 CC  CC  1  +  x.  1  +  1  +  +  +  x.
 
Theorem1p1times 6904 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  1  +  1  x.  +
 
Theorempeano2cn 6905 A theorem for complex numbers analogous the second Peano postulate peano2 4261. (Contributed by NM, 17-Aug-2005.)
 CC  +  1  CC
 
Theorempeano2re 6906 A theorem for reals analogous the second Peano postulate peano2 4261. (Contributed by NM, 5-Jul-2005.)
 RR  +  1  RR
 
Theoremaddcom 6907 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
 CC  CC  +  +
 
Theoremaddid1 6908  0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
 CC  +  0
 
Theoremaddid2 6909  0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 CC  0  +
 
Theoremreaddcan 6910 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
 RR  RR  C  RR  C  +  C  +
 
Theorem00id 6911  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
 0  +  0  0
 
Theoremaddid1i 6912  0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 CC   =>     +  0
 
Theoremaddid2i 6913  0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
 CC   =>     0  +
 
Theoremaddcomi 6914 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 CC   &     CC   =>     +  +
 
Theoremaddcomli 6915 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
 CC   &     CC   &     +  C   =>     +  C
 
Theoremmul12i 6916 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.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  x.  x.  C
 
Theoremmul32i 6917 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  x.  C  x.
 
Theoremmul4i 6918 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
 CC   &     CC   &     C  CC   &     D  CC   =>     x.  x.  C  x.  D  x.  C  x.  x.  D
 
Theoremaddid1d 6919  0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     +  0
 
Theoremaddid2d 6920  0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>    
 0  +
 
Theoremaddcomd 6921 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  +
 
Theoremmul12d 6922 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  x.  x.  C
 
Theoremmul32d 6923 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  x.  C  x.
 
Theoremmul31d 6924 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  C  x.  x.
 
Theoremmul4d 6925 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     x.  x.  C  x.  D  x.  C  x.  x.  D
 
3.3  Real and complex numbers - basic operations
 
3.3.1  Addition
 
Theoremadd12 6926 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
 CC  CC  C  CC  +  +  C  +  +  C
 
Theoremadd32 6927 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
 CC  CC  C  CC  +  +  C  +  C  +
 
Theoremadd32r 6928 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
 CC  CC  C  CC  +  +  C  +  C  +
 
Theoremadd4 6929 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 CC  CC  C  CC  D  CC  +  +  C  +  D  +  C  +  +  D
 
Theoremadd42 6930 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
 CC  CC  C  CC  D  CC  +  +  C  +  D  +  C  +  D  +
 
Theoremadd12i 6931 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 CC   &     CC   &     C  CC   =>     +  +  C  +  +  C
 
Theoremadd32i 6932 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 CC   &     CC   &     C  CC   =>     +  +  C  +  C  +
 
Theoremadd4i 6933 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  +  C  +  D  +  C  +  +  D
 
Theoremadd42i 6934 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  +  C  +  D  +  C  +  D  +
 
Theoremadd12d 6935 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  +  C  +  +  C
 
Theoremadd32d 6936 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  +  C  +  C  +
 
Theoremadd4d 6937 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  +  C  +  D  +  C  +  +  D
 
Theoremadd42d 6938 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  +  C  +  D  +  C  +  D  +
 
3.3.2  Subtraction
 
Syntaxcmin 6939 Extend class notation to include subtraction.
 -
 
Syntaxcneg 6940 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 6939 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 5455, if we used the same symbol then "  -  - " could mean either " - " minus "", or it could represent the (meaningless) operation of classes " - " and " - " connected with "operation" "". On the other hand, " -u  - " is unambiguous.
 -u
 
Definitiondf-sub 6941* Define subtraction. Theorem subval 6960 shows its value (and describes how this definition works), theorem subaddi 7054 relates it to addition, and theorems subcli 7043 and resubcli 7030 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

 -  CC ,  CC  |->  iota_ 
 CC  +
 
Definitiondf-neg 6942 Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 6940 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
 -u  0  -
 
Theoremcnegexlem1 6943 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 6946. (Contributed by Eric Schmidt, 22-May-2007.)
 RR  CC  C  CC  +  +  C  C
 
Theoremcnegexlem2 6944 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 6946. (Contributed by Eric Schmidt, 22-May-2007.)
 RR  _i  x.  RR
 
Theoremcnegexlem3 6945* Existence of real number difference. Lemma for cnegex 6946. (Contributed by Eric Schmidt, 22-May-2007.)
 b 
 RR  RR  c  RR  b  +  c
 
Theoremcnegex 6946* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
 CC  CC  +  0
 
Theoremcnegex2 6947* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 CC  CC  +  0
 
Theoremaddcan 6948 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.)
 CC  CC  C  CC  +  +  C  C
 
Theoremaddcan2 6949 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
 CC  CC  C  CC  +  C  +  C
 
Theoremaddcani 6950 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 CC   &     CC   &     C  CC   =>     +  +  C  C
 
Theoremaddcan2i 6951 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
 CC   &     CC   &     C  CC   =>     +  C  +  C
 
Theoremaddcand 6952 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  +  C  C
 
Theoremaddcan2d 6953 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  C  +  C
 
Theoremaddcanad 6954 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 6952. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     +  +  C   =>     C
 
Theoremaddcan2ad 6955 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 6953. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     +  C  +  C   =>   
 
Theoremaddneintrd 6956 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 6954. Consequence of addcand 6952. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     =/=  C   =>     +  =/=  +  C
 
Theoremaddneintr2d 6957 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 6955. Consequence of addcan2d 6953. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     =/=    =>     +  C  =/=  +  C
 
Theorem0cnALT 6958 Alternate proof of 0cn 6777. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 0  CC
 
Theoremnegeu 6959* 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.)
 CC  CC  CC  +
 
Theoremsubval 6960* Value of subtraction, which is the (unique) element such that  + . (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
 CC  CC  -  iota_  CC  +
 
Theoremnegeq 6961 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
 -u  -u
 
Theoremnegeqi 6962 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
   =>     -u  -u
 
Theoremnegeqd 6963 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
   =>     -u  -u
 
Theoremnfnegd 6964 Deduction version of nfneg 6965. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_   =>     F/_ -u
 
Theoremnfneg 6965 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/_   =>     F/_ -u
 
Theoremcsbnegg 6966 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.)
 V  [_  ]_ -u  -u [_  ]_
 
Theoremsubcl 6967 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
 CC  CC  -  CC
 
Theoremnegcl 6968 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
 CC  -u  CC
 
Theoremnegicn 6969  -u _i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 -u _i  CC
 
Theoremsubf 6970 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 6971 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
 CC  CC  C  CC  -  C  +  C
 
Theoremsubadd2 6972 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  -  C  C  +
 
Theoremsubsub23 6973 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
 CC  CC  C  CC  -  C  -  C
 
Theorempncan 6974 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  +  -
 
Theorempncan2 6975 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
 CC  CC  +  -
 
Theorempncan3 6976 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
 CC  CC  +  -
 
Theoremnpcan 6977 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC 
 -  +
 
Theoremaddsubass 6978 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  +  -  C  +  -  C
 
Theoremaddsub 6979 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 CC  CC  C  CC  +  -  C  -  C  +
 
Theoremsubadd23 6980 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
 CC  CC  C  CC  -  +  C  +  C  -
 
Theoremaddsub12 6981 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
 CC  CC  C  CC  +  -  C  +  -  C
 
Theorem2addsub 6982 Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
 CC  CC  C  CC  D  CC  +  +  C  -  D  +  C  -  D  +
 
Theoremaddsubeq4 6983 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
 CC  CC  C  CC  D  CC  +  C  +  D  C  -  -  D
 
Theorempncan3oi 6984 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 7044 and pncan 6974, 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 7079. (Contributed by David A. Wheeler, 11-Oct-2018.)
 CC   &     CC   =>     +  -
 
Theoremmvlladdi 6985 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 CC   &     CC   &     +  C   =>     C  -
 
Theoremsubid 6986 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  -  0
 
Theoremsubid1 6987 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  -  0
 
Theoremnpncan 6988 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 CC  CC  C  CC  -  +  -  C  -  C
 
Theoremnppcan 6989 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
 CC  CC  C  CC 
 -  +  C  +  +  C
 
Theoremnnpcan 6990 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.)
 CC  CC  C  CC 
 -  -  C  +  -  C
 
Theoremnppcan3 6991 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
 CC  CC  C  CC  -  +  C  +  +  C
 
Theoremsubcan2 6992 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 CC  CC  C  CC  -  C  -  C
 
Theoremsubeq0 6993 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
 CC  CC 
 -  0
 
Theoremnpncan2 6994 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
 CC  CC 
 -  +  -  0
 
Theoremsubsub2 6995 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  -  -  C  +  C  -
 
Theoremnncan 6996 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 CC  CC  -  -
 
Theoremsubsub 6997 Law for double subtraction. (Contributed by NM, 13-May-2004.)
 CC  CC  C  CC  -  -  C 
 -  +  C
 
Theoremnppcan2 6998 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
 CC  CC  C  CC  -  +  C  +  C  -
 
Theoremsubsub3 6999 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
 CC  CC  C  CC  -  -  C  +  C  -
 
Theoremsubsub4 7000 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  -  -  C  -  +  C
    < 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 9301-9381
  Copyright terms: Public domain < Previous  Next >