Home Metamath Proof ExplorerTheorem List (p. 90 of 309) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)

Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmul12d 8901 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul32d 8902 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul31d 8903 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul4d 8904 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)

5.3  Real and complex numbers - basic operations

Theoremadd12 8905 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)

Theoremadd32 8906 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)

Theoremadd4 8907 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremadd42 8908 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)

Theoremadd12i 8909 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd32i 8910 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd4i 8911 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)

Theoremadd42i 8912 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)

Theoremadd12d 8913 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd32d 8914 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd4d 8915 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd42d 8916 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.2  Subtraction

Syntaxcmin 8917 Extend class notation to include subtraction.

Syntaxcneg 8918 Extend class notation to include unary minus. The symbol 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 () and subtraction cmin 8917 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5710, 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, " " is unambiguous.

Definitiondf-sub 8919* Define subtraction. Theorem subval 8923 shows it value (and describes how this definition works), theorem subaddi 9013 relates it to addition, and theorems subcli 9002 and resubcli 8989 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Definitiondf-neg 8920 Define the negative of a number (unary minus). We use different symbols for unary minus () and subtraction () to prevent syntax ambiguity. See cneg 8918 for a discussion of this. (Contributed by NM, 10-Feb-1995.)

Theorem0cnALT 8921 0 is a complex number. (Proved without referencing ax-1cn 8675. Compare 0cn 8711.) (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnegeu 8922* 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.)

Theoremsubval 8923* Value of subtraction, which is the (unique) element such that . (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremnegeq 8924 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)

Theoremnegeqi 8925 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)

Theoremnegeqd 8926 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)

Theoremnfnegd 8927 Deduction version of nfneg 8928. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfneg 8928 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbnegg 8929 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.)

Theoremnegex 8930 A negative is a set. (Contributed by NM, 4-Apr-2005.)

Theoremsubcl 8931 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremnegcl 8932 Closure law for negative. (Contributed by NM, 6-Aug-2003.)

Theoremsubf 8933 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremsubadd 8934 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremsubadd2 8935 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubsub23 8936 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)

Theorempncan 8937 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempncan2 8938 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)

Theorempncan3 8939 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)

Theoremnpcan 8940 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsubass 8941 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsub 8942 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremsubadd23 8943 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)

Theoremaddsub12 8944 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremaddsubeq4 8946 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremsubid 8947 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubid1 8948 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnpncan 8949 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremnppcan 8950 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)

Theoremnppcan3 8951 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremsubcan2 8952 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremsubeq0 8953 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)

Theoremnpncan2 8954 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)

Theoremsubsub2 8955 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnncan 8956 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubsub 8957 Law for double subtraction. (Contributed by NM, 13-May-2004.)

Theoremnppcan2 8958 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)

Theoremsubsub3 8959 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremsubsub4 8960 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsub32 8961 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)

Theoremnnncan 8962 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)

Theoremnnncan1 8963 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremnnncan2 8964 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)

Theoremnpncan3 8965 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorempnpcan 8966 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempnpcan2 8967 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)

Theorempnncan 8968 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremppncan 8969 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)

Theoremaddsub4 8970 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)

Theoremsubadd4 8971 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)

Theoremsub4 8972 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)

Theoremneg0 8973 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)

Theoremnegid 8974 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)

Theoremnegsub 8975 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubneg 8976 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegneg 8977 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremneg11 8978 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegcon1 8979 Negative contraposition law. (Contributed by NM, 9-May-2004.)

Theoremnegcon2 8980 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)

Theoremnegeq0 8981 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubcan 8982 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegsubdi 8983 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremnegdi 8984 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremnegdi2 8985 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)

Theoremnegsubdi2 8986 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)

Theoremneg2sub 8987 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)

Theoremrenegcli 8988 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8990 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremresubcli 8989 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremrenegcl 8990 Closure law for negative of reals. The weak deduction theorem dedth 3511 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 8988, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)

Theoremresubcl 8991 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)

Theoremnegreb 8992 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theorempeano2rem 8993 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)

Theoremnegcli 8994 Closure law for negative. (Contributed by NM, 26-Nov-1994.)

Theoremnegidi 8995 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)

Theoremnegnegi 8996 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremsubidi 8997 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)

Theoremsubid1i 8998 Identity law for subtraction. (Contributed by NM, 29-May-1999.)

Theoremnegne0bi 8999 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)

Theoremnegrebi 9000 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)

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 30801-30843
 Copyright terms: Public domain < Previous  Next >