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Theorem List for Metamath Proof Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlesubadd2i 9301 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)

Theoremltaddsubi 9302 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)

Theoremlt2addi 9303 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremle2addi 9304 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0d 9305 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0d 9306 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremleidd 9307 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmsqgt0d 9308 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmsqge0d 9309 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg1d 9310 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg2d 9311 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg1d 9312 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg2d 9313 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgegt0d 9314 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgt0d 9315 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge0d 9316 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulge0d 9317 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegd 9318 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegd 9319 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon1d 9320 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon2d 9321 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon1d 9322 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon2d 9323 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddposd 9324 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddpos2d 9325 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubposd 9326 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremposdifd 9327 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge01d 9328 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge02d 9329 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge0d 9330 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsuble0d 9331 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge02d 9332 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1d 9333 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd1d 9334 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd2d 9335 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubaddd 9336 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubaddd 9337 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2d 9338 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2d 9339 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsubd 9340 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsub2d 9341 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremleaddsub2d 9342 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubled 9343 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubd 9344 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub23d 9345 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub13d 9346 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub1d 9347 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub2d 9348 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub1d 9349 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub2d 9350 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1dd 9351 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub1dd 9352 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub2dd 9353 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd1dd 9354 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd2dd 9355 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub1dd 9356 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub2dd 9357 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremle2addd 9358 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle2subd 9359 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltleaddd 9360 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltaddd 9361 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2addd 9362 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2subd 9363 Adding both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theorem1le1 9364 . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

5.3.5  Reciprocals

Theoremixi 9365 times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrecextlem1 9366 Lemma for recex 9368. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecextlem2 9367 Lemma for recex 9368. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecex 9368* Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremmulcand 9369 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcan2d 9370 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcanad 9371 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 9369. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan2ad 9372 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 9370. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan 9373 Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcan2 9374 Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcani 9375 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.)

Theoremmul0or 9376 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulne0b 9377 The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmulne0 9378 The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.)

Theoremmulne0i 9379 The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.)

Theoremmuleqadd 9380 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)

Theoremreceu 9381* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremmulnzcnopr 9382 Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)

Theoremmsq0i 9383 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.)

Theoremmul0ori 9384 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 7-Oct-1999.)

Theoremmsq0d 9385 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul0ord 9386 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0bd 9387 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0d 9388 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0bad 9389 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9388 and consequence of mulne0bd 9387. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulne0bbd 9390 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9388 and consequence of mulne0bd 9387. (Contributed by David Moews, 28-Feb-2017.)

5.3.6  Division

Syntaxcdiv 9391 Extend class notation to include division.

Definitiondf-div 9392* Define division. Theorem divmuli 9482 relates it to multiplication, and divcli 9470 and redivcli 9495 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.)

Theorem1div0 9393 You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.)

Theoremdivval 9394* Value of division: the (unique) element such that . This is meaningful only when is nonzero. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivmul 9395 Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivmul2 9396 Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006.)

Theoremdivmul3 9397 Relationship between division and multiplication. (Contributed by NM, 13-Feb-2006.)

Theoremdivcl 9398 Closure law for division. (Contributed by NM, 21-Jul-2001.) (Proof shortened by Mario Carneiro, 17-Feb-2014.)

Theoremreccl 9399 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremdivcan2 9400 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Revised by Mario Carneiro, 27-May-2016.)

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