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Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubsub23i 7301 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)

Theoremaddsubassi 7302 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)

Theoremsubcani 7304 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremsubcan2i 7305 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theorempnncani 7306 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)

Theoremaddsub4i 7307 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)

Theorem0reALT 7308 Alternate proof of 0re 7027. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnegcld 7309 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubidd 7310 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubid1d 7311 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegidd 7312 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegnegd 7313 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegeq0d 7314 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegne0bd 7315 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegcon1d 7316 Contraposition law for unary minus. Deduction form of negcon1 7263. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegcon1ad 7317 Contraposition law for unary minus. One-way deduction form of negcon1 7263. (Contributed by David Moews, 28-Feb-2017.)

Theoremneg11ad 7318 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7262. Generalization of neg11d 7334. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegned 7319 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7334. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegne0d 7320 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegrebd 7321 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsubcld 7322 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncand 7323 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncan2d 7324 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncan3d 7325 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpcand 7326 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncand 7327 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubd 7328 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubnegd 7329 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubeq0d 7330 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubne0d 7331 Two unequal numbers have nonzero difference. See also subap0d 7631 which is the same thing for apartness rather than negated equality. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremsubeq0ad 7332 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 7237. Generalization of subeq0d 7330. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubne0ad 7333 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7331. Contrapositive of subeq0bd 7377. (Contributed by David Moews, 28-Feb-2017.)

Theoremneg11d 7334 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegdid 7335 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegdi2d 7336 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubdid 7337 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubdi2d 7338 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremneg2subd 7339 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubaddd 7340 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd2d 7341 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubassd 7342 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubd 7343 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd23d 7344 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsub12d 7345 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpncand 7346 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcand 7347 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcan2d 7348 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcan3d 7349 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsubd 7350 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub2d 7351 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub3d 7352 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub4d 7353 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsub32d 7354 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncand 7355 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncan1d 7356 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncan2d 7357 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpncan3d 7358 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnpcand 7359 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnpcan2d 7360 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnncand 7361 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremppncand 7362 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubcand 7363 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubcan2d 7364 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)

Theoremsubcanad 7365 Cancellation law for subtraction. Deduction form of subcan 7266. Generalization of subcand 7363. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubneintrd 7366 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7363. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubcan2ad 7367 Cancellation law for subtraction. Deduction form of subcan2 7236. Generalization of subcan2d 7364. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubneintr2d 7368 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7364. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddsub4d 7369 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd4d 7370 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsub4d 7371 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorem2addsubd 7372 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubeq4d 7373 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubeqrev 7374 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)

Theorempncan1 7375 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)

Theoremnpcan1 7376 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)

Theoremsubeq0bd 7377 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 7332. Converse of subeq0d 7330. Contrapositive of subne0ad 7333. (Contributed by David Moews, 28-Feb-2017.)

Theoremrenegcld 7378 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremresubcld 7379 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)

3.3.3  Multiplication

Theoremkcnktkm1cn 7380 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)

Theoremmuladd 7381 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubdi 7382 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)

Theoremsubdir 7383 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)

Theoremmul02 7384 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)

Theoremmul02lem2 7385 Zero times a real is zero. Although we prove it as a corollary of mul02 7384, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 7384. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul01 7386 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul02i 7387 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)

Theoremmul01i 7388 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul02d 7389 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul01d 7390 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremine0 7391 The imaginary unit is not zero. (Contributed by NM, 6-May-1999.)

Theoremmulneg1 7392 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.)

Theoremmulneg2 7393 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)

Theoremmulneg12 7394 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)

Theoremmul2neg 7395 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.)

Theoremsubmul2 7396 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)

Theoremmulm1 7397 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)

Theoremmulsub 7398 Product of two differences. (Contributed by NM, 14-Jan-2006.)

Theoremmulsub2 7399 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremmulm1i 7400 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)

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