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Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrec11i 9501 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)

Theoremdivcli 9502 Closure law for division. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivcan2i 9503 A cancellation law for division. (Contributed by NM, 9-Feb-1995.)

Theoremdivcan1i 9504 A cancellation law for division. (Contributed by NM, 18-May-1999.)

Theoremdivreci 9505 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)

Theoremdivcan3i 9506 A cancellation law for division. (Contributed by NM, 16-Feb-1995.)

Theoremdivcan4i 9507 A cancellation law for division. (Contributed by NM, 18-May-1999.)

Theoremdivne0i 9508 The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995.)

Theoremrec11ii 9509 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)

Theoremdivasszi 9510 An associative law for division. (Contributed by NM, 12-Aug-1999.)

Theoremdivmulzi 9511 Relationship between division and multiplication. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivdirzi 9512 Distribution of division over addition. (Contributed by NM, 31-Jul-2004.)

Theoremdivdiv23zi 9513 Swap denominators in a division. (Contributed by NM, 15-Sep-1999.)

Theoremdivmuli 9514 Relationship between division and multiplication. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivdiv32i 9515 Swap denominators in a division. (Contributed by NM, 15-Sep-1999.)

Theoremdivassi 9516 An associative law for division. (Contributed by NM, 15-Feb-1995.)

Theoremdivdiri 9517 Distribution of division over addition. (Contributed by NM, 16-Feb-1995.)

Theoremdiv23i 9518 A commutative/associative law for division. (Contributed by NM, 3-Sep-1999.)

Theoremdiv11i 9519 One-to-one relationship for division. (Contributed by NM, 20-Aug-2001.)

Theoremdivmuldivi 9520 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 16-Feb-1995.)

Theoremdivmul13i 9521 Swap denominators of two ratios. (Contributed by NM, 6-Aug-1999.)

Theoremdivadddivi 9522 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 21-Feb-1995.)

Theoremdivdivdivi 9523 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 22-Feb-1995.)

Theoremrerecclzi 9524 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremrereccli 9525 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremredivclzi 9526 Closure law for division of reals. (Contributed by NM, 9-May-1999.)

Theoremredivcli 9527 Closure law for division of reals. (Contributed by NM, 9-May-1999.)

Theoremdiv1d 9528 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremreccld 9529 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecne0d 9530 The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecidd 9531 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecid2d 9532 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecrecd 9533 A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdividd 9534 A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv0d 9535 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcld 9536 Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan1d 9537 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan2d 9538 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrecd 9539 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrec2d 9540 Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan3d 9541 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan4d 9542 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq0d 9543 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1d 9544 Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1ad 9545 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 9454. Generalization of diveq1d 9544. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiveq0ad 9546 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 9434. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivne1d 9547 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 9544. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivne0bd 9548 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivnegd 9549 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivneg2d 9550 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv2negd 9551 Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivne0d 9552 The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecdivd 9553 The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecdiv2d 9554 Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan6d 9555 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremddcand 9556 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrec11d 9557 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuld 9558 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv32d 9559 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv13d 9560 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv32d 9561 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5d 9562 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5rd 9563 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdivcan7d 9564 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcand 9565 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcan2d 9566 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivdiv1d 9567 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv2d 9568 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul2d 9569 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul3d 9570 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivassd 9571 An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv12d 9572 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv23d 9573 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdird 9574 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdird 9575 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv11d 9576 One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuldivd 9577 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul13d 9578 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul24d 9579 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivadddivd 9580 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdivd 9581 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuleqd 9582 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdivdivd 9583 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1bd 9584 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 9454. Converse of diveq1d 9544. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiv2sub 9585 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremdiv2subd 9586 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 9585. (Contributed by David Moews, 28-Feb-2017.)

Theoremrereccld 9587 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremredivcld 9588 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubrec 9589 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)

Theoremsubreci 9590 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

Theoremsubrecd 9591 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

5.3.7  Ordering on reals (cont.)

Theoremelimgt0 9592 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremelimge0 9593 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 30-Jul-1999.)

Theoremltp1 9594 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremlep1 9595 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)

Theoremltm1 9596 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)

Theoremlem1 9597 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremletrp1 9598 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)

Theoremp1le 9599 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)

Theoremrecgt0 9600 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)

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