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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrecdivap2 7701 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremddcanap 7702 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivsubdivap 7704 Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremconjmulap 7705 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremrerecclap 7706 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremredivclap 7707 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremeqneg 7708 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremeqnegd 7709 A complex number equals its negative iff it is zero. Deduction form of eqneg 7708. (Contributed by David Moews, 28-Feb-2017.)

Theoremeqnegad 7710 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 7708. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiv2negap 7711 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivneg2ap 7712 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrecclapzi 7713 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrecap0apzi 7714 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrecidapzi 7715 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdiv1i 7716 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)

Theoremeqnegi 7717 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)

Theoremrecclapi 7718 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
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Theoremrecidapi 7719 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
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Theoremrecrecapi 7720 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
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Theoremdividapi 7721 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
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Theoremdiv0api 7722 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
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Theoremdivclapzi 7723 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap1zi 7724 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap2zi 7725 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivrecapzi 7726 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap3zi 7727 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap4zi 7728 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrec11api 7729 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivclapi 7730 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap2i 7731 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap1i 7732 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivrecapi 7733 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap3i 7734 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap4i 7735 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivap0i 7736 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremrec11apii 7737 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivassapzi 7738 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivmulapzi 7739 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivdirapzi 7740 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivdiv23apzi 7741 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivmulapi 7742 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivdiv32api 7743 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivassapi 7744 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivdirapi 7745 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdiv23api 7746 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdiv11api 7747 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivmuldivapi 7748 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivmul13api 7749 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivdivdivapi 7751 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremrerecclapzi 7752 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremrerecclapi 7753 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremredivclapzi 7754 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremredivclapi 7755 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdiv1d 7756 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecclapd 7757 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecap0d 7758 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecidapd 7759 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecidap2d 7760 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecrecapd 7761 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdividapd 7762 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdiv0apd 7763 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremapmul1 7764 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
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Theoremdivclapd 7765 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap1d 7766 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap2d 7767 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivrecapd 7768 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivrecap2d 7769 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap3d 7770 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap4d 7771 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdiveqap0d 7772 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiveqap1d 7773 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiveqap1ad 7774 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7682. Generalization of diveqap1d 7773. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiveqap0ad 7775 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7661. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivap1d 7776 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
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Theoremdivap0bd 7777 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivnegapd 7778 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivneg2apd 7779 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiv2negapd 7780 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivap0d 7781 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecdivapd 7782 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecdivap2d 7783 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdivcanap6d 7784 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremddcanapd 7785 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrec11apd 7786 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdivmulapd 7787 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdiv32apd 7788 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdiv13apd 7789 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdiv32apd 7790 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivcanap5d 7791 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivcanap5rd 7792 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivcanap7d 7793 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdmdcanapd 7794 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdmdcanap2d 7795 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdivap1d 7796 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdivap2d 7797 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivmulap2d 7798 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivmulap3d 7799 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivassapd 7800 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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