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**Statement List for Metamath Proof Explorer -** 6901-7000 - Page 70 of 175
Type	Label	Description
Statement

Theorem	divcl 6901	Closure law for division.
$/\$ $/\$

Theorem	reccli 6902	Closure law for reciprocal.


Theorem	recclzi 6903	Closure law for reciprocal.


Theorem	reccl 6904	Closure law for reciprocal.
$/\$

Theorem	divcan2i 6905	A cancellation law for division.


Theorem	divcan1i 6906	A cancellation law for division.


Theorem	divcan1zi 6907	A cancellation law for division.


Theorem	divcan2zi 6908	A cancellation law for division. We eliminate the third hypothesis of divcan2i 6905 using the weak deduction theorem dedth 3011 and keep the other two. Because the first hypothesis shares the class variable with the hypothesis we're eliminating, we need to use keepel 3030 in order to keep the first hypothesis.


Theorem	divcan1 6909	A cancellation law for division.
$/\$ $/\$

Theorem	divcan2 6910	A cancellation law for division.
$/\$ $/\$

Theorem	divne0b 6911	The ratio of nonzero numbers is nonzero.
$/\$ $/\$

Theorem	divne0 6912	The ratio of nonzero numbers is nonzero.
$/\$ $/\$ $/\$

Theorem	divne0i 6913	The ratio of nonzero numbers is nonzero.


Theorem	recne0zi 6914	The reciprocal of a nonzero number is nonzero.


Theorem	recne0 6915	The reciprocal of a nonzero number is nonzero.
$/\$

Theorem	recidi 6916	Multiplication of a number and its reciprocal.


Theorem	recidzi 6917	Multiplication of a number and its reciprocal.


Theorem	recid 6918	Multiplication of a number and its reciprocal.
$/\$

Theorem	recid2 6919	Multiplication of a number and its reciprocal.
$/\$

Theorem	divreci 6920	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18.


Theorem	divreczi 6921	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18.


Theorem	divrec 6922	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18.
$/\$ $/\$

Theorem	divrec2 6923	Relationship between division and reciprocal.
$/\$ $/\$

Theorem	divass 6924	An associative law for division.
$/\$ $/\$ $/\$

Theorem	div23 6925	A commutative/associative law for division.
$/\$ $/\$ $/\$

Theorem	div13 6926	A commutative/associative law for division.
$/\$ $/\$ $/\$

Theorem	div12 6927	A commutative/associative law for division.
$/\$ $/\$ $/\$

Theorem	divasszi 6928	An associative law for division.


Theorem	divassi 6929	An associative law for division.


Theorem	divdiri 6930	Distribution of division over addition.


Theorem	div23i 6931	A commutative/associative law for division.


Theorem	divdirzi 6932	Distribution of division over addition.


Theorem	divdir 6933	Distribution of division over addition.
$/\$ $/\$ $/\$

Theorem	divcan3i 6934	A cancellation law for division.


Theorem	divcan4i 6935	A cancellation law for division.


Theorem	divcan3zi 6936	A cancellation law for division. (Eliminates a hypothesis of divcan3i 6934 with the weak deduction theorem.)


Theorem	divcan4zi 6937	A cancellation law for division.


Theorem	divcan3 6938	A cancellation law for division.
$/\$ $/\$

Theorem	divcan4 6939	A cancellation law for division.
$/\$ $/\$

Theorem	div11i 6940	One-to-one relationship for division.


Theorem	div11 6941	One-to-one relationship for division.
$/\$ $/\$ $/\$

Theorem	divid 6942	A number divided by itself is one.
$/\$

Theorem	div0 6943	Division into zero is zero.
$/\$

Theorem	diveq0 6944	A ratio is zero iff the numerator is zero.
$/\$ $/\$

Theorem	recreci 6945	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18.


Theorem	dividi 6946	A number divided by itself is one.


Theorem	div0i 6947	Division into zero is zero.


Theorem	div1i 6948	A number divided by 1 is itself.


Theorem	div1 6949	A number divided by 1 is itself.


Theorem	divneg 6950	Move negative sign inside of a division.
$/\$ $/\$

Theorem	divsubdir 6951	Distribution of division over subtraction.
$/\$ $/\$ $/\$

Theorem	recrec 6952	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18.
$/\$

Theorem	rec11ii 6953	Reciprocal is one-to-one.


Theorem	rec11i 6954	Reciprocal is one-to-one.
$/\$

Theorem	rec11r 6955	Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)
$/\$ $/\$ $/\$

Theorem	divmuldiv 6956	Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18.
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	divcan5 6957	Cancellation of common factor in a ratio.
$/\$ $/\$ $/\$ $/\$

Theorem	divmul13 6958	Swap the denominators in the product of two ratios.
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	divmul24 6959	Swap the numerators in the product of two ratios.
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	divadddiv 6960	Addition of two ratios. Theorem I.13 of [Apostol] p. 18.
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	divdivdiv 6961	Division of two ratios. Theorem I.15 of [Apostol] p. 18.
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	divdivdivOLD 6962	Division of two ratios. Theorem I.15 of [Apostol] p. 18.
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	divmuldivi 6963	Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18.


Theorem	divmul13i 6964	Swap denominators of two ratios.


Theorem	divadddivi 6965	Addition of two ratios. Theorem I.13 of [Apostol] p. 18.


Theorem	divdivdivi 6966	Division of two ratios. Theorem I.15 of [Apostol] p. 18.


Theorem	recdiv 6967	The reciprocal of a ratio.
$/\$ $/\$ $/\$

Theorem	divcan6 6968	Cancellation of inverted fractions.
$/\$ $/\$ $/\$

Theorem	divdiv23 6969	Swap denominators in a division.
$/\$ $/\$ $/\$ $/\$

Theorem	divdiv23i 6970	Swap denominators in a division.


Theorem	divdiv23zi 6971	Swap denominators in a division.
$/\$

Theorem	divdiv1 6972	Division into a fraction.
$/\$ $/\$ $/\$ $/\$

Theorem	divdiv2 6973	Division by a fraction.
$/\$ $/\$ $/\$ $/\$

Theorem	recdiv2 6974	Division into a reciprocal.
$/\$ $/\$ $/\$

Theorem	conjmul 6975	Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12.
$/\$ $/\$ $/\$

Theorem	redivcli 6976	Closure law for division of reals.


Theorem	redivclzi 6977	Closure law for division of reals.


Theorem	redivcl 6978	Closure law for division of reals.
$/\$ $/\$

Theorem	rereccli 6979	Closure law for reciprocal.


Theorem	rerecclzi 6980	Closure law for reciprocal.


Theorem	rereccl 6981	Closure law for reciprocal.
$/\$

Theorem	eqnegi 6982	A number equal to its negative is zero.


Theorem	eqneg 6983	A number equal to its negative is zero.


Theorem	negeq0 6984	A number is zero iff its negative is zero.


Theorem	negne0bi 6985	A number is nonzero iff its negative is nonzero.


Theorem	negne0i 6986	The negative of a nonzero number is nonzero.


Ordering on reals (cont.)

Theorem	elimgt0 6987	Hypothesis for weak deduction theorem to eliminate .


Theorem	elimge0 6988	Hypothesis for weak deduction theorem to eliminate .


Theorem	ltp1 6989	A number is less than itself plus 1.


Theorem	lep1 6990	A number is less than or equal to itself plus 1.


Theorem	ltp1i 6991	A number is less than itself plus 1.


Theorem	recgt0ii 6992	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21.


Theorem	ltm1 6993	A number minus 1 is less than itself.


Theorem	letrp1 6994	A transitive property of 'less than or equal' and plus 1.
$/\$ $/\$

Theorem	p1le 6995	A transitive property of plus 1 and 'less than or equal'.
$/\$ $/\$

Theorem	prodgt0lem 6996	Lemma for prodgt0i 6997.

Theorem	prodgt0i 6997	Infer that a multiplicand is positive from a nonnegative muliplier and positive product.
$/\$

Theorem	prodge0i 6998	Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product.
$/\$

Theorem	ltmul1ii 6999	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Proof shortened by Paul Chapman, 25-Jan-2008.)


Theorem	ltmul1i 7000	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.

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