mmtheorems58 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5701-5800 - Page 58 of 107

Type	Label	Description
Statement
Theorem	recclz 5701	Closure law for reciprocal.
|- A e. CC   =>   |- (A =/= 0 -> (1 / A) e. CC)
Theorem	recclt 5702	Closure law for reciprocal.
|- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
Theorem	divcan2 5703	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- (A x. (B / A)) = B
Theorem	divcan1 5704	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- ((B / A) x. A) = B
Theorem	divcan1z 5705	A cancellation law for division.
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> ((B / A) x. A) = B)
Theorem	divcan2z 5706	A cancellation law for division. We eliminate the third hypothesis of divcan2 5703 using the weak deduction theorem dedth 2379 and keep the other two. Because the first hypothesis shares the class variable A with the hypothesis we're eliminating, we need to use keepel 2395 in order to keep the first hypothesis.
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> (A x. (B / A)) = B)
Theorem	divcan1t 5707	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> ((B / A) x. A) = B)
Theorem	divcan2t 5708	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> (A x. (B / A)) = B)
Theorem	divne0bt 5709	The ratio of non-zero numbers is non-zero.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A =/= 0 <-> (A / B) =/= 0))
Theorem	divne0t 5710	The ratio of non-zero numbers is non-zero.
|- (((A e. CC /\ A =/= 0) /\ (B e. CC /\ B =/= 0)) -> (A / B) =/= 0)
Theorem	divne0 5711	The ratio of non-zero numbers is non-zero.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   &   |- B =/= 0   =>   |- (A / B) =/= 0
Theorem	recne0z 5712	The reciprocal of a non-zero number is non-zero.
|- A e. CC   =>   |- (A =/= 0 -> (1 / A) =/= 0)
Theorem	recne0t 5713	The reciprocal of a non-zero number is non-zero.
|- ((A e. CC /\ A =/= 0) -> (1 / A) =/= 0)
Theorem	recid 5714	Multiplication of a number and its reciprocal.
|- A e. CC   &   |- A =/= 0   =>   |- (A x. (1 / A)) = 1
Theorem	recidz 5715	Multiplication of a number and its reciprocal.
|- A e. CC   =>   |- (A =/= 0 -> (A x. (1 / A)) = 1)
Theorem	recidt 5716	Multiplication of a number and its reciprocal.
|- ((A e. CC /\ A =/= 0) -> (A x. (1 / A)) = 1)
Theorem	recid2t 5717	Multiplication of a number and its reciprocal.
|- ((A e. CC /\ A =/= 0) -> ((1 / A) x. A) = 1)
Theorem	divrec 5718	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- B =/= 0   =>   |- (A / B) = (A x. (1 / B))
Theorem	divrecz 5719	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (B =/= 0 -> (A / B) = (A x. (1 / B)))
Theorem	divrect 5720	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) = (A x. (1 / B)))
Theorem	divrec2t 5721	Relationship between division and reciprocal.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) = ((1 / B) x. A))
Theorem	divasst 5722	An associative law for division.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ C =/= 0) -> ((A x. B) / C) = (A x. (B / C)))
Theorem	div23t 5723	A commutative/associative law for division.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ C =/= 0) -> ((A x. B) / C) = ((A / C) x. B))
Theorem	div13t 5724	A commutative/associative law for division.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ B =/= 0) -> ((A / B) x. C) = ((C / B) x. A))
Theorem	div12t 5725	A commutative/associative law for division.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ C =/= 0) -> (A x. (B / C)) = (B x. (A / C)))
Theorem	divassz 5726	An associative law for division.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (C =/= 0 -> ((A x. B) / C) = (A x. (B / C)))
Theorem	divass 5727	An associative law for division.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- C =/= 0   =>   |- ((A x. B) / C) = (A x. (B / C))
Theorem	divdir 5728	Distribution of division over addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- C =/= 0   =>   |- ((A + B) / C) = ((A / C) + (B / C))
Theorem	div23 5729	A commutative/associative law for division.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- C =/= 0   =>   |- ((A x. B) / C) = ((A / C) x. B)
Theorem	divdirz 5730	Distribution of division over addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (C =/= 0 -> ((A + B) / C) = ((A / C) + (B / C)))
Theorem	divdirt 5731	Distribution of division over addition.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ C =/= 0) -> ((A + B) / C) = ((A / C) + (B / C)))
Theorem	divcan3 5732	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- ((A x. B) / A) = B
Theorem	divcan4 5733	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- ((B x. A) / A) = B
Theorem	divcan3z 5734	A cancellation law for division. (Eliminates a hypothesis of divcan3 5732 with the weak deduction theorem.)
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> ((A x. B) / A) = B)
Theorem	divcan4z 5735	A cancellation law for division.
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> ((B x. A) / A) = B)
Theorem	divcan3t 5736	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> ((A x. B) / A) = B)
Theorem	divcan4t 5737	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> ((B x. A) / A) = B)
Theorem	div11 5738	One-to-one relationship for division.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- C =/= 0   =>   |- ((A / C) = (B / C) <-> A = B)
Theorem	div11t 5739	One-to-one relationship for division.
|- ((A e. CC /\ B e. CC /\ (C e. CC /\ C =/= 0)) -> ((A / C) = (B / C) <-> A = B))
Theorem	dividt 5740	A number divided by itself is one.
|- ((A e. CC /\ A =/= 0) -> (A / A) = 1)
Theorem	div0t 5741	Division into zero is zero.
|- ((A e. CC /\ A =/= 0) -> (0 / A) = 0)
Theorem	diveq0t 5742	A ratio is zero iff the numerator is zero.
|- ((A e. CC /\ C e. CC /\ C =/= 0) -> ((A / C) = 0 <-> A = 0))
Theorem	recrec 5743	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18.
|- A e. CC   &   |- A =/= 0   =>   |- (1 / (1 / A)) = A
Theorem	divid 5744	A number divided by itself is one.
|- A e. CC   &   |- A =/= 0   =>   |- (A / A) = 1
Theorem	div0 5745	Division into zero is zero.
|- A e. CC   &   |- A =/= 0   =>   |- (0 / A) = 0
Theorem	div1 5746	A number divided by 1 is itself.
|- A e. CC   =>   |- (A / 1) = A
Theorem	div1t 5747	A number divided by 1 is itself.
|- (A e. CC -> (A / 1) = A)
Theorem	divnegt 5748	Move negative sign inside of a division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> -u(A / B) = (-uA / B))
Theorem	divsubdirt 5749	Distribution of division over subtraction.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ C =/= 0) -> ((A - B) / C) = ((A / C) - (B / C)))
Theorem	recrect 5750	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18.
|- ((A e. CC /\ A =/= 0) -> (1 / (1 / A)) = A)
Theorem	rec11i 5751	Reciprocal is one-to-one.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   &   |- B =/= 0   =>   |- ((1 / A) = (1 / B) <-> A = B)
Theorem	rec11 5752	Reciprocal is one-to-one.
|- A e. CC   &   |- B e. CC   =>   |- ((A =/= 0 /\ B =/= 0) -> ((1 / A) = (1 / B) <-> A = B))
Theorem	rec11rt 5753	Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)
|- (((A e. CC /\ B e. CC) /\ (A =/= 0 /\ B =/= 0)) -> ((1 / A) = B <-> (1 / B) = A))
Theorem	divmuldivt