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	reccl 5701	Closure law for reciprocal.
|- A e. CC   &   |- A =/= 0   =>   |- (1 / A) e. CC
Theorem	recclz 5702	Closure law for reciprocal.
|- A e. CC   =>   |- (A =/= 0 -> (1 / A) e. CC)
Theorem	recclt 5703	Closure law for reciprocal.
|- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
Theorem	divcan2 5704	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- (A x. (B / A)) = B
Theorem	divcan1 5705	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- ((B / A) x. A) = B
Theorem	divcan1z 5706	A cancellation law for division.
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> ((B / A) x. A) = B)
Theorem	divcan2z 5707	A cancellation law for division. We eliminate the third hypothesis of divcan2 5704 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 5708	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> ((B / A) x. A) = B)
Theorem	divcan2t 5709	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> (A x. (B / A)) = B)
Theorem	divne0bt 5710	The ratio of non-zero numbers is non-zero.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A =/= 0 <-> (A / B) =/= 0))
Theorem	divne0t 5711	The ratio of non-zero numbers is non-zero.
|- (((A e. CC /\ A =/= 0) /\ (B e. CC /\ B =/= 0)) -> (A / B) =/= 0)
Theorem	divne0 5712	The ratio of non-zero numbers is non-zero.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   &   |- B =/= 0   =>   |- (A / B) =/= 0
Theorem	recne0z 5713	The reciprocal of a non-zero number is non-zero.
|- A e. CC   =>   |- (A =/= 0 -> (1 / A) =/= 0)
Theorem	recne0t 5714	The reciprocal of a non-zero number is non-zero.
|- ((A e. CC /\ A =/= 0) -> (1 / A) =/= 0)
Theorem	recid 5715	Multiplication of a number and its reciprocal.
|- A e. CC   &   |- A =/= 0   =>   |- (A x. (1 / A)) = 1
Theorem	recidz 5716	Multiplication of a number and its reciprocal.
|- A e. CC   =>   |- (A =/= 0 -> (A x. (1 / A)) = 1)
Theorem	recidt 5717	Multiplication of a number and its reciprocal.
|- ((A e. CC /\ A =/= 0) -> (A x. (1 / A)) = 1)
Theorem	recid2t 5718	Multiplication of a number and its reciprocal.
|- ((A e. CC /\ A =/= 0) -> ((1 / A) x. A) = 1)
Theorem	divrec 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	divrecz 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	divrect 5721	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 5722	Relationship between division and reciprocal.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) = ((1 / B) x. A))
Theorem	divasst 5723	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 5724	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 5725	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 5726	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 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	divass 5728	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 5729	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 5730	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 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	divdirt 5732	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 5733	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- ((A x. B) / A) = B
Theorem	divcan4 5734	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- ((B x. A) / A) = B
Theorem	divcan3z 5735	A cancellation law for division. (Eliminates a hypothesis of divcan3 5733 with the weak deduction theorem.)
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> ((A x. B) / A) = B)
Theorem	divcan4z 5736	A cancellation law for division.
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> ((B x. A) / A) = B)
Theorem	divcan3t 5737	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> ((A x. B) / A) = B)
Theorem	divcan4t 5738	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> ((B x. A) / A) = B)
Theorem	div11 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	div11t 5740	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 5741	A number divided by itself is one.
|- ((A e. CC /\ A =/= 0) -> (A / A) = 1)
Theorem	div0t 5742	Division into zero is zero.
|- ((A e. CC /\ A =/= 0) -> (0 / A) = 0)
Theorem	diveq0t 5743	A ratio is zero iff the numerator is zero.
|- ((A e. CC /\ C e. CC /\ C =/= 0) -> ((A / C) = 0 <-> A = 0))
Theorem	recrec 5744	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 5745	A number divided by itself is one.
|- A e. CC   &   |- A =/= 0   =>   |- (A / A) = 1
Theorem	div0 5746	Division into zero is zero.
|- A e. CC   &   |- A =/= 0   =>   |- (0 / A) = 0
Theorem	div1 5747	A number divided by 1 is itself.
|- A e. CC   =>   |- (A / 1) = A
Theorem	div1t 5748	A number divided by 1 is itself.
|- (A e. CC -> (A / 1) = A)
Theorem	divnegt 5749	Move negative sign inside of a division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> -u(A / B) = (-uA / B))
Theorem	divsubdirt 5750	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 5751	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 5752	Reciprocal is one-to-one.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   &   |- B =/= 0   =>   |- ((1 / A) = (1 / B) <-> A = B)
Theorem	rec11 5753	Reciprocal is one-to-one.
|- A e. CC   &   |- B e. CC   =>   |- ((A =/= 0 /\ B =/= 0) -> ((1 / A) = (1 / B) <-> A = B))
Theorem	rec11rt 5754	Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)
|- (((A e. CC /\ B e. CC) /\ (A =/= 0 /\ B =/=