mmtheorems58 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5701-5800 - Page 58 of 108

Type	Label	Description
Statement
Theorem	muln0 5701	The product of two non-zero numbers is non-zero.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   &   |- B =/= 0   =>   |- (A x. B) =/= 0
Theorem	muleqaddt 5702	Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12.
|- ((A e. CC /\ B e. CC) -> ((A x. B) = (A + B) <-> ((A - 1) x. (B - 1)) = 1))
Theorem	receu 5703	Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- E!x e. CC (A x. x) = B
Theorem	mulnzcnopr 5704	Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)
|- ( x. |` ((CC \ {0}) X. (CC \ {0}))):((CC \ {0}) X. (CC \ {0}))-->(CC \ {0})
Division
Definition	df-div 5705	Define division. Theorem divmul 5707 relates it to multiplication, and divcl 5712 and redivcl 5801 prove its closure laws.
|- / = {<.<.x, y>., z>. | ((x e. CC /\ y e. (CC \ {0})) /\ z = U.{w e. CC | (y x. w) = x})}
Theorem	divval 5706	Value of division: the (unique) element x such that (B x. x) = A. This is meaningful only when B is nonzero.
|- A e. CC   &   |- B e. CC   &   |- B =/= 0   =>   |- (A / B) = U.{x e. CC | (B x. x) = A}
Theorem	divmul 5707	Relationship between division and multiplication.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- B =/= 0   =>   |- ((A / B) = C <-> (B x. C) = A)
Theorem	divmulz 5708	Relationship between division and multiplication.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (B =/= 0 -> ((A / B) = C <-> (B x. C) = A))
Theorem	divmult 5709	Relationship between division and multiplication.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ B =/= 0) -> ((A / B) = C <-> (B x. C) = A))
Theorem	divmul2t 5710	Relationship between division and multiplication.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ B =/= 0) -> ((A / B) = C <-> A = (B x. C)))
Theorem	divmul3t 5711	Relationship between division and multiplication.
|- (((A e. CC /\ B e. CC /\ C e. CC) /\ B =/= 0) -> ((A / B) = C <-> A = (C x. B)))
Theorem	divcl 5712	Closure law for division.
|- A e. CC   &   |- B e. CC   &   |- B =/= 0   =>   |- (A / B) e. CC
Theorem	divclz 5713	Closure law for division.
|- A e. CC   &   |- B e. CC   =>   |- (B =/= 0 -> (A / B) e. CC)
Theorem	divclt 5714	Closure law for division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) e. CC)
Theorem	reccl 5715	Closure law for reciprocal.
|- A e. CC   &   |- A =/= 0   =>   |- (1 / A) e. CC
Theorem	recclz 5716	Closure law for reciprocal.
|- A e. CC   =>   |- (A =/= 0 -> (1 / A) e. CC)
Theorem	recclt 5717	Closure law for reciprocal.
|- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
Theorem	divcan2 5718	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- B =/= 0   =>   |- (B x. (A / B)) = A
Theorem	divcan1 5719	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- B =/= 0   =>   |- ((A / B) x. B) = A
Theorem	divcan1z 5720	A cancellation law for division.
|- A e. CC   &   |- B e. CC   =>   |- (B =/= 0 -> ((A / B) x. B) = A)
Theorem	divcan2z 5721	A cancellation law for division. We eliminate the third hypothesis of divcan2 5718 using the weak deduction theorem dedth 2383 and keep the other two. Because the first hypothesis shares the class variable B with the hypothesis we're eliminating, we need to use keepel 2399 in order to keep the first hypothesis.
|- A e. CC   &   |- B e. CC   =>   |- (B =/= 0 -> (B x. (A / B)) = A)
Theorem	divcan2OLD 5722	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- (A x. (B / A)) = B
Theorem	divcan1OLD 5723	A cancellation law for division.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   =>   |- ((B / A) x. A) = B
Theorem	divcan1zOLD 5724	A cancellation law for division.
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> ((B / A) x. A) = B)
Theorem	divcan2zOLD 5725	A cancellation law for division. We eliminate the third hypothesis of divcan2OLD 5722 using the weak deduction theorem dedth 2383 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 2399 in order to keep the first hypothesis.
|- A e. CC   &   |- B e. CC   =>   |- (A =/= 0 -> (A x. (B / A)) = B)
Theorem	divcan1t 5726	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> ((A / B) x. B) = A)
Theorem	divcan1tOLD 5727	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> ((B / A) x. A) = B)
Theorem	divcan2t 5728	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (B x. (A / B)) = A)
Theorem	divcan2tOLD 5729	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ A =/= 0) -> (A x. (B / A)) = B)
Theorem	divne0bt 5730	The ratio of non-zero numbers is non-zero.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A =/= 0 <-> (A / B) =/= 0))
Theorem	divne0t 5731	The ratio of non-zero numbers is non-zero.
|- (((A e. CC /\ A =/= 0) /\ (B e. CC /\ B =/= 0)) -> (A / B) =/= 0)
Theorem	divne0 5732	The ratio of non-zero numbers is non-zero.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   &   |- B =/= 0   =>   |- (A / B) =/= 0
Theorem	recne0z 5733	The reciprocal of a non-zero number is non-zero.
|- A e. CC   =>   |- (A =/= 0 -> (1 / A) =/= 0)
Theorem	recne0t 5734	The reciprocal of a non-zero number is non-zero.
|- ((A e. CC /\ A =/= 0) -> (1 / A) =/= 0)
Theorem	recid 5735	Multiplication of a number and its reciprocal.
|- A e. CC   &   |- A =/= 0   =>   |- (A x. (1 / A)) = 1
Theorem	recidz 5736	Multiplication of a number and its reciprocal.
|- A e. CC   =>   |- (A =/= 0 -> (A x. (1 / A)) = 1)
Theorem	recidt 5737	Multiplication of a number and its reciprocal.
|- ((A e. CC /\ A =/= 0) -> (A x. (1 / A)) = 1)
Theorem	recid2t 5738	Multiplication of a number and its reciprocal.
|- ((A e. CC /\ A =/= 0) -> ((1 / A) x. A) = 1)
Theorem	divrec 5739	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 5740	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 5741	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 5742	Relationship between division and reciprocal.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) = ((1 / B) x. A))
Theorem	divasst 5743	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 5744	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 5745	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 5746	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 5747	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 5748	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 5749	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 5750	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 5751	Distribution of division over addition.
|- A e. CC