mmtheorems66 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6501-6600 - Page 66 of 175

Type	Label	Description
Statement
Theorem	cnegexlem3 6501	Lemma for cnegex 6502.
Theorem	cnegex 6502	Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
|- (A e. CC -> E.x e. CC (A + x) = 0)
Theorem	cnegexi 6503	Existence of negatives.
|- A e. CC   =>   |- E.x e. CC (A + x) = 0
Theorem	0cnALT 6504	0 is a complex number. (Proved without referencing ax1cn 6422 by Eric Schmidt, 11-Apr-2007. Compare 0cn 6481.)
|- 0 e. CC
Theorem	addcani 6505	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) = (A + C) <-> B = C)
Theorem	addcaniOLD 6506	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) = (A + C) <-> B = C)
Theorem	addcan 6507	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. This proof illustrates how dedth3h 3016 can be used to convert the assumptions of addcani 6505 into antecedents. This general method can be used to convert deductions into theorems as needed.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) = (A + C) <-> B = C))
Theorem	addcan2 6508	Cancellation law for addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + C) = (B + C) <-> A = B))
Theorem	addcan2i 6509	Cancellation law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + C) = (B + C) <-> A = B)
Theorem	negeui 6510	Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- E!x e. CC (A + x) = B
Definition	df-sub 6511	Define subtraction. Theorem subval 6512 shows it value (and describes how this definition works), theorem subaddi 6528 relates it to addition, and theorems subcli 6523 and resubcli 6602 prove its closure laws.
|- - = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ z = (iota_w e. CC(y + w) = x))}
Theorem	subval 6512	Value of subtraction, which is the (unique) element x such that B + x = A.
|- ((A e. CC /\ B e. CC) -> (A - B) = (iota_x e. CC(B + x) = A))
Definition	df-neg 6513	Define the negative of a number (unary minus). We use different symbols for unary minus (-u) and subtraction (-) to prevent syntax ambiguity. See cneg 6446 for a discussion of this.
|- -uA = (0 - A)
Theorem	negeq 6514	Equality theorem for negatives.
|- (A = B -> -uA = -uB)
Theorem	negeqi 6515	Equality inference for negatives.
|- A = B   =>   |- -uA = -uB
Theorem	negeqd 6516	Equality deduction for negatives.
|- (ph -> A = B)   =>   |- (ph -> -uA = -uB)
Theorem	hbneg 6517	Bound-variable hypothesis builder for the negative of a complex number.
|- (y e. A -> A.x y e. A)   =>   |- (y e. -uA -> A.x y e. -uA)
Theorem	hbnegd 6518	Deduction version of hbneg 6517. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. -uA -> A.x y e. -uA))
Theorem	hbnegdOLD 6519	Deduction version of hbneg 6517.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. -uA -> A.x y e. -uA))
Theorem	csbnegg 6520	Move class substitution in and out of the negative of a number. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)
Theorem	csbneggOLD 6521	Move class substitution in and out of the negative of a number.
|- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)
Theorem	negex 6522	A negative is a set.
|- -uA e. _V
Theorem	subcli 6523	Closure law for subtraction.
|- A e. CC   &   |- B e. CC   =>   |- (A - B) e. CC
Theorem	subcl 6524	Closure law for subtraction.
|- ((A e. CC /\ B e. CC) -> (A - B) e. CC)
Theorem	negcl 6525	Closure law for negative.
|- (A e. CC -> -uA e. CC)
Theorem	negcli 6526	Closure law for negative.
|- A e. CC   =>   |- -uA e. CC
Theorem	subopr 6527	Subtraction is an operation on the complex numbers.
|- - :(CC X. CC)-->CC
Theorem	subaddi 6528	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (B + C) = A)
Theorem	subaddrii 6529	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- (B + C) = A   =>   |- (A - B) = C
Theorem	subadd2i 6530	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (C + B) = A)
Theorem	subsub23i 6531	Swap subtrahend and result of subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (A - C) = B)
Theorem	subadd 6532	Relationship between subtraction and addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = C <-> (B + C) = A))
Theorem	subsub23 6533	Swap subtrahend and result of subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = C <-> (A - C) = B))
Theorem	pncan3 6534	Subtraction and addition of equals.
|- ((A e. CC /\ B e. CC) -> (A + (B - A)) = B)
Theorem	pncan3i 6535	Subtraction and addition of equals.
|- A e. CC   &   |- B e. CC   =>   |- (A + (B - A)) = B
Theorem	negid 6536	Addition of a number and its negative.
|- (A e. CC -> (A + -uA) = 0)
Theorem	negidi 6537	Addition of a number and its negative.
|- A e. CC   =>   |- (A + -uA) = 0
Theorem	negsubi 6538	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. CC   &   |- B e. CC   =>   |- (A + -uB) = (A - B)
Theorem	negsubiOLD 6539	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (A + -uB) = (A - B)
Theorem	negsub 6540	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC) -> (A + -uB) = (A - B))
Theorem	addsubass 6541	Associative-type law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) - C) = (A + (B - C)))
Theorem	addsub 6542	Law for addition and subtraction. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) - C) = ((A - C) + B))
Theorem	addsubOLD 6543	Law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) - C) = ((A - C) + B))
Theorem	subadd23 6544	Commutative/associative law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) + C) = (A + (C - B)))
Theorem	addsub12 6545	Commutative/associative law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A + (B - C)) = (B + (A - C)))
Theorem	addsubassi 6546	Associative-type law for subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) - C) = (A + (B - C))
Theorem	addsubi 6547	Law for subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) - C) = ((A - C) + B)
Theorem	2addsub 6548	Law for subtraction and addition.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A + B) + C) - D) = (((A + C) - D) + B))
Theorem	negnegi 6549	A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. CC   =>   |- -u-uA = A
Theorem	negnegiOLD 6550	A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18.
|- A e. CC   =>   |- -u-uA = A
Theorem	subidi 6551	Subtraction of a number from itself.
|- A e. CC   =>   |- (A - A) = 0
Theorem	subid1i 6552	Identity law for subtraction.
|- A e. CC   =>   |- (A - 0) = A
Theorem	negneg 6553	A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18.
|- (A e. CC -> -u-uA = A)
Theorem	subneg 6554	Relationship between subtraction and negative.
|- ((A e. CC /\ B e. CC) -> (A - -uB) = (A + B))
Theorem	subid 6555	Subtraction of a number from itself.
|- (A e. CC -> (A - A) = 0)
Theorem	subid1 6556	Identity law for subtraction.
|- (A e. CC -> (A - 0) = A)
Theorem	pncan 6557	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A + B) - B) = A)
Theorem	pncan2 6558	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A + B) - A) = B)
Theorem	npcan 6559	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A - B) + B) = A)
Theorem	npncan 6560	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) + (B - C)) = (A - C))
Theorem	nppcan 6561	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (((A - B) + C) + B) = (A + C))
Theorem	subcan2 6562	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - C) = (B - C) <-> A = B))
Theorem	subeq0 6563	If the difference between two numbers is zero, they are equal.
|- ((A e. CC /\ B e. CC) -> ((A - B) = 0 <-> A = B))
Theorem	subnegi 6564	Relationship between subtraction and negative.
|- A e. CC   &   |- B e. CC   =>   |- (A - -uB) = (A + B)
Theorem	subeq0i 6565	If the difference between two numbers is zero, they are equal.
|- A e. CC   &   |- B e. CC   =>   |- ((A - B) = 0 <-> A = B)
Theorem	neg11i 6566	Negative is one-to-one.
|- A e. CC   &   |- B e. CC   =>   |- (-uA = -uB <-> A = B)
Theorem	negcon1i 6567	Negative contraposition law.
|- A e. CC   &   |- B e. CC   =>   |- (-uA = B <-> -uB = A)
Theorem	negcon2i 6568	Negative contraposition law.
|- A e. CC   &   |- B e. CC   =>   |- (A = -uB <-> B = -uA)
Theorem	neg11 6569	Negative is one-to-one.
|- ((A e. CC /\ B e. CC) -> (-uA = -uB <-> A = B))
Theorem	negcon1 6570	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (-uA = B <-> -uB = A))
Theorem	negcon2 6571	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (A = -uB <-> B = -uA))
Theorem	subcan 6572	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = (A - C) <-> B = C))
Theorem	subcani 6573	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = (A - C) <-> B = C)
Theorem	subcan2i 6574	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - C) = (B - C) <-> A = B)
Theorem	neg0 6575	Minus 0 equals 0.
|- -u0 = 0
Theorem	renegcli 6576	Closure law for negative of reals. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. RR   =>   |- -uA e. RR
Theorem	renegcliOLD 6577	Closure law for negative of reals.
|- A e. RR   =>   |- -uA e. RR
Multiplication
Theorem	mulid2 6578	Identity law for multiplication. Note: see ax1id 6435 for commuted version.
|- (A e. CC -> (1 x. A) = A)
Theorem	mul12 6579	Commutative/associative law for multiplication.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B x. C)) = (B x. (A x. C)))
Theorem	mul23 6580	Commutative/associative law.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = ((A x. C) x. B))
Theorem	mul4 6581	Rearrangement of 4 factors.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. B) x. (C x. D)) = ((A x. C) x. (B x. D)))
Theorem	muladd 6582	Product of two sums. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. (C + D)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
Theorem	muladdOLD 6583	Product of two sums.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. (C + D)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
Theorem	muladd11 6584	A simple product of sums expansion.
|- ((A e. CC /\ B e. CC) -> ((1 + A) x. (1 + B)) = ((1 + A) + (B + (A x. B))))
Theorem	mul12i 6585	Commutative/associative law that swaps the first two factors in a triple product. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B x. C)) = (B x. (A x. C))
Theorem	mul12iOLD 6586	Commutative/associative law that swaps the first two factors in a triple product.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B x. C)) = (B x. (A x. C))
Theorem	mul23i 6587	Commutative/associative law that swaps the last two factors in a triple product.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A x. B) x. C) = ((A x. C) x. B)
Theorem	mul4i 6588	Rearrangement of 4 factors.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A x. B) x. (C x. D)) = ((A x. C) x. (B x. D))
Theorem	muladdi 6589	Product of two sums.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A + B) x. (C + D)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B)))
Theorem	subdi 6590	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B - C)) = ((A x. B) - (A x. C)))
Theorem	subdir 6591	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) x. C) = ((A x. C) - (B x. C)))
Theorem	subdii 6592	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B - C)) = ((A x. B) - (A x. C))
Theorem	subdiri 6593	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) x. C) = ((A x. C) - (B x. C))
Theorem	mul01i 6594	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (A x. 0) = 0
Theorem	mul02i 6595	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (0 x. A) = 0
Theorem	1p1timesi 6596	Two times a number.
|- A e. CC   =>   |- ((1 + 1) x. A) = (A + A)
Theorem	ine0 6597	The imaginary unit _i is not zero.
|- _i =/= 0
Theorem	1re 6598	1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax1cn 6422, by exploiting properties of the imaginary unit _i. (Contributed by Eric Schmidt, 11-Apr-2007.)
|- 1 e. RR
Theorem	peano2re 6599	A theorem for reals analogous the second Peano postulate peano2nn 7118.
|- (A e. RR -> (A + 1) e. RR)
Theorem	renegcl 6600	Closure law for negative of reals. The weak deduction theorem dedth 3011 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 6576, to an antecedent.
|- (A e. RR -> -uA e. RR)