mmtheorems56 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5501-5600 - Page 56 of 123

Type	Label	Description
Statement
Theorem	cnegexlem3 5501	Lemma for cnegex 5502.
Theorem	cnegex 5502	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 5503	Existence of negatives.
|- A e. CC   =>   |- E.x e. CC (A + x) = 0
Theorem	0cnALT 5504	0 is a complex number. (Proved without referencing ax1cn 5423 by Eric Schmidt, 11-Apr-2007. Compare 0cn 5482.)
|- 0 e. CC
Theorem	addcani 5505	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 5506	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. This proof illustrates how dedth3h 2442 can be used to convert the assumptions of addcani 5505 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 5507	Cancellation law for addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + C) = (B + C) <-> A = B))
Theorem	addcan2i 5508	Cancellation law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + C) = (B + C) <-> A = B)
Theorem	negeui 5509	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 5510	Define subtraction. Theorem subval 5511 shows it value (and describes how this definition works), theorem subaddi 5525 relates it to addition, and theorems subcli 5520 and resubcli 5593 prove its closure laws.
|- - = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ z = U.{w e. CC | (y + w) = x})}
Theorem	subval 5511	Value of subtraction, which is the (unique) element x such that B + x = A. The notation U.{x e. CC | (B + x) = A} may at first seem cryptic but is actually a way of saying "the element x such that B + x = A" (see Theorem 8.17 of [Quine] p. 56); this works because there is only one such x as shown by negeui 5509, allowing us to exploit eusn 2507 and unisn 2583 (which you will find if you trace back the proof of subcli 5520).
|- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})
Definition	df-neg 5512	Define the negative of a number (unary minus). We use different symbols for unary minus (-u) and subtraction (-) to prevent syntax ambiguity. See cneg 5447 for a discussion of this.
|- -uA = (0 - A)
Theorem	negeq 5513	Equality theorem for negatives.
|- (A = B -> -uA = -uB)
Theorem	negeqi 5514	Equality inference for negatives.
|- A = B   =>   |- -uA = -uB
Theorem	negeqd 5515	Equality deduction for negatives.
|- (ph -> A = B)   =>   |- (ph -> -uA = -uB)
Theorem	hbneg 5516	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 5517	Deduction version of hbneg 5516.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. -uA -> A.x y e. -uA))
Theorem	csbnegg 5518	Move class substitution in and out of the negative of a number.
|- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)
Theorem	negex 5519	A negative is a set.
|- -uA e. V
Theorem	subcli 5520	Closure law for subtraction.
|- A e. CC   &   |- B e. CC   =>   |- (A - B) e. CC
Theorem	subcl 5521	Closure law for subtraction.
|- ((A e. CC /\ B e. CC) -> (A - B) e. CC)
Theorem	negcl 5522	Closure law for negative.
|- (A e. CC -> -uA e. CC)
Theorem	negcli 5523	Closure law for negative.
|- A e. CC   =>   |- -uA e. CC
Theorem	subopr 5524	Subtraction is an operation on the complex numbers.
|- - :(CC X. CC)-->CC
Theorem	subaddi 5525	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (B + C) = A)
Theorem	subaddrii 5526	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- (B + C) = A   =>   |- (A - B) = C
Theorem	subadd2i 5527	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (C + B) = A)
Theorem	subsub23i 5528	Swap subtrahend and result of subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (A - C) = B)
Theorem	subadd 5529	Relationship between subtraction and addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = C <-> (B + C) = A))
Theorem	subsub23 5530	Swap subtrahend and result of subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = C <-> (A - C) = B))
Theorem	pncan3 5531	Subtraction and addition of equals.
|- ((A e. CC /\ B e. CC) -> (A + (B - A)) = B)
Theorem	pncan3i 5532	Subtraction and addition of equals.
|- A e. CC   &   |- B e. CC   =>   |- (A + (B - A)) = B
Theorem	negid 5533	Addition of a number and its negative.
|- (A e. CC -> (A + -uA) = 0)
Theorem	negidi 5534	Addition of a number and its negative.
|- A e. CC   =>   |- (A + -uA) = 0
Theorem	negsubi 5535	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (A + -uB) = (A - B)
Theorem	negsub 5536	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC) -> (A + -uB) = (A - B))
Theorem	addsubass 5537	Associative-type law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) - C) = (A + (B - C)))
Theorem	addsub 5538	Law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) - C) = ((A - C) + B))
Theorem	subadd23 5539	Commutative/associative law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) + C) = (A + (C - B)))
Theorem	addsub12 5540	Commutative/associative law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A + (B - C)) = (B + (A - C)))
Theorem	addsubassi 5541	Associative-type law for subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) - C) = (A + (B - C))
Theorem	addsubi 5542	Law for subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) - C) = ((A - C) + B)
Theorem	2addsub 5543	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 5544	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 5545	Subtraction of a number from itself.
|- A e. CC   =>   |- (A - A) = 0
Theorem	subid1i 5546	Identity law for subtraction.
|- A e. CC   =>   |- (A - 0) = A
Theorem	negneg 5547	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 5548	Relationship between subtraction and negative.
|- ((A e. CC /\ B e. CC) -> (A - -uB) = (A + B))
Theorem	subid 5549	Subtraction of a number from itself.
|- (A e. CC -> (A - A) = 0)
Theorem	subid1 5550	Identity law for subtraction.
|- (A e. CC -> (A - 0) = A)
Theorem	pncan 5551	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A + B) - B) = A)
Theorem	pncan2 5552	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A + B) - A) = B)
Theorem	npcan 5553	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A - B) + B) = A)
Theorem	npncan 5554	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) + (B - C)) = (A - C))
Theorem	nppcan 5555	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (((A - B) + C) + B) = (A + C))
Theorem	subcan2 5556	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - C) = (B - C) <-> A = B))
Theorem	subeq0 5557	If the difference between two numbers is zero, they are equal.
|- ((A e. CC /\ B e. CC) -> ((A - B) = 0 <-> A = B))
Theorem	subnegi 5558	Relationship between subtraction and negative.
|- A e. CC   &   |- B e. CC   =>   |- (A - -uB) = (A + B)
Theorem	subeq0i 5559	If the difference between two numbers is zero, they are equal.
|- A e. CC   &   |- B e. CC   =>   |- ((A - B) = 0 <-> A = B)
Theorem	neg11i 5560	Negative is one-to-one.
|- A e. CC   &   |- B e. CC   =>   |- (-uA = -uB <-> A = B)
Theorem	negcon1i 5561	Negative contraposition law.
|- A e. CC   &   |- B e. CC   =>   |- (-uA = B <-> -uB = A)
Theorem	negcon2i 5562	Negative contraposition law.
|- A e. CC   &   |- B e. CC   =>   |- (A = -uB <-> B = -uA)
Theorem	neg11 5563	Negative is one-to-one.
|- ((A e. CC /\ B e. CC) -> (-uA = -uB <-> A = B))
Theorem	negcon1 5564	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (-uA = B <-> -uB = A))
Theorem	negcon2 5565	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (A = -uB <-> B = -uA))
Theorem	subcan 5566	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = (A - C) <-> B = C))
Theorem	subcani 5567	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = (A - C) <-> B = C)
Theorem	subcan2i 5568	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - C) = (B - C) <-> A = B)
Theorem	neg0 5569	Minus 0 equals 0.
|- -u0 = 0
Theorem	renegcli 5570	Closure law for negative of reals.
|- A e. RR   =>   |- -uA e. RR
Multiplication
Theorem	mulid2 5571	Identity law for multiplication. Note: see ax1id 5436 for commuted version.
|- (A e. CC -> (1 x. A) = A)
Theorem	mul12 5572	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 5573	Commutative/associative law.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = ((A x. C) x. B))
Theorem	mul4 5574	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 5575	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 5576	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 5577	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 5578	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 5579	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 5580	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 5581	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 5582	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 5583	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 5584	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 5585	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (A x. 0) = 0
Theorem	mul02i 5586	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (0 x. A) = 0
Theorem	1p1timesi 5587	Two times a number.
|- A e. CC   =>   |- ((1 + 1) x. A) = (A + A)
Theorem	ine0 5588	The imaginary unit i is not zero.
|- i =/= 0
Theorem	1re 5589	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 5423, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.)
|- 1 e. RR
Theorem	peano2re 5590	A theorem for reals analogous the second Peano postulate peano2nn 6080.
|- (A e. RR -> (A + 1) e. RR)
Theorem	renegcl 5591	Closure law for negative of reals. The weak deduction theorem dedth 2437 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 5570, to an antecedent.
|- (A e. RR -> -uA e. RR)
Theorem	resubcl 5592	Closure law for subtraction of reals.
|- ((A e. RR /\ B e. RR) -> (A - B) e. RR)
Theorem	resubcli 5593	Closure law for subtraction of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A - B) e. RR
Theorem	0re 5594	0 is a real number. Proved without referencing 1re 5589. (Contributed by Eric Schmidt, 21-May-2007.)
|- 0 e. RR
Theorem	0reALT 5595	0 is a real number.
|- 0 e. RR
Theorem	peano2rem 5596	"Reverse" second Peano postulate analog for reals.
|- (N e. RR -> (N - 1) e. RR)
Theorem	mul01 5597	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (A x. 0) = 0)
Theorem	mul02 5598	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (0 x. A) = 0)
Theorem	mulneg1i 5599	Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (-uA x. B) = -u(A x. B)
Theorem	mulneg2i 5600	Product with negative is negative of product.
|- A e. CC   &   |- B e. CC   =>   |- (A x. -uB) = -u(A x. B)