mmtheorems54 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5301-5400 - Page 54 of 107

Type	Label	Description
Statement
Theorem	addcl 5301	Closure law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) e. CC
Theorem	mulcl 5302	Closure law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) e. CC
Theorem	addcom 5303	Commutative law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) = (B + A)
Theorem	mulcom 5304	Commutative law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) = (B x. A)
Theorem	addass 5305	Associative law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = (A + (B + C))
Theorem	mulass 5306	Associative law for multiplication.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A x. B) x. C) = (A x. (B x. C))
Theorem	adddi 5307	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B + C)) = ((A x. B) + (A x. C))
Theorem	adddir 5308	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) x. C) = ((A x. C) + (B x. C))
Theorem	0cn 5309	0 is a complex number.
|- 0 e. CC
Theorem	addid2t 5310	Identity law for addition.
|- (A e. CC -> (0 + A) = A)
Theorem	addid1 5311	Identity law for addition.
|- A e. CC   =>   |- (A + 0) = A
Theorem	addid2 5312	Identity law for addition.
|- A e. CC   =>   |- (0 + A) = A
Theorem	mulid1 5313	Identity law for multiplication.
|- A e. CC   =>   |- (A x. 1) = A
Theorem	mulid2 5314	Identity law for multiplication.
|- A e. CC   =>   |- (1 x. A) = A
Theorem	readdcl 5315	Closure law for addition of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A + B) e. RR
Theorem	remulcl 5316	Closure law for multiplication of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A x. B) e. RR
Addition
Theorem	add12t 5317	Commutative/associative law that swaps the first two terms in a triple sum.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A + (B + C)) = (B + (A + C)))
Theorem	add23t 5318	Commutative/associative law that swaps the last two terms in a triple sum.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) + C) = ((A + C) + B))
Theorem	add4t 5319	Rearrangement of 4 terms in a sum.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) + (C + D)) = ((A + C) + (B + D)))
Theorem	add42t 5320	Rearrangement of 4 terms in a sum.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) + (C + D)) = ((A + C) + (D + B)))
Theorem	add12 5321	Commutative/associative law that swaps the first two terms in a triple sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A + (B + C)) = (B + (A + C))
Theorem	add23 5322	Commutative/associative law that swaps the last two terms in a triple sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = ((A + C) + B)
Theorem	add4 5323	Rearrangement of 4 terms in a sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A + B) + (C + D)) = ((A + C) + (B + D))
Theorem	add42 5324	Rearrangement of 4 terms in a sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A + B) + (C + D)) = ((A + C) + (D + B))
Theorem	peano2cn 5325	A theorem for complex numbers analogous the second Peano postulate peano2nn 5892.
|- (A e. CC -> (A + 1) e. CC)
Subtraction
Theorem	cnegextlem1 5326	Lemma for cnegext 5329.
Theorem	cnegextlem2 5327	Lemma for cnegext 5329.
Theorem	cnegextlem3 5328	Lemma for cnegext 5329.
Theorem	cnegext 5329	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	cnegex 5330	Existence of negatives.
|- A e. CC   =>   |- E.x e. CC (A + x) = 0
Theorem	0cnALT 5331	0 is a complex number. (Proved without referencing ax1cn 5250 by Eric Schmidt, 11-Apr-2007. Compare 0cn 5309.)
|- 0 e. CC
Theorem	addcan 5332	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	addcant 5333	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. This proof illustrates how dedth3h 2384 can be used to convert the assumptions of addcan 5332 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	addcan2t 5334	Cancellation law for addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + C) = (B + C) <-> A = B))
Theorem	addcan2 5335	Cancellation law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + C) = (B + C) <-> A = B)
Theorem	negeu 5336	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 5337	Define subtraction. Theorem subvalt 5338 shows it value (and describes how this definition works), theorem subadd 5352 relates it to addition, and theorems subcl 5347 and resubcl 5420 prove its closure laws.
|- - = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ z = U.{w e. CC | (y + w) = x})}
Theorem	subvalt 5338	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 negeu 5336, allowing us to exploit eusn 2442 and unisn 2512 (which you will find if you trace back the proof of subcl 5347).
|- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})
Definition	df-neg 5339	Define the negative of a number (unary minus). We use different symbols for unary minus (-u) and subtraction (-) to prevent syntax ambiguity. See cneg 5274 for a discussion of this.
|- -uA = (0 - A)
Theorem	negeq 5340	Equality theorem for negatives.
|- (A = B -> -uA = -uB)
Theorem	negeqi 5341	Equality inference for negatives.
|- A = B   =>   |- -uA = -uB
Theorem	negeqd 5342	Equality deduction for negatives.
|- (ph -> A = B)   =>   |- (ph -> -uA = -uB)
Theorem	hbneg 5343	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 5344	Deduction version of hbneg 5343.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. -uA -> A.x y e. -uA))
Theorem	csbnegg 5345	Move class substitution in and out of the negative of a number.
|- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)
Theorem	negex 5346	A negative is a set.
|- -uA e. V
Theorem	subcl 5347	Closure law for subtraction.
|- A e. CC   &   |- B e. CC   =>   |- (A - B) e. CC
Theorem	subclt 5348	Closure law for subtraction.
|- ((A e. CC /\ B e. CC) -> (A - B) e. CC)
Theorem	negclt 5349	Closure law for negative.
|- (A e. CC -> -uA e. CC)
Theorem	negcl 5350	Closure law for negative.
|- A e. CC   =>   |- -uA e. CC
Theorem	subopr 5351	Subtraction is an operation on the complex numbers.
|- - :(CC X. CC)-->CC
Theorem	subadd 5352	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (B + C) = A)
Theorem	subaddri 5353	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- (B + C) = A   =>   |- (A - B) = C
Theorem	subadd2 5354	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (C + B) = A)
Theorem	subsub23 5355	Swap subtrahend and result of subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (A - C) = B)
Theorem	subaddt 5356	Relationship between subtraction and addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = C <-> (B + C) = A))
Theorem	subsub23t 5357	Swap subtrahend and result of subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = C <-> (A - C) = B))
Theorem	pncan3t 5358	Subtraction and addition of equals.
|- ((A e. CC /\ B e. CC) -> (A + (B - A)) = B)
Theorem	pncan3 5359	Subtraction and addition of equals.
|- A e. CC   &   |- B e. CC   =>   |- (A + (B - A)) = B
Theorem	negidt 5360	Addition of a number and its negative.
|- (A e. CC -> (A + -uA) = 0)
Theorem	negid 5361	Addition of a number and its negative.
|- A e. CC   =>   |- (A + -uA) = 0
Theorem	negsub 5362	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (A + -uB) = (A -