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	addid1t 5301	Alias for ax0id 5272, for naming consistency with addid1 5321.
|- (A e. CC -> (A + 0) = A)
Theorem	mulid1t 5302	Alias for ax1id 5273, for naming consistency with mulid1 5323.
|- (A e. CC -> (A x. 1) = A)
Theorem	reex 5303	The set of real numbers exists.
|- RR e. V
Theorem	recnt 5304	A real number is a complex number.
|- (A e. RR -> A e. CC)
Theorem	recn 5305	A real number is a complex number.
|- A e. RR   =>   |- A e. CC
Theorem	recnd 5306	Deduction from real number to complex number.
|- (ph -> A e. RR)   =>   |- (ph -> A e. CC)
Theorem	elimne0 5307	Hypothesis for weak deduction theorem to eliminate A =/= 0.
|- if(A =/= 0, A, 1) =/= 0
Theorem	addex 5308	The addition operation is a set.
|- + e. V
Theorem	mulex 5309	The multiplication operation is a set.
|- x. e. V
Theorem	adddirt 5310	Distributive law for complex numbers.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) x. C) = ((A x. C) + (B x. C)))
Theorem	addcl 5311	Closure law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) e. CC
Theorem	mulcl 5312	Closure law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) e. CC
Theorem	addcom 5313	Commutative law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) = (B + A)
Theorem	mulcom 5314	Commutative law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) = (B x. A)
Theorem	addass 5315	Associative law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = (A + (B + C))
Theorem	mulass 5316	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 5317	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B + C)) = ((A x. B) + (A x. C))
Theorem	adddir 5318	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) x. C) = ((A x. C) + (B x. C))
Theorem	0cn 5319	0 is a complex number.
|- 0 e. CC
Theorem	addid2t 5320	Identity law for addition.
|- (A e. CC -> (0 + A) = A)
Theorem	addid1 5321	Identity law for addition.
|- A e. CC   =>   |- (A + 0) = A
Theorem	addid2 5322	Identity law for addition.
|- A e. CC   =>   |- (0 + A) = A
Theorem	mulid1 5323	Identity law for multiplication.
|- A e. CC   =>   |- (A x. 1) = A
Theorem	mulid2 5324	Identity law for multiplication.
|- A e. CC   =>   |- (1 x. A) = A
Theorem	readdcl 5325	Closure law for addition of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A + B) e. RR
Theorem	remulcl 5326	Closure law for multiplication of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A x. B) e. RR
Addition
Theorem	add12t 5327	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 5328	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 5329	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 5330	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 5331	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 5332	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 5333	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 5334	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 5335	A theorem for complex numbers analogous the second Peano postulate peano2nn 5902.
|- (A e. CC -> (A + 1) e. CC)
Subtraction
Theorem	cnegextlem1 5336	Lemma for cnegext 5339.
Theorem	cnegextlem2 5337	Lemma for cnegext 5339.
Theorem	cnegextlem3 5338	Lemma for cnegext 5339.
Theorem	cnegext 5339	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 5340	Existence of negatives.
|- A e. CC   =>   |- E.x e. CC (A + x) = 0
Theorem	0cnALT 5341	0 is a complex number. (Proved without referencing ax1cn 5260 by Eric Schmidt, 11-Apr-2007. Compare 0cn 5319.)
|- 0 e. CC
Theorem	addcan 5342	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 5343	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 5342 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 5344	Cancellation law for addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + C) = (B + C) <-> A = B))
Theorem	addcan2 5345	Cancellation law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + C) = (B + C) <-> A = B)
Theorem	negeu 5346	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 5347	Define subtraction. Theorem subvalt 5348 shows it value (and describes how this definition works), theorem subadd 5362 relates it to addition, and theorems subcl 5357 and resubcl 5430 prove its closure laws.
|- - = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ z = U.{w e. CC | (y + w) = x})}
Theorem	subvalt 5348	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 5346, allowing us to exploit eusn 2442 and unisn 2513 (which you will find if you trace back the proof of subcl 5357).
|- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})
Definition	df-neg 5349	Define the negative of a number (unary minus). We use different symbols for unary minus (-u) and subtraction (-) to prevent syntax ambiguity. See cneg 5284 for a discussion of this.
|- -uA = (0 - A)
Theorem	negeq 5350	Equality theorem for negatives.
|- (A = B -> -uA = -uB)
Theorem	negeqi 5351	Equality inference for negatives.
|- A = B   =>   |- -uA = -uB
Theorem	negeqd 5352	Equality deduction for negatives.
|- (ph -> A = B)   =>   |- (ph -> -uA = -uB)
Theorem	hbneg 5353	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 5354	Deduction version of hbneg 5353.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. -uA -> A.x y e. -uA))
Theorem	csbnegg 5355	Move class substitution in and out of the negative of a number.
|- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)
Theorem	negex 5356	A negative is a set.
|- -uA e. V
Theorem	subcl 5357	Closure law for subtraction.
|- A e. CC   &   |- B e. CC   =>   |- (A - B) e. CC
Theorem	subclt 5358	Closure law for subtraction.
|- ((A e. CC /\ B e. CC) -> (A - B) e. CC)
Theorem	negclt 5359	Closure law for negative.
|- (A e. CC -> -uA e. CC)
Theorem	negcl 5360	Closure law for negative.
|- A e. CC   =>   |- -uA e. CC
Theorem	subopr 5361	Subtraction is an operation on the complex numbers.
|- - :(CC X. CC)-->CC
Theorem	subadd 5362	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (B + C) = A)
Theorem	subaddri 5363	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- (B + C) = A   =>   |- (A - B) = C
Theorem	subadd2 5364	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (C + B) = A)
Theorem	subsub23 5365	Swap subtrahend and result of subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (A - C) = B)
Theorem	subaddt 5366	Relationship between subtraction and addition.
|- ((A e. CC /\