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	mulid1t 5301	Alias for ax1id 5272, for naming consistency with mulid1 5322.
|- (A e. CC -> (A x. 1) = A)
Theorem	reex 5302	The set of real numbers exists.
|- RR e. V
Theorem	recnt 5303	A real number is a complex number.
|- (A e. RR -> A e. CC)
Theorem	recn 5304	A real number is a complex number.
|- A e. RR   =>   |- A e. CC
Theorem	recnd 5305	Deduction from real number to complex number.
|- (ph -> A e. RR)   =>   |- (ph -> A e. CC)
Theorem	elimne0 5306	Hypothesis for weak deduction theorem to eliminate A =/= 0.
|- if(A =/= 0, A, 1) =/= 0
Theorem	addex 5307	The addition operation is a set.
|- + e. V
Theorem	mulex 5308	The multiplication operation is a set.
|- x. e. V
Theorem	adddirt 5309	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 5310	Closure law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) e. CC
Theorem	mulcl 5311	Closure law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) e. CC
Theorem	addcom 5312	Commutative law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) = (B + A)
Theorem	mulcom 5313	Commutative law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) = (B x. A)
Theorem	addass 5314	Associative law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = (A + (B + C))
Theorem	mulass 5315	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 5316	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B + C)) = ((A x. B) + (A x. C))
Theorem	adddir 5317	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) x. C) = ((A x. C) + (B x. C))
Theorem	0cn 5318	0 is a complex number.
|- 0 e. CC
Theorem	addid2t 5319	Identity law for addition.
|- (A e. CC -> (0 + A) = A)
Theorem	addid1 5320	Identity law for addition.
|- A e. CC   =>   |- (A + 0) = A
Theorem	addid2 5321	Identity law for addition.
|- A e. CC   =>   |- (0 + A) = A
Theorem	mulid1 5322	Identity law for multiplication.
|- A e. CC   =>   |- (A x. 1) = A
Theorem	mulid2 5323	Identity law for multiplication.
|- A e. CC   =>   |- (1 x. A) = A
Theorem	readdcl 5324	Closure law for addition of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A + B) e. RR
Theorem	remulcl 5325	Closure law for multiplication of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A x. B) e. RR
Addition
Theorem	add12t 5326	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 5327	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 5328	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 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) + (D + B)))
Theorem	add12 5330	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 5331	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 5332	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 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) + (D + B))
Theorem	peano2cn 5334	A theorem for complex numbers analogous the second Peano postulate peano2nn 5901.
|- (A e. CC -> (A + 1) e. CC)
Subtraction
Theorem	cnegextlem1 5335	Lemma for cnegext 5338.
Theorem	cnegextlem2 5336	Lemma for cnegext 5338.
Theorem	cnegextlem3 5337	Lemma for cnegext 5338.
Theorem	cnegext 5338	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 5339	Existence of negatives.
|- A e. CC   =>   |- E.x e. CC (A + x) = 0
Theorem	0cnALT 5340	0 is a complex number. (Proved without referencing ax1cn 5259 by Eric Schmidt, 11-Apr-2007. Compare 0cn 5318.)
|- 0 e. CC
Theorem	addcan 5341	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 5342	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 5341 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 5343	Cancellation law for addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + C) = (B + C) <-> A = B))
Theorem	addcan2 5344	Cancellation law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + C) = (B + C) <-> A = B)
Theorem	negeu 5345	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 5346	Define subtraction. Theorem subvalt 5347 shows it value (and describes how this definition works), theorem subadd 5361 relates it to addition, and theorems subcl 5356 and resubcl 5429 prove its closure laws.
|- - = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ z = U.{w e. CC | (y + w) = x})}
Theorem	subvalt 5347	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 5345, allowing us to exploit eusn 2442 and unisn 2513 (which you will find if you trace back the proof of subcl 5356).
|- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})
Definition	df-neg 5348	Define the negative of a number (unary minus). We use different symbols for unary minus (-u) and subtraction (-) to prevent syntax ambiguity. See cneg 5283 for a discussion of this.
|- -uA = (0 - A)
Theorem	negeq 5349	Equality theorem for negatives.
|- (A = B -> -uA = -uB)
Theorem	negeqi 5350	Equality inference for negatives.
|- A = B   =>   |- -uA = -uB
Theorem	negeqd 5351	Equality deduction for negatives.
|- (ph -> A = B)   =>   |- (ph -> -uA = -uB)
Theorem	hbneg 5352	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 5353	Deduction version of hbneg 5352.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. -uA -> A.x y e. -uA))
Theorem	csbnegg 5354	Move class substitution in and out of the negative of a number.
|- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)
Theorem	negex 5355	A negative is a set.
|- -uA e. V
Theorem	subcl 5356	Closure law for subtraction.
|- A e. CC   &   |- B e. CC   =>   |- (A - B) e. CC
Theorem	subclt 5357	Closure law for subtraction.
|- ((A e. CC /\ B e. CC) -> (A - B) e. CC)
Theorem	negclt 5358	Closure law for negative.
|- (A e. CC -> -uA e. CC)
Theorem	negcl 5359	Closure law for negative.
|- A e. CC   =>   |- -uA e. CC
Theorem	subopr 5360	Subtraction is an operation on the complex numbers.
|- - :(CC X. CC)-->CC
Theorem	subadd 5361	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (B + C) = A)
Theorem	subaddri 5362	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- (B + C) = A   =>   |- (A - B) = C
Theorem	subadd2 5363	Relationship between subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (C + B) = A)
Theorem	subsub23 5364	Swap subtrahend and result of subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = C <-> (A - C) = B)
Theorem	subaddt 5365	Relationship between subtraction and addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = C <-> (B + C