mmtheorems54 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5301-5400 - Page 54 of 108

Type	Label	Description
Statement
Some deductions from the field axioms for complex numbers
Theorem	addclt 5301	Alias for axaddcl 5271, for naming consistency with addcl 5320.
|- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
Theorem	readdclt 5302	Alias for axaddrcl 5272, for naming consistency with readdcl 5334.
|- ((A e. RR /\ B e. RR) -> (A + B) e. RR)
Theorem	mulclt 5303	Alias for axmulcl 5273, for naming consistency with mulcl 5321.
|- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
Theorem	remulclt 5304	Alias for axmulrcl 5274, for naming consistency with remulcl 5335.
|- ((A e. RR /\ B e. RR) -> (A x. B) e. RR)
Theorem	addcomt 5305	Alias for axaddcom 5275, for naming consistency with addcom 5322.
|- ((A e. CC /\ B e. CC) -> (A + B) = (B + A))
Theorem	mulcomt 5306	Alias for axmulcom 5276, for naming consistency with mulcom 5323.
|- ((A e. CC /\ B e. CC) -> (A x. B) = (B x. A))
Theorem	addasst 5307	Alias for axaddass 5277, for naming consistency with addass 5324.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) + C) = (A + (B + C)))
Theorem	mulasst 5308	Alias for axmulass 5278, for naming consistency with mulass 5325.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = (A x. (B x. C)))
Theorem	adddit 5309	Alias for axdistr 5279, for naming consistency with adddi 5326.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B + C)) = ((A x. B) + (A x. C)))
Theorem	addid1t 5310	Alias for ax0id 5281, for naming consistency with addid1 5330.
|- (A e. CC -> (A + 0) = A)
Theorem	mulid1t 5311	Alias for ax1id 5282, for naming consistency with mulid1 5332.
|- (A e. CC -> (A x. 1) = A)
Theorem	reex 5312	The set of real numbers exists.
|- RR e. V
Theorem	recnt 5313	A real number is a complex number.
|- (A e. RR -> A e. CC)
Theorem	recn 5314	A real number is a complex number.
|- A e. RR   =>   |- A e. CC
Theorem	recnd 5315	Deduction from real number to complex number.
|- (ph -> A e. RR)   =>   |- (ph -> A e. CC)
Theorem	elimne0 5316	Hypothesis for weak deduction theorem to eliminate A =/= 0.
|- if(A =/= 0, A, 1) =/= 0
Theorem	addex 5317	The addition operation is a set.
|- + e. V
Theorem	mulex 5318	The multiplication operation is a set.
|- x. e. V
Theorem	adddirt 5319	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 5320	Closure law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) e. CC
Theorem	mulcl 5321	Closure law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) e. CC
Theorem	addcom 5322	Commutative law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) = (B + A)
Theorem	mulcom 5323	Commutative law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) = (B x. A)
Theorem	addass 5324	Associative law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = (A + (B + C))
Theorem	mulass 5325	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 5326	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B + C)) = ((A x. B) + (A x. C))
Theorem	adddir 5327	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) x. C) = ((A x. C) + (B x. C))
Theorem	0cn 5328	0 is a complex number.
|- 0 e. CC
Theorem	addid2t 5329	Identity law for addition.
|- (A e. CC -> (0 + A) = A)
Theorem	addid1 5330	Identity law for addition.
|- A e. CC   =>   |- (A + 0) = A
Theorem	addid2 5331	Identity law for addition.
|- A e. CC   =>   |- (0 + A) = A
Theorem	mulid1 5332	Identity law for multiplication.
|- A e. CC   =>   |- (A x. 1) = A
Theorem	mulid2 5333	Identity law for multiplication.
|- A e. CC   =>   |- (1 x. A) = A
Theorem	readdcl 5334	Closure law for addition of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A + B) e. RR
Theorem	remulcl 5335	Closure law for multiplication of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A x. B) e. RR
Addition
Theorem	add12t 5336	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 5337	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 5338	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 5339	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 5340	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 5341	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 5342	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 5343	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 5344	A theorem for complex numbers analogous the second Peano postulate peano2nn 5935.
|- (A e. CC -> (A + 1) e. CC)
Subtraction
Theorem	cnegextlem1 5345	Lemma for cnegext 5348.
Theorem	cnegextlem2 5346	Lemma for cnegext 5348.
Theorem	cnegextlem3 5347	Lemma for cnegext 5348.
Theorem	cnegext 5348	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 5349	Existence of negatives.
|- A e. CC   =>   |- E.x e. CC (A + x) = 0
Theorem	0cnALT 5350	0 is a complex number. (Proved without referencing ax1cn 5269 by Eric Schmidt, 11-Apr-2007. Compare 0cn 5328.)
|- 0 e. CC
Theorem	addcan 5351	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 5352	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. This proof illustrates how dedth3h 2388 can be used to convert the assumptions of addcan 5351 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 5353	Cancellation law for addition.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + C) = (B + C) <-> A = B))
Theorem	addcan2 5354	Cancellation law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + C) = (B + C) <-> A = B)
Theorem	negeu 5355	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 5356	Define subtraction. Theorem subvalt 5357 shows it value (and describes how this definition works), theorem subadd 5371 relates it to addition, and theorems subcl 5366 and resubcl 5439 prove its closure laws.
|- - = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ z = U.{w e. CC | (y + w) = x})}
Theorem	subvalt 5357	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 5355, allowing us to exploit eusn 2446 and unisn 2517 (which you will find if you trace back the proof of subcl 5366).
|- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})
Definition	df-neg 5358	Define the negative of a number (unary minus). We use different symbols for unary minus (-u) and subtraction (-) to prevent syntax ambiguity. See cneg 5293 for a discussion of this.
|- -uA = (0 - A)
Theorem	negeq 5359	Equality theorem for negatives.
|- (A = B -> -uA = -uB)
Theorem	negeqi 5360	Equality inference for negatives.
|- A = B   =>   |- -uA = -uB
Theorem	negeqd 5361	Equality deduction for negatives.
|- (ph -> A = B)   =>   |- (ph -> -uA = -uB)
Theorem	hbneg 5362	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 5363	Deduction version of hbneg 5362.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. -uA -> A.