mmtheorems53 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10670

Statement List for Metamath Proof Explorer - 5201-5300 - Page 53 of 107

Type	Label	Description
Statement
Theorem	mappsrpr 5201	Mapping from positive signed reals to positive reals.
|- A e. V   =>   |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)
Theorem	ltpsrpr 5202	Mapping of order from positive signed reals to positive reals.
|- A e. V   &   |- B e. V   =>   |- ([<.(A +P. 1P), 1P>.] ~R <R [<.(B +P. 1P), 1P>.] ~R <-> A <P B)
Theorem	map2psrpr 5203	Equivalence for positive signed real.
|- A e. V   =>   |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
Theorem	suppsrlem 5204	Mapping of non-empty subset from positive reals to positive signed reals.
|- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}   =>   |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> (B (_ P. /\ -. B = (/)))
Theorem	suppsr 5205	A non-empty, bounded set of positive signed reals has a supremum.
|- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}   =>   |- (((A.x(x e. A -> 0R <R x) /\ -. A = (/)) /\ E.x(0R <R x /\ A.y(0R <R y -> (y e. A -> y <R x)))) -> E.x(0R <R x /\ A.y(0R <R y -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(0R <R z /\ (z e. A /\ y <R z)))))))
Theorem	suppsr2 5206	A non-empty, bounded set of positive signed reals has a supremum. (Converts quantifier restrictions to all reals.)
|- (((A.x(x e. A -> 0R <R x) /\ -. A = (/)) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Theorem	suppsr3 5207	A non-empty, bounded set with at least one positive real has a supremum.
|- B = {y | (y e. A /\ 0R <R y)}   =>   |- ((E.y(y e. A /\ 0R <R y) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Theorem	supsrlem1 5208	Lemma for supremum theorem.
Theorem	supsrlem2 5209	Lemma for supremum theorem.
Theorem	supsrlem3 5210	Lemma for supremum theorem.
Theorem	supsrlem4 5211	Lemma for supremum theorem.
Theorem	supsrlem5 5212	Lemma for supremum theorem.
Theorem	supsrlem6 5213	Lemma for supremum theorem.
Theorem	supsr 5214	A non-empty, bounded set of signed reals has a supremum.
|- (((A (_ R. /\ -. A = (/)) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Syntax	cc 5215	Class of complex numbers.
class CC
Syntax	cr 5216	Class of real numbers.
class RR
Syntax	cc0 5217	Extend class notation to include the complex number 0.
class 0
Syntax	c1 5218	Extend class notation to include the complex number 1.
class 1
Syntax	ci 5219	Extend class notation to include the complex number i.
class i
Syntax	caddc 5220	Addition on complex numbers.
class +
Syntax	cltrr 5221	'Less than' predicate (defined over real subset of complex numbers).
class <R
Syntax	cmul 5222	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 5223	Define the set of complex numbers. The 25 axioms for complex numbers start at axcnex 5250.
|- CC = (R. X. R.)
Definition	df-0 5224	Define the complex number 0 (base 10).
|- 0 = <.0R, 0R>.
Definition	df-1 5225	Define the complex number 1 (base 10).
|- 1 = <.1R, 0R>.
Definition	df-i 5226	Define the complex number i (the imaginary unit).
|- i = <.0R, 1R>.
Definition	df-r 5227	Define the set of real numbers.
|- RR = (R. X. {0R})
Definition	df-plus 5228	Define addition over complex numbers.
|- + = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +R u), (v +R f)>.))}
Definition	df-mul 5229	Define multiplication over complex numbers.
|- x. = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.))}
Definition	df-lt 5230	Define 'less than' on the real subset of complex numbers.
|- <R = {<.x, y>. | ((x e. RR /\ y e. RR) /\ E.zE.w((x = <.z, 0R>. /\ y = <.w, 0R>.) /\ z <R w))}
Theorem	opelcn 5231	Ordered pair membership in the class of complex numbers.
|- B e. V   =>   |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
Theorem	opelreal 5232	Ordered pair membership in class of real subset of complex numbers.
|- (<.A, 0R>. e. RR <-> A e. R.)
Theorem	elreal 5233	Membership in class of real numbers.
|- (A e. RR <-> E.x(x e. R. /\ <.x, 0R>. = A))
Theorem	0ncn 5234	The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property.
|- -. (/) e. CC
Theorem	ltrelre 5235	'Less than' is a relation on real numbers.
|- <R (_ (RR X. RR)
Theorem	addcnsr 5236	Addition of complex numbers in terms of signed reals.
|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> (<.A, B>. + <.C, D>.) = <.(A +R C), (B +R D)>.)
Theorem	mulcnsr 5237	Multiplication of complex numbers in terms of signed reals.
|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> (<.A, B>. x. <.C, D>.) = <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.)
Theorem	eqresr 5238	Equality of real numbers in terms of intermediate signed reals.
|- A e. V   =>   |- (<.A, 0R>. = <.B, 0R>. <-> A = B)
Theorem	addresr 5239	Addition of real numbers in terms of intermediate signed reals.
|- ((A e. R. /\ B e. R.) -> (<.A, 0R>. + <.B, 0R>.) = <.(A +R B), 0R>.)
Theorem	mulresr 5240	Multiplication of real numbers in terms of intermediate signed reals.
|- B e. V   =>   |- ((A e. R. /\ B e. R.) -> (<.A, 0R>. x. <.B, 0R>.) = <.(A .R B), 0R>.)
Theorem	ltresr 5241	Ordering of real subset of complex numbers in terms of signed reals.
|- A e. V   &   |- B e. V   =>   |- (<.A, 0R>. <R <.B, 0R>. <-> A <R B)
Theorem	suprelem 5242	Mapping of non-empty subset from signed reals to reals.
|- B = {w | <.w, 0R>. e. A}   =>   |- ((A (_ RR /\ -. A = (/)) -> (B (_ R. /\ -. B = (/)))
Theorem	supre 5243	A non-empty, bounded-above set of reals has a supremum.
|- B = {w | <.w, 0R>. e. A}   =>   |- (((A (_ RR /\ -. A = (/)) /\ E.x(x e. RR /\ A.y(y e. RR -> (y e. A -> y <R x)))) -> E.x(x e. RR /\ A.y(y e. RR -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. RR /\ (z e. A /\ y <R z)))))))
Theorem	ltsor 5244	'Less than' is a strict ordering on real subset of complex numbers. Note: use ltso 5495 and not this one after the complex number postulates are derived, in order to maintain a "clean" derivation of complex number theorems directly from postulates. The artificial right conjunct is intended to help discourage its accidental use in place of ltso 5495.
|- ( <R Or RR /\ RR = RR)
Theorem	dfcnqs 5245	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 4294, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that CC is a quotient set, even though it is not (compare df-c 5223), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc.
|- CC = ((R. X. R.)/.`'E)
Theorem	addcnsrec 5246	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 5245 and mulcnsrec 5247.
|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> ([<.A, B>.]`'E + [<.C, D>.]`'E) = [<.(A +R C), (B +R D)>.]`'E)
Theorem	mulcnsrec 5247	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4293, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 5245. Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 4955.
|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> ([<.A, B>.]`'E x. [<.C, D>.]`'E) = [<.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.]`'E)
Real and complex number postulates
Theorem	axaddopr 5248	Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 5254.
|- + :(CC X. CC)-->CC
Theorem	axmulopr 5249	Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 5256.
|- x. :(CC X. CC)-->CC
Theorem	axcnex 5250	The class of complex numbers is a set, i.e. it is a member of the universe of sets V (see isset 1811). Axiom 1 of 25 for real and complex numbers, derived from ZF set theory.
|- CC e. V
Theorem	axresscn 5251	The real numbers are a subset of the complex numbers. Axiom 2 of 25 for real and complex numbers, derived from ZF set theory.
|- RR (_ CC
Theorem	ax1cn 5252	1 is a complex number. Axiom 3 of 25 for real and complex numbers, derived from ZF set theory.
|- 1 e. CC
Theorem	axicn 5253	i is a complex number. Axiom 4 of 25 for real and complex numbers, derived from ZF set theory.
|- i e. CC
Theorem	axaddcl 5254	Closure law for addition of complex numbers. Axiom 5 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC) -> (A + B) e. CC