mmtheorems50 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4901-5000 - Page 50 of 107

Type	Label	Description
Statement
Theorem	cardcf 4901	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.
|- (card` (cf` A)) = (cf` A)
Theorem	cflecard 4902	Cofinality is bounded by the cardinality of its argument.
|- (cf` A) (_ (card` A)
Theorem	cfle 4903	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.
|- (cf` A) (_ A
Theorem	cfeq0 4904	Only the ordinal zero has cofinality zero.
|- (A e. On -> ((cf` A) = (/) <-> A = (/)))
Theorem	cfsuc 4905	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102.
|- (A e. On -> (cf` suc A) = 1o)
Theorem	cfom 4906	Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102.
|- (cf` om) = om
Cardinal number arithmetic
Syntax	ccda 4907	Extend class definition to include cardinal number addition.
class +c
Definition	df-cda 4908	Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 4910 for its value and a description.
|- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
Theorem	cdavalt 4909	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.
|- ((A e. C /\ B e. D) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
Theorem	cdaval 4910	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 4821, carddom 4826, and cardsdom 4827. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.
|- A e. V   &   |- B e. V   =>   |- (A +c B) = ((A X. {(/)}) u. (B X. {1o}))
Theorem	uncdadom 4911	Cardinal addition dominates union.
|- A e. V   &   |- B e. V   =>   |- (A u. B) ~<_ (A +c B)
Theorem	cdaun 4912	Cardinal addition is equinumerous to union for disjoint sets.
|- A e. V   &   |- B e. V   =>   |- ((A i^i B) = (/) -> (A +c B) ~~ (A u. B))
Theorem	pm110.643 4913	1+1=2 for cardinal number addition. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Unlike us, Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 4716), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4562. The comment for cdaval 4910 explains why we use ~~ instead of =.
|- (1o +c 1o) ~~ 2o
Theorem	cdaen 4914	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))
Theorem	cda0en 4915	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143.
|- A e. V   =>   |- (A +c (/)) ~~ A
Theorem	cda1en 4916	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.
|- A e. V   =>   |- (A +c 1o) ~~ suc (card` A)
Theorem	xp1en 4917	One times a cardinal number.
|- A e. V   =>   |- (A X. 1o) ~~ A
Theorem	xp2cda 4918	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.
|- A e. V   =>   |- (A X. 2o) = (A +c A)
Theorem	cdacomen 4919	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   =>   |- (A +c B) ~~ (B +c A)
Theorem	cdaassen 4920	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A +c B) +c C) ~~ (A +c (B +c C))
Theorem	xpcdaen 4921	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))
Theorem	mapcdaen 4922	Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ^m (B +c C)) ~~ ((A ^m B) X. (A ^m C))
Theorem	cdadom1 4923	Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (A +c C) ~<_ (B +c C))
Theorem	cdadom2 4924	Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (C +c A) ~<_ (C +c B))
Theorem	cdadom3 4925	A set is dominated by its cardinal sum with another.
|- A e. V   &   |- B e. V   =>   |- A ~<_ (A +c B)
Theorem	cdafi 4926	The cardinal sum of two finite sets is finite.
|- ((A ~< om /\ B ~< om) -> (A +c B) ~< om)
Theorem	cdainf 4927	A set is infinite iff the cardinal sum with itself is infinite.
|- A e. V   =>   |- (om ~<_ A <-> om ~<_ (A +c A))
ZFC Axioms with no distinct variable requirements
Theorem	nd1 4928	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x y e. z)
Theorem	nd2 4929	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x z e. y)
Theorem	nd3 4930	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z x e. y)
Theorem	nd4 4931	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z y e. x)
Theorem	nd5 4932	A lemma for proving conditionless ZFC axioms.
|- (-. A.y y = x -> (z = y -> A.x z = y))
Theorem	axextnd 4933	A version of the Axiom of Extensionality with no distinct variable conditions.
|- E.x((x e. y <-> x e. z) -> y = z)
Theorem	axrepndlem1 4934	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepndlem2 4935	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepnd 4936	A version of the Axiom of Replacement with no distinct variable conditions.
|- E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
Theorem	axunndlem1 4937	Lemma for the Axiom of Union with no distinct variable conditions.
Theorem	axunnd 4938	A version of the Axiom of Union with no distinct variable conditions.
|- E.xA.y(E.x(y e. x /\ x e. z) -> y e. x)
Theorem	axpowndlem1 4939	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem2 4940	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem3 4941	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem4 4942	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpownd 4943	A version of the Axiom of Power Sets with no distinct variable conditions.
|- (-. x = y -> E.xA.y(A.x(E.z x e. y -> A.y x e. z) -> y e. x))
Theorem	axregndlem1 4944	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregndlem2 4945	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregnd 4946	A version of the Axiom of Regularity with no distinct variable conditions.
|- (x e. y -> E.x(x e. y /\ A.z(z e. x -> -. z e. y)))
Theorem	axinfndlem1 4947	Lemma for the Axiom of Infinity with no distinct variable conditions.
Theorem	axinfnd 4948	A version of the Axiom of Infinity with no distinct variable conditions.
|- E.x(y e. z -> (y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
Theorem	axacndlem1 4949	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem2 4950	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem3 4951	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem4 4952	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem5 4953	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacnd 4954	A version of the Axiom of Choice with no distinct variable conditions.
|- E.xA.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w))
Theorem	zfcndext 4955	Axiom of Extensionality, reproved from conditionless ZFC version and predicate calculus.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	zfcndrep 4956	Axiom of Replacement, reproved from conditionless ZFC axioms.
|- (A.wE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.w(w e. x /\ A.yph)))
Theorem	zfcndun 4957	Axiom of Union, reproved from conditionless ZFC axioms.
|- E.yA.z(E.w(z e. w /\ w e. x) -> z e. y)
Theorem	zfcndpow 4958	Axiom of Power Sets, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 2768.
|- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
Theorem	zfcndreg 4959	Axiom of Regularity, reproved from conditionless ZFC axioms..
|- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)))
Theorem	zfcndinf 4960	Axiom of Infinity, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets, we are justified in referencing theorem el 2747 in the proof.
|- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))
Theorem	zfcndac 4961	Axiom of Choice, reproved from conditionless ZFC axioms.
|- E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
Real and complex numbers
Dedekind-cut construction of real and complex numbers
Syntax	cnpi 4962	The set of positive integers, which is the set of natural numbers om with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 5254. The actual set of Dedekind cuts is defined by df-np 5076.
class N.
Syntax	cpli 4963	Positive integer addition.
class +N
Syntax	cmi 4964	Positive integer multiplication.
class .N
Syntax	clti 4965	Positive integer ordering relation.
class <N
Syntax	cplpq 4966	Positive fraction pre-addition.
class +pQ
Syntax	cmpq 4967	Positive fraction pre-multiplication.
class .pQ
Syntax	ceq 4968	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 4969	Set of positive fractions.
class Q.
Syntax	c1q 4970	The positive fraction constant 1.
class 1Q
Syntax	cplq 4971	Positive fraction addition.
class +Q
Syntax	cmq 4972	Positive fraction multiplication.
class .Q
Syntax	crq 4973	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 4974	Positive fraction ordering relation.
class <Q
Syntax	cnp 4975	Set of positive reals.
class P.
Syntax	c1p 4976	Positive real constant 1.
class 1P
Syntax	cpp 4977	Positive real addition.
class +P.
Syntax	cmp 4978	Positive real multiplication.
class .P.
Syntax	cltp 4979	Positive real ordering relation.
class <P
Syntax	cplpr 4980	Signed real pre-addition.
class +pR
Syntax	cmpr 4981	Signed real pre-multiplication.
class .pR
Syntax	cer 4982	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 4983	Set of signed reals.
class R.
Syntax	c0r 4984	The signed real constant 0.
class 0R
Syntax	c1r 4985	The signed real constant 1.
class 1R
Syntax	cm1r 4986	The signed real constant -1.
class -1R
Syntax	cplr 4987	Signed real addition.
class +R
Syntax	cmr 4988	Signed real multiplication.
class .R
Syntax	cltr 4989	Signed real ordering relation.
class <R
Definition	df-ni 4990	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5230, and is intended to be used only by the construction.
|- N. = (om \ {(/)})
Definition	df-pli 4991	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5230, and is intended to be used only by the construction.
|- +N = ( +o |` (N. X. N.))
Definition	df-mi 4992	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5230, and is intended to be used only by the construction.
|- .N = ( .o |` (N. X. N.))
Definition	df-lti 4993	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5230, and is intended to be used only by the construction.
|- <N = (E i^i (N. X. N.))
Theorem	elni 4994	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ A =/= (/)))
Theorem	elni2 4995	Membership in the class of positive integers.
|- (A e. N. <-> (A e.