mmtheorems50 - Metamath Proof Explorer

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

Type	Label	Description
Statement
Definition	df-cda 4901	Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 4903 for its value and a description.
⊢ +c = {⟨⟨x, y⟩, z⟩∣z = ((x × {∅}) ∪ (y × {1o}))}
Theorem	cdavalt 4902	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.
⊢ ((A ∈ C ⋀ B ∈ D) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))
Theorem	cdaval 4903	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 4814, carddom 4819, and cardsdom 4820. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A +c B) = ((A × {∅}) ∪ (B × {1o}))
Theorem	uncdadom 4904	Cardinal addition dominates union.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A ∪ B) ≼ (A +c B)
Theorem	cdaun 4905	Cardinal addition is equinumerous to union for disjoint sets.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ((A ∩ B) = ∅ → (A +c B) ≈ (A ∪ B))
Theorem	pm110.643 4906	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 4709), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4555. The comment for cdaval 4903 explains why we use ≈ instead of =.
⊢ (1o +c 1o) ≈ 2o
Theorem	cdaen 4907	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ D ∈ V    ⇒   ⊢ ((A ≈ B ⋀ C ≈ D) → (A +c C) ≈ (B +c D))
Theorem	cda0en 4908	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143.
⊢ A ∈ V    ⇒   ⊢ (A +c ∅) ≈ A
Theorem	cda1en 4909	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.
⊢ A ∈ V    ⇒   ⊢ (A +c 1o) ≈ suc (card ‘A)
Theorem	xp1en 4910	One times a cardinal number.
⊢ A ∈ V    ⇒   ⊢ (A × 1o) ≈ A
Theorem	xp2cda 4911	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.
⊢ A ∈ V    ⇒   ⊢ (A × 2o) = (A +c A)
Theorem	cdacomen 4912	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A +c B) ≈ (B +c A)
Theorem	cdaassen 4913	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ ((A +c B) +c C) ≈ (A +c (B +c C))
Theorem	xpcdaen 4914	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A × (B +c C)) ≈ ((A × B) +c (A × C))
Theorem	mapcdaen 4915	Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ↑m (B +c C)) ≈ ((A ↑m B) × (A ↑m C))
Theorem	cdadom1 4916	Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ≼ B → (A +c C) ≼ (B +c C))
Theorem	cdadom2 4917	Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ≼ B → (C +c A) ≼ (C +c B))
Theorem	cdadom3 4918	A set is dominated by its cardinal sum with another.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ A ≼ (A +c B)
Theorem	cdafi 4919	The cardinal sum of two finite sets is finite.
⊢ ((A ≺ ω ⋀ B ≺ ω) → (A +c B) ≺ ω)
Theorem	cdainf 4920	A set is infinite iff the cardinal sum with itself is infinite.
⊢ A ∈ V    ⇒   ⊢ (ω ≼ A ↔ ω ≼ (A +c A))
ZFC Axioms with no distinct variable requirements
Theorem	nd1 4921	A lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀x y ∈ z)
Theorem	nd2 4922	A lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀x z ∈ y)
Theorem	nd3 4923	A lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀z x ∈ y)
Theorem	nd4 4924	A lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀z y ∈ x)
Theorem	nd5 4925	A lemma for proving conditionless ZFC axioms.
⊢ (¬ ∀y y = x → (z = y → ∀x z = y))
Theorem	axextnd 4926	A version of the Axiom of Extensionality with no distinct variable conditions.
⊢ ∃x((x ∈ y ↔ x ∈ z) → y = z)
Theorem	axrepndlem1 4927	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepndlem2 4928	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepnd 4929	A version of the Axiom of Replacement with no distinct variable conditions.
⊢ ∃x(∃y∀z(φ → z = y) → ∀z(∀y z ∈ x ↔ ∃x(∀z x ∈ y ⋀ ∀yφ)))
Theorem	axunndlem1 4930	Lemma for the Axiom of Union with no distinct variable conditions.
Theorem	axunnd 4931	A version of the Axiom of Union with no distinct variable conditions.
⊢ ∃x∀y(∃x(y ∈ x ⋀ x ∈ z) → y ∈ x)
Theorem	axpowndlem1 4932	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem2 4933	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem3 4934	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem4 4935	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpownd 4936	A version of the Axiom of Power Sets with no distinct variable conditions.
⊢ (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x))
Theorem	axregndlem1 4937	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregndlem2 4938	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregnd 4939	A version of the Axiom of Regularity with no distinct variable conditions.
⊢ (x ∈ y → ∃x(x ∈ y ⋀ ∀z(z ∈ x → ¬ z ∈ y)))
Theorem	axinfndlem1 4940	Lemma for the Axiom of Infinity with no distinct variable conditions.
Theorem	axinfnd 4941	A version of the Axiom of Infinity with no distinct variable conditions.
⊢ ∃x(y ∈ z → (y ∈ x ⋀ ∀y(y ∈ x → ∃z(y ∈ z ⋀ z ∈ x))))
Theorem	axacndlem1 4942	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem2 4943	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem3 4944	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem4 4945	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem5 4946	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacnd 4947	A version of the Axiom of Choice with no distinct variable conditions.
⊢ ∃x∀y∀z(∀x(y ∈ z ⋀ z ∈ w) → ∃w∀y(∃w((y ∈ z ⋀ z ∈ w) ⋀ (y ∈ w ⋀ w ∈ x)) ↔ y = w))
Theorem	zfcndext 4948	Axiom of Extensionality, reproved from conditionless ZFC version and predicate calculus.
⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)
Theorem	zfcndrep 4949	Axiom of Replacement, reproved from conditionless ZFC axioms.
⊢ (∀w∃y∀z(∀yφ → z = y) → ∃y∀z(z ∈ y ↔ ∃w(w ∈ x ⋀ ∀yφ)))
Theorem	zfcndun 4950	Axiom of Union, reproved from conditionless ZFC axioms.
⊢ ∃y∀z(∃w(z ∈ w ⋀ w ∈ x) → z ∈ y)
Theorem	zfcndpow 4951	Axiom of Power Sets, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 2768.
⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y)
Theorem	zfcndreg 4952	Axiom of Regularity, reproved from conditionless ZFC axioms..
⊢ (∃y y ∈ x → ∃y(y ∈ x ⋀ ∀z(z ∈ y → ¬ z ∈ x)))
Theorem	zfcndinf 4953	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.
⊢ ∃y(x ∈ y ⋀ ∀z(z ∈ y → ∃w(z ∈ w ⋀ w ∈ y)))
Theorem	zfcndac 4954	Axiom of Choice, reproved from conditionless ZFC axioms.
⊢ ∃y∀z∀w((z ∈ w ⋀ w ∈ x) → ∃v∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v))
Real and complex numbers
Dedekind-cut construction of real and complex numbers
Syntax	cnpi 4955	The set of positive integers, which is the set of natural numbers ω 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 5247. The actual set of Dedekind cuts is defined by df-np 5069.
class N
Syntax	cpli 4956	Positive integer addition.
class +N
Syntax	cmi 4957	Positive integer multiplication.
class ·N
Syntax	clti 4958	Positive integer ordering relation.
class <N
Syntax	cplpq 4959	Positive fraction pre-addition.
class +pQ
Syntax	cmpq 4960	Positive fraction pre-multiplication.
class ·pQ
Syntax	ceq 4961	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 4962	Set of positive fractions.
class Q
Syntax	c1q 4963	The positive fraction constant 1.
class 1Q
Syntax	cplq 4964	Positive fraction addition.
class +Q
Syntax	cmq 4965	Positive fraction multiplication.
class ·Q
Syntax	crq 4966	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 4967	Positive fraction ordering relation.
class <Q
Syntax	cnp 4968	Set of positive reals.
class P
Syntax	c1p 4969	Positive real constant 1.
class 1P
Syntax	cpp 4970	Positive real addition.
class +P
Syntax	cmp 4971	Positive real multiplication.
class ·P
Syntax	cltp 4972	Positive real ordering relation.
class <P
Syntax	cplpr 4973	Signed real pre-addition.
class +pR
Syntax	cmpr 4974	Signed real pre-multiplication.
class ·pR
Syntax	cer 4975	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 4976	Set of signed reals.
class R
Syntax	c0r 4977	The signed real constant 0.
class 0R
Syntax	c1r 4978	The signed real constant 1.
class 1R
Syntax	cm1r 4979	The signed real constant -1.
class -1R
Syntax	cplr 4980	Signed real addition.
class +R
Syntax	cmr 4981	Signed real multiplication.
class ·R
Syntax	cltr 4982	Signed real ordering relation.
class <R
Definition	df-ni 4983	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5223, and is intended to be used only by the construction.
⊢ N = (ω ∖ {∅})
Definition	df-pli 4984	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5223, and is intended to be used only by the construction.
⊢ +N = ( +o ↾ (N × N))
Definition	df-mi 4985	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5223, and is intended to be used only by the construction.
⊢ ·N = ( ·o ↾ (N × N))
Definition	df-lti 4986	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5223, and is intended to be used only by the construction.
⊢ <N = (E ∩ (N × N))
Theorem	elni 4987	Membership in the class of positive integers.
⊢ (A ∈ N ↔ (A ∈ ω ⋀ A ≠ ∅))
Theorem	elni2 4988	Membership in the class of positive integers.
⊢ (A ∈ N ↔ (A ∈ ω ⋀ ∅ ∈ A))
Theorem	pinn 4989	A positive integer is a natural number.
⊢ (A ∈ N → A ∈ ω)
Theorem	pion 4990	A positive integer is an ordinal number.
⊢ (A ∈ N → A ∈ On)
Theorem	piord 4991	A positive integer is ordinal.
⊢ (A ∈ N → Ord A)
Theorem	niex 4992	The class of positive integers is a set.
⊢ N ∈ V
Theorem	0npi 4993	The empty set is not a positive integer.
⊢ ¬ ∅ ∈ N
Theorem	1pi 4994	Ordinal 'one' is a positive integer.
⊢ 1o ∈ N
Theorem	addpiord 4995	Positive integer addition in terms of ordinal addition.
⊢ ((A ∈ N ⋀ B ∈ N) → (A +N B) = (A +o B))
Theorem	mulpiord 4996	Positive integer multiplication in terms of ordinal multiplication.
⊢ ((A ∈ N ⋀ B ∈ N) → (A ·N B) = (A ·o B))
Theorem	mulidpi 4997	1 is an identity element for multiplication on positive integers.
⊢ (A ∈ N → (A ·N 1o) = A)
Theorem	ltpiord 4998	Positive integer 'less than' in terms of ordinal membership.
⊢ ((A ∈ N ⋀ B ∈ N) → (A <N B ↔ A ∈ B))
Theorem	ltsopi 4999	Positive integer 'less than' is a strict ordering.
⊢ <N Or N
Theorem	ltrelpi 5000	Positive integer 'less than' is a relation on positive integers.
⊢ <N ⊆ (N × N)