P. 330 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	znumd 32901	Numerator of an integer. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ (𝜑 → 𝑍 ∈ ℤ)    ⇒   ⊢ (𝜑 → (numer‘𝑍) = 𝑍)
Theorem	zdend 32902	Denominator of an integer. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ (𝜑 → 𝑍 ∈ ℤ)    ⇒   ⊢ (𝜑 → (denom‘𝑍) = 1)
Theorem	numdenneg 32903	Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.)
⊢ (𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄)))
Theorem	divnumden2 32904	Calculate the reduced form of a quotient using gcd. This version extends divnumden 16709 for the negative integers. (Contributed by Thierry Arnoux, 25-Oct-2017.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ -𝐵 ∈ ℕ) → ((numer‘(𝐴 / 𝐵)) = -(𝐴 / (𝐴 gcd 𝐵)) ∧ (denom‘(𝐴 / 𝐵)) = -(𝐵 / (𝐴 gcd 𝐵))))
Theorem	expgt0b 32905	A real number 𝐴 raised to an odd integer power is positive iff it is positive. (Contributed by SN, 4-Mar-2023.) Use the more standard ¬ 2 ∥ 𝑁 (Revised by Thierry Arnoux, 14-Jun-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝑁)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴↑𝑁)))
21.3.5.10  Integers
Theorem	nn0split01 32906	Split 0 and 1 from the nonnegative integers. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ ℕ0 = ({0, 1} ∪ (ℤ≥‘2))
Theorem	nn0disj01 32907	The pair {0, 1} does not overlap the rest of the nonnegative integers. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ ({0, 1} ∩ (ℤ≥‘2)) = ∅
Theorem	nnindf 32908*	Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜏)
Theorem	nn0min 32909*	Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12615. (Contributed by Thierry Arnoux, 6-May-2018.)
⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏))    &   ⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏))
Theorem	subne0nn 32910	A nonnegative difference is positive if the two numbers are not equal. (Contributed by Thierry Arnoux, 17-Dec-2023.)
⊢ (𝜑 → 𝑀 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℂ)    &   ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ≠ 𝑁)    ⇒   ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℕ)
Theorem	ltesubnnd 32911	Subtracting an integer number from another number decreases it. See ltsubrpd 13009. (Contributed by Thierry Arnoux, 18-Apr-2017.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀)
Theorem	fprodeq02 32912*	If one of the factors is zero the product is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 = 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0)
Theorem	fprodex01 32913*	A product of factors equal to zero or one is zero exactly when one of the factors is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝑘 = 𝑙 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {0, 1})    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = if(∀𝑙 ∈ 𝐴 𝐶 = 1, 1, 0))
Theorem	prodpr 32914*	A product over a pair is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸)    &   ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐸 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹))
Theorem	prodtp 32915*	A product over a triple is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸)    &   ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐸 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺))
Theorem	fsumub 32916*	An upper bound for a term of a positive finite sum. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ (𝑘 = 𝐾 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐷 ≤ 𝐶)
Theorem	fsumiunle 32917*	Upper bound for a sum of nonnegative terms over an indexed union. The inequality may be strict if the indexed union is non-disjoint, since in the right hand side, a summand may be counted several times. (Contributed by Thierry Arnoux, 1-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 0 ≤ 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)
21.3.5.11  Decimal numbers
Theorem	dfdec100 32918	Split the hundreds from a decimal value. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶)
21.3.6  Real and complex functions
21.3.6.1  Signum (sgn or sign) function - misc. additions
Theorem	sgncl 32919	Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.)
⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1})
Theorem	sgnclre 32920	Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ)
Theorem	sgnneg 32921	Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.)
⊢ (𝐴 ∈ ℝ → (sgn‘-𝐴) = -(sgn‘𝐴))
Theorem	sgn3da 32922	A conditional containing a signum is true if it is true in all three possible cases. (Contributed by Thierry Arnoux, 1-Oct-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ ((sgn‘𝐴) = 0 → (𝜓 ↔ 𝜒))    &   ⊢ ((sgn‘𝐴) = 1 → (𝜓 ↔ 𝜃))    &   ⊢ ((sgn‘𝐴) = -1 → (𝜓 ↔ 𝜏))    &   ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝜒)    &   ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝜏)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sgnmul 32923	Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)))
Theorem	sgnmulrp2 32924	Multiplication by a positive number does not affect signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (sgn‘(𝐴 · 𝐵)) = (sgn‘𝐴))
Theorem	sgnsub 32925	Subtraction of a number of opposite sign. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → (sgn‘(𝐴 − 𝐵)) = (sgn‘𝐴))
Theorem	sgnnbi 32926	Negative signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0))
Theorem	sgnpbi 32927	Positive signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 ↔ 0 < 𝐴))
Theorem	sgn0bi 32928	Zero signum. (Contributed by Thierry Arnoux, 10-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0))
Theorem	sgnsgn 32929	Signum is idempotent. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → (sgn‘(sgn‘𝐴)) = (sgn‘𝐴))
Theorem	sgnmulsgn 32930	If two real numbers are of different signs, so are their signs. (Contributed by Thierry Arnoux, 12-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) < 0 ↔ ((sgn‘𝐴) · (sgn‘𝐵)) < 0))
Theorem	sgnmulsgp 32931	If two real numbers are of same signs, so are their signs. (Contributed by Thierry Arnoux, 12-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < (𝐴 · 𝐵) ↔ 0 < ((sgn‘𝐴) · (sgn‘𝐵))))
21.3.6.2  Integer powers - misc. additions
Theorem	nexple 32932	A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))
Theorem	2exple2exp 32933*	If a nonnegative integer 𝑋 is a multiple of a power of two, but less than the next power of two, it is itself a power of two. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (2↑𝐾) ∥ 𝑋)    &   ⊢ (𝜑 → 𝑋 ≤ (2↑(𝐾 + 1)))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 𝑋 = (2↑𝑛))
Theorem	expevenpos 32934	Even powers are positive. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 2 ∥ 𝑁)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴↑𝑁))
Theorem	oexpled 32935	Odd power monomials are monotonic. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝑁)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
21.3.6.3  Indicator Functions (continued)
Theorem	indsumin 32936*	Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑂)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶)
Theorem	prodindf 32937*	The product of indicators is one if and only if all values are in the set. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑂)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (((𝟭‘𝑂)‘𝐵)‘(𝐹‘𝑘)) = if(ran 𝐹 ⊆ 𝐵, 1, 0))
Theorem	indsn 32938*	The indicator function of a singleton. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝑋 ∈ 𝑂) → ((𝟭‘𝑂)‘{𝑋}) = (𝑥 ∈ 𝑂 ↦ if(𝑥 = 𝑋, 1, 0)))
Theorem	indf1o 32939	The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂))
Theorem	indpreima 32940	A function with range {0, 1} as an indicator of the preimage of {1}. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐹:𝑂⟶{0, 1}) → 𝐹 = ((𝟭‘𝑂)‘(◡𝐹 “ {1})))
Theorem	indf1ofs 32941*	The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin})
Theorem	indsupp 32942	The support of the indicator function. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (((𝟭‘𝑂)‘𝐴) supp 0) = 𝐴)
Theorem	indfsd 32943	The indicator function of a finite set has finite support. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ((𝟭‘𝑂)‘𝐴) finSupp 0)
Theorem	indfsid 32944	Conditions for a function to be an indicator function. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑂⟶{0, 1})    ⇒   ⊢ (𝜑 → 𝐹 = ((𝟭‘𝑂)‘(𝐹 supp 0)))
21.3.7  Decimal expansion Define a decimal expansion constructor. The decimal expansions built with this constructor are not meant to be used alone outside of this chapter. Rather, they are meant to be used exclusively as part of a decimal number with a decimal fraction, for example (3._1_4_1_59). That decimal point operator is defined in the next section. The bulk of these constructions have originally been proposed by David A. Wheeler on 12-May-2015, and discussed with Mario Carneiro in this thread: https://groups.google.com/g/metamath/c/2AW7T3d2YiQ.
Syntax	cdp2 32945	Constant used for decimal fraction constructor. See df-dp2 32946.
class _𝐴𝐵
Definition	df-dp2 32946	Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 12636. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
Theorem	dp2eq1 32947	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)
Theorem	dp2eq2 32948	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)
Theorem	dp2eq1i 32949	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐶
Theorem	dp2eq2i 32950	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐶𝐴 = _𝐶𝐵
Theorem	dp2eq12i 32951	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐷
Theorem	dp20u 32952	Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ _𝐴0 = 𝐴
Theorem	dp20h 32953	Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ _0𝐴 = (𝐴 / ;10)
Theorem	dp2cl 32954	Closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ)
Theorem	dp2clq 32955	Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℚ    ⇒   ⊢ _𝐴𝐵 ∈ ℚ
Theorem	rpdp2cl 32956	Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ _𝐴𝐵 ∈ ℝ+
Theorem	rpdp2cl2 32957	Closure for a decimal fraction with no decimal expansion in the positive real numbers. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ _𝐴0 ∈ ℝ+
Theorem	dp2lt10 32958	Decimal fraction builds real numbers less than 10. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐴 < ;10    &   ⊢ 𝐵 < ;10    ⇒   ⊢ _𝐴𝐵 < ;10
Theorem	dp2lt 32959	Comparing two decimal fractions (equal unit places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℝ+    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ _𝐴𝐵 < _𝐴𝐶
Theorem	dp2ltsuc 32960	Comparing a decimal fraction with the next integer. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐵 < ;10    &   ⊢ (𝐴 + 1) = 𝐶    ⇒   ⊢ _𝐴𝐵 < 𝐶
Theorem	dp2ltc 32961	Comparing two decimal expansions (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ+    &   ⊢ 𝐵 < ;10    &   ⊢ 𝐴 < 𝐶    ⇒   ⊢ _𝐴𝐵 < _𝐶𝐷
21.3.7.1  Decimal point Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 32963 and df-dp2 32946 for more information; dpval2 32967 and dpfrac1 32966 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 12636.
Syntax	cdp 32962	Decimal point operator. See df-dp 32963.
class .
Definition	df-dp 32963*	Define the . (decimal point) operator. For example, (1.5) = (3 / 2), and -(;32._7_18) = -(;;;;32718 / ;;;1000) Unary minus, if applied, should normally be applied in front of the parentheses. Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that Metamath has no built-in way to parse and handle decimal numbers as traditionally written, e.g., "2.54". Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers. The RHS is ℝ, not ℚ; this should simplify some proofs. The LHS is ℕ0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression -(𝐴.𝐵) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)
⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
Theorem	dpval 32964	Define the value of the decimal point operator. See df-dp 32963. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
Theorem	dpcl 32965	Prove that the closure of the decimal point is ℝ as we have defined it. See df-dp 32963. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
Theorem	dpfrac1 32966	Prove a simple equivalence involving the decimal point. See df-dp 32963 and dpcl 32965. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10))
Theorem	dpval2 32967	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10))
Theorem	dpval3 32968	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dpmul10 32969	Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵
Theorem	decdiv10 32970	Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵)
Theorem	dpmul100 32971	Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶
Theorem	dp3mul10 32972	Multiply by 10 a decimal expansion with 3 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴._𝐵𝐶) · ;10) = (;𝐴𝐵.𝐶)
Theorem	dpmul1000 32973	Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ    ⇒   ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷
Theorem	dpval3rp 32974	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dp0u 32975	Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (𝐴.0) = 𝐴
Theorem	dp0h 32976	Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ (0.𝐴) = (𝐴 / ;10)
Theorem	rpdpcl 32977	Closure of the decimal point in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ (𝐴.𝐵) ∈ ℝ+
Theorem	dplt 32978	Comparing two decimal expansions (equal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℝ+    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ (𝐴.𝐵) < (𝐴.𝐶)
Theorem	dplti 32979	Comparing a decimal expansions with the next higher integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐵 < ;10    &   ⊢ (𝐴 + 1) = 𝐶    ⇒   ⊢ (𝐴.𝐵) < 𝐶
Theorem	dpgti 32980	Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ 𝐴 < (𝐴.𝐵)
Theorem	dpltc 32981	Comparing two decimal integers (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ+    &   ⊢ 𝐴 < 𝐶    &   ⊢ 𝐵 < ;10    ⇒   ⊢ (𝐴.𝐵) < (𝐶.𝐷)
Theorem	dpexpp1 32982	Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ (𝑃 + 1) = 𝑄    &   ⊢ 𝑃 ∈ ℤ    &   ⊢ 𝑄 ∈ ℤ    ⇒   ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄))
Theorem	0dp2dp 32983	Multiply by 10 a decimal expansion which starts with a zero. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ ((0._𝐴𝐵) · ;10) = (𝐴.𝐵)
Theorem	dpadd2 32984	Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ+    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℝ+    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐻 ∈ ℕ0    &   ⊢ (𝐺 + 𝐻) = 𝐼    &   ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)    ⇒   ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)
Theorem	dpadd 32985	Addition with one decimal. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ (;𝐴𝐵 + ;𝐶𝐷) = ;𝐸𝐹    ⇒   ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)
Theorem	dpadd3 32986	Addition with two decimals. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝐻 ∈ ℕ0    &   ⊢ 𝐼 ∈ ℕ0    &   ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼    ⇒   ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼)
Theorem	dpmul 32987	Multiplication with one decimal point. (Contributed by Thierry Arnoux, 26-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐽 ∈ ℕ0    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ (𝐴 · 𝐶) = 𝐹    &   ⊢ (𝐴 · 𝐷) = 𝑀    &   ⊢ (𝐵 · 𝐶) = 𝐿    &   ⊢ (𝐵 · 𝐷) = ;𝐸𝐾    &   ⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽    &   ⊢ (𝐹 + 𝐺) = 𝐼    ⇒   ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾)
Theorem	dpmul4 32988	An upper bound to multiplication of decimal numbers with 4 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐽 ∈ ℕ0    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝐻 ∈ ℕ0    &   ⊢ 𝐼 ∈ ℕ0    &   ⊢ 𝐿 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑂 ∈ ℕ0    &   ⊢ 𝑃 ∈ ℕ0    &   ⊢ 𝑄 ∈ ℕ0    &   ⊢ 𝑅 ∈ ℕ0    &   ⊢ 𝑆 ∈ ℕ0    &   ⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝑈 ∈ ℕ0    &   ⊢ 𝑊 ∈ ℕ0    &   ⊢ 𝑋 ∈ ℕ0    &   ⊢ 𝑌 ∈ ℕ0    &   ⊢ 𝑍 ∈ ℕ0    &   ⊢ 𝑈 < ;10    &   ⊢ 𝑃 < ;10    &   ⊢ 𝑄 < ;10    &   ⊢ (;;𝐿𝑀𝑁 + 𝑂) = ;;;𝑅𝑆𝑇𝑈    &   ⊢ ((𝐴.𝐵) · (𝐸.𝐹)) = (𝐼._𝐽𝐾)    &   ⊢ ((𝐶.𝐷) · (𝐺.𝐻)) = (𝑂._𝑃𝑄)    &   ⊢ (;;;𝐼𝐽𝐾1 + ;;𝑅𝑆𝑇) = ;;;𝑊𝑋𝑌𝑍    &   ⊢ (((𝐴.𝐵) + (𝐶.𝐷)) · ((𝐸.𝐹) + (𝐺.𝐻))) = (((𝐼._𝐽𝐾) + (𝐿._𝑀𝑁)) + (𝑂._𝑃𝑄))    ⇒   ⊢ ((𝐴._𝐵_𝐶𝐷) · (𝐸._𝐹_𝐺𝐻)) < (𝑊._𝑋_𝑌𝑍)
Theorem	threehalves 32989	Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ (3 / 2) = (1.5)
Theorem	1mhdrd 32990	Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ ((0._99) + (0._01)) = 1
21.3.7.2  Division in the extended real number system
Syntax	cxdiv 32991	Extend class notation to include division of extended reals.
class /𝑒
Definition	df-xdiv 32992*	Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
Theorem	xdivval 32993*	Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
Theorem	xrecex 32994*	Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 ·e 𝑥) = 1)
Theorem	xmulcand 32995	Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐶 ·e 𝐴) = (𝐶 ·e 𝐵) ↔ 𝐴 = 𝐵))
Theorem	xreceu 32996*	Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)
Theorem	xdivcld 32997	Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*)
Theorem	xdivcl 32998	Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)
Theorem	xdivmul 32999	Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /𝑒 𝐶) = 𝐵 ↔ (𝐶 ·e 𝐵) = 𝐴))
Theorem	rexdiv 33000	The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 / 𝐵))