P. 329 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzsplit3 32801	Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁)))
Theorem	bcm1n 32802	The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)
⊢ ((𝐾 ∈ (0...(𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (((𝑁 − 1)C𝐾) / (𝑁C𝐾)) = ((𝑁 − 𝐾) / 𝑁))
21.3.5.7  Half-open integer ranges - misc additions
Theorem	iundisjfi 32803*	Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 25596. (Contributed by Thierry Arnoux, 15-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    ⇒   ⊢ ∪ 𝑛 ∈ (1..^𝑁)𝐴 = ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
Theorem	iundisj2fi 32804*	A disjoint union is disjoint, finite version. Cf. iundisj2 25597. (Contributed by Thierry Arnoux, 16-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    ⇒   ⊢ Disj 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
Theorem	iundisjcnt 32805*	Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    ⇒   ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
Theorem	iundisj2cnt 32806*	A countable disjoint union is disjoint. Cf. iundisj2 25597. (Contributed by Thierry Arnoux, 16-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    ⇒   ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
Theorem	fzone1 32807	Elementhood in a half-open interval, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)..^𝑁))
Theorem	fzom1ne1 32808	Elementhood in a half-open interval, except the lower bound, shifted by one. (Contributed by Thierry Arnoux, 1-Jan-2024.)
⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀..^(𝑁 − 1)))
Theorem	f1ocnt 32809*	Given a countable set 𝐴, number its elements by providing a one-to-one mapping either with ℕ or an integer range starting from 1. The domain of the function can then be used with iundisjcnt 32805 or iundisj2cnt 32806. (Contributed by Thierry Arnoux, 25-Jul-2020.)
⊢ (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
Theorem	fz1nnct 32810	NN and integer ranges starting from 1 are countable. (Contributed by Thierry Arnoux, 25-Jul-2020.)
⊢ ((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) → 𝐴 ≼ ω)
Theorem	fz1nntr 32811	NN and integer ranges starting from 1 are a transitive family of set. (Contributed by Thierry Arnoux, 25-Jul-2020.)
⊢ (((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) ∧ 𝑁 ∈ 𝐴) → (1..^𝑁) ⊆ 𝐴)
Theorem	fzo0opth 32812	Equality for a half open integer range starting at zero is the same as equality of its upper bound, analogous to fzopth 13597 and fzoopth 13797. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((0..^𝑀) = (0..^𝑁) ↔ 𝑀 = 𝑁))
Theorem	nn0difffzod 32813	A nonnegative integer that is not in the half-open range from 0 to 𝑁 is at least 𝑁. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ (ℕ0 ∖ (0..^𝑁)))    ⇒   ⊢ (𝜑 → ¬ 𝑀 < 𝑁)
Theorem	suppssnn0 32814*	Show that the support of a function is contained in an half-open nonnegative integer range. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ (𝜑 → 𝐹 Fn ℕ0)    &   ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑁 ≤ 𝑘) → (𝐹‘𝑘) = 𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0..^𝑁))
21.3.5.8  The ` # ` (set size) function - misc additions
Theorem	hashunif 32815*	The cardinality of a disjoint finite union of finite sets. Cf. hashuni 15858. (Contributed by Thierry Arnoux, 17-Feb-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ Fin)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥)    ⇒   ⊢ (𝜑 → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
Theorem	hashxpe 32816	The size of the Cartesian product of two finite sets is the product of their sizes. This is a version of hashxp 14469 valid for infinite sets, which uses extended real numbers. (Contributed by Thierry Arnoux, 27-May-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) ·e (♯‘𝐵)))
Theorem	hashgt1 32817	Restate "set contains at least two elements" in terms of elementhood. (Contributed by Thierry Arnoux, 21-Nov-2023.)
⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ (◡♯ “ {0, 1}) ↔ 1 < (♯‘𝐴)))
21.3.5.9  The greatest common divisor operator - misc. additions
Theorem	znumd 32818	Numerator of an integer. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ (𝜑 → 𝑍 ∈ ℤ)    ⇒   ⊢ (𝜑 → (numer‘𝑍) = 𝑍)
Theorem	zdend 32819	Denominator of an integer. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ (𝜑 → 𝑍 ∈ ℤ)    ⇒   ⊢ (𝜑 → (denom‘𝑍) = 1)
Theorem	numdenneg 32820	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 32821	Calculate the reduced form of a quotient using gcd. This version extends divnumden 16781 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 32822	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 32823	Split 0 and 1 from the nonnegative integers. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ ℕ0 = ({0, 1} ∪ (ℤ≥‘2))
Theorem	nn0disj01 32824	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 32825*	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 32826*	Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12710. (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 32827	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 32828	Subtracting an integer number from another number decreases it. See ltsubrpd 13106. (Contributed by Thierry Arnoux, 18-Apr-2017.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀)
Theorem	fprodeq02 32829*	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	pr01ssre 32830	The range of the indicator function is a subset of ℝ. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ {0, 1} ⊆ ℝ
Theorem	fprodex01 32831*	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 32832*	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 32833*	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 32834*	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 32835*	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 32836	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  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 32837	Constant used for decimal fraction constructor. See df-dp2 32838.
class _𝐴𝐵
Definition	df-dp2 32838	Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 12731. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
Theorem	dp2eq1 32839	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)
Theorem	dp2eq2 32840	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)
Theorem	dp2eq1i 32841	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐶
Theorem	dp2eq2i 32842	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐶𝐴 = _𝐶𝐵
Theorem	dp2eq12i 32843	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐷
Theorem	dp20u 32844	Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ _𝐴0 = 𝐴
Theorem	dp20h 32845	Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ _0𝐴 = (𝐴 / ;10)
Theorem	dp2cl 32846	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 32847	Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℚ    ⇒   ⊢ _𝐴𝐵 ∈ ℚ
Theorem	rpdp2cl 32848	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 32849	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 32850	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 32851	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 32852	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 32853	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
21.3.6.1  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 32855 and df-dp2 32838 for more information; dpval2 32859 and dpfrac1 32858 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 12731.
Syntax	cdp 32854	Decimal point operator. See df-dp 32855.
class .
Definition	df-dp 32855*	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 32856	Define the value of the decimal point operator. See df-dp 32855. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
Theorem	dpcl 32857	Prove that the closure of the decimal point is ℝ as we have defined it. See df-dp 32855. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
Theorem	dpfrac1 32858	Prove a simple equivalence involving the decimal point. See df-dp 32855 and dpcl 32857. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10))
Theorem	dpval2 32859	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10))
Theorem	dpval3 32860	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dpmul10 32861	Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵
Theorem	decdiv10 32862	Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵)
Theorem	dpmul100 32863	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 32864	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 32865	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 32866	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dp0u 32867	Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (𝐴.0) = 𝐴
Theorem	dp0h 32868	Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ (0.𝐴) = (𝐴 / ;10)
Theorem	rpdpcl 32869	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 32870	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 32871	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 32872	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 32873	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 32874	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 32875	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 32876	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 32877	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 32878	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 32879	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 32880	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 32881	Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ (3 / 2) = (1.5)
Theorem	1mhdrd 32882	Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ ((0._99) + (0._01)) = 1
21.3.6.2  Division in the extended real number system
Syntax	cxdiv 32883	Extend class notation to include division of extended reals.
class /𝑒
Definition	df-xdiv 32884*	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 32885*	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 32886*	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 32887	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 32888*	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 32889	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 32890	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 32891	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 32892	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))
Theorem	xdivrec 32893	Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 ·e (1 /𝑒 𝐵)))
Theorem	xdivid 32894	A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
Theorem	xdiv0 32895	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
Theorem	xdiv0rp 32896	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
Theorem	eliccioo 32897	Membership in a closed interval of extended reals versus the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)))
Theorem	elxrge02 32898	Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞))
Theorem	xdivpnfrp 32899	Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
Theorem	rpxdivcld 32900	Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ+)