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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.3.5.8  The ` # ` (set size) function - misc additions
 
Theoremhashunif 30701* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 15274. (Contributed by Thierry Arnoux, 17-Feb-2017.)
𝑥𝜑    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ Fin)    &   (𝜑Disj 𝑥𝐴 𝑥)       (𝜑 → (♯‘ 𝐴) = Σ𝑥𝐴 (♯‘𝑥))
 
Theoremhashxpe 30702 The size of the Cartesian product of two finite sets is the product of their sizes. This is a version of hashxp 13887 valid for infinite sets, which uses extended real numbers. (Contributed by Thierry Arnoux, 27-May-2023.)
((𝐴𝑉𝐵𝑊) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) ·e (♯‘𝐵)))
 
Theoremhashgt1 30703 Restate "set contains at least two elements" in terms of elementhood. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝐴)))
 
20.3.5.9  The greatest common divisor operator - misc. add
 
Theoremdvdszzq 30704 Divisibility for an integer quotient. (Contributed by Thierry Arnoux, 17-Sep-2023.)
𝑁 = (𝐴 / 𝐵)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑃𝐴)    &   (𝜑 → ¬ 𝑃𝐵)       (𝜑𝑃𝑁)
 
Theoremprmdvdsbc 30705 Condition for a prime number to divide a binomial coefficient. (Contributed by Thierry Arnoux, 17-Sep-2023.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑁))
 
Theoremnumdenneg 30706 Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.)
(𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄)))
 
Theoremdivnumden2 30707 Calculate the reduced form of a quotient using gcd. This version extends divnumden 16188 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))))
 
20.3.5.10  Integers
 
Theoremnnindf 30708* Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
𝑦𝜑    &   (𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ → (𝜒𝜃))       (𝐴 ∈ ℕ → 𝜏)
 
Theoremnn0min 30709* Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12158. (Contributed by Thierry Arnoux, 6-May-2018.)
(𝑛 = 0 → (𝜓𝜒))    &   (𝑛 = 𝑚 → (𝜓𝜃))    &   (𝑛 = (𝑚 + 1) → (𝜓𝜏))    &   (𝜑 → ¬ 𝜒)    &   (𝜑 → ∃𝑛 ∈ ℕ 𝜓)       (𝜑 → ∃𝑚 ∈ ℕ0𝜃𝜏))
 
Theoremsubne0nn 30710 A nonnegative difference is positive if the two numbers are not equal. (Contributed by Thierry Arnoux, 17-Dec-2023.)
(𝜑𝑀 ∈ ℂ)    &   (𝜑𝑁 ∈ ℂ)    &   (𝜑 → (𝑀𝑁) ∈ ℕ0)    &   (𝜑𝑀𝑁)       (𝜑 → (𝑀𝑁) ∈ ℕ)
 
Theoremltesubnnd 30711 Subtracting an integer number from another number decreases it. See ltsubrpd 12546. (Contributed by Thierry Arnoux, 18-Apr-2017.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀)
 
Theoremfprodeq02 30712* If one of the factors is zero the product is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.)
(𝑘 = 𝐾𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐾𝐴)    &   (𝜑𝐶 = 0)       (𝜑 → ∏𝑘𝐴 𝐵 = 0)
 
Theorempr01ssre 30713 The range of the indicator function is a subset of . (Contributed by Thierry Arnoux, 14-Aug-2017.)
{0, 1} ⊆ ℝ
 
Theoremfprodex01 30714* 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 = 𝑙𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ {0, 1})       (𝜑 → ∏𝑘𝐴 𝐵 = if(∀𝑙𝐴 𝐶 = 1, 1, 0))
 
Theoremprodpr 30715* A product over a pair is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝑘 = 𝐴𝐷 = 𝐸)    &   (𝑘 = 𝐵𝐷 = 𝐹)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐹 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹))
 
Theoremprodtp 30716* A product over a triple is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝑘 = 𝐴𝐷 = 𝐸)    &   (𝑘 = 𝐵𝐷 = 𝐹)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐹 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   (𝑘 = 𝐶𝐷 = 𝐺)    &   (𝜑𝐶𝑋)    &   (𝜑𝐺 ∈ ℂ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺))
 
Theoremfsumub 30717* An upper bound for a term of a positive finite sum. (Contributed by Thierry Arnoux, 27-Dec-2021.)
(𝑘 = 𝐾𝐵 = 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → Σ𝑘𝐴 𝐵 = 𝐶)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ+)    &   (𝜑𝐾𝐴)       (𝜑𝐷𝐶)
 
Theoremfsumiunle 30718* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)       (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
 
20.3.5.11  Decimal numbers
 
Theoremdfdec100 30719 Split the hundreds from a decimal value. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℝ       𝐴𝐵𝐶 = ((100 · 𝐴) + 𝐵𝐶)
 
20.3.6  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 (3.14159).

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.

 
Syntaxcdp2 30720 Constant used for decimal fraction constructor. See df-dp2 30721.
class 𝐴𝐵
 
Definitiondf-dp2 30721 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 12180. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
𝐴𝐵 = (𝐴 + (𝐵 / 10))
 
Theoremdp2eq1 30722 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 
Theoremdp2eq2 30723 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 
Theoremdp2eq1i 30724 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶
 
Theoremdp2eq2i 30725 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵
 
Theoremdp2eq12i 30726 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷
 
Theoremdp20u 30727 Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       𝐴0 = 𝐴
 
Theoremdp20h 30728 Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       0𝐴 = (𝐴 / 10)
 
Theoremdp2cl 30729 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)
 
Theoremdp2clq 30730 Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℚ       𝐴𝐵 ∈ ℚ
 
Theoremrpdp2cl 30731 Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴𝐵 ∈ ℝ+
 
Theoremrpdp2cl2 30732 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+
 
Theoremdp2lt10 30733 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
 
Theoremdp2lt 30734 Comparing two decimal fractions (equal unit places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+    &   𝐶 ∈ ℝ+    &   𝐵 < 𝐶       𝐴𝐵 < 𝐴𝐶
 
Theoremdp2ltsuc 30735 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
 
Theoremdp2ltc 30736 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
 
20.3.6.1  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 30738 and df-dp2 30721 for more information; dpval2 30742 and dpfrac1 30741 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 12180.

 
Syntaxcdp 30737 Decimal point operator. See df-dp 30738.
class .
 
Definitiondf-dp 30738* Define the . (decimal point) operator. For example, (1.5) = (3 / 2), and -(32.718) = -(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 , not ; this should simplify some proofs. The LHS is 0, 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)
 
Theoremdpval 30739 Define the value of the decimal point operator. See df-dp 30738. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
 
Theoremdpcl 30740 Prove that the closure of the decimal point is as we have defined it. See df-dp 30738. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
 
Theoremdpfrac1 30741 Prove a simple equivalence involving the decimal point. See df-dp 30738 and dpcl 30740. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))
 
Theoremdpval2 30742 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = (𝐴 + (𝐵 / 10))
 
Theoremdpval3 30743 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = 𝐴𝐵
 
Theoremdpmul10 30744 Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       ((𝐴.𝐵) · 10) = 𝐴𝐵
 
Theoremdecdiv10 30745 Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴𝐵 / 10) = (𝐴.𝐵)
 
Theoremdpmul100 30746 Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℝ       ((𝐴.𝐵𝐶) · 100) = 𝐴𝐵𝐶
 
Theoremdp3mul10 30747 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)
 
Theoremdpmul1000 30748 Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℝ       ((𝐴.𝐵𝐶𝐷) · 1000) = 𝐴𝐵𝐶𝐷
 
Theoremdpval3rp 30749 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       (𝐴.𝐵) = 𝐴𝐵
 
Theoremdp0u 30750 Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       (𝐴.0) = 𝐴
 
Theoremdp0h 30751 Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       (0.𝐴) = (𝐴 / 10)
 
Theoremrpdpcl 30752 Closure of the decimal point in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       (𝐴.𝐵) ∈ ℝ+
 
Theoremdplt 30753 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)
 
Theoremdplti 30754 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
 
Theoremdpgti 30755 Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴 < (𝐴.𝐵)
 
Theoremdpltc 30756 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)
 
Theoremdpexpp1 30757 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))
 
Theorem0dp2dp 30758 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)
 
Theoremdpadd2 30759 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)
 
Theoremdpadd 30760 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)
 
Theoremdpadd3 30761 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)
 
Theoremdpmul 30762 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)
 
Theoremdpmul4 30763 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.𝐵𝐶𝐷) · (𝐸.𝐹𝐺𝐻)) < (𝑊.𝑋𝑌𝑍)
 
Theoremthreehalves 30764 Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
(3 / 2) = (1.5)
 
Theorem1mhdrd 30765 Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
((0.99) + (0.01)) = 1
 
20.3.6.2  Division in the extended real number system
 
Syntaxcxdiv 30766 Extend class notation to include division of extended reals.
class /𝑒
 
Definitiondf-xdiv 30767* 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))
 
Theoremxdivval 30768* 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))
 
Theoremxrecex 30769* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 ·e 𝑥) = 1)
 
Theoremxmulcand 30770 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 ·e 𝐴) = (𝐶 ·e 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremxreceu 30771* 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)
 
Theoremxdivcld 30772 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*)
 
Theoremxdivcl 30773 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)
 
Theoremxdivmul 30774 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))
 
Theoremrexdiv 30775 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))
 
Theoremxdivrec 30776 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)))
 
Theoremxdivid 30777 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
 
Theoremxdiv0 30778 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
 
Theoremxdiv0rp 30779 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
 
Theoremeliccioo 30780 Membership in a closed interval of extended reals versus the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)))
 
Theoremelxrge02 30781 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+𝐴 = +∞))
 
Theoremxdivpnfrp 30782 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
 
Theoremrpxdivcld 30783 Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ+)
 
Theoremxrpxdivcld 30784 Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞))
 
20.3.7  Words over a set - misc additions
 
Theoremwrdfd 30785 A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021.)
(𝜑𝑁 = (♯‘𝑊))    &   (𝜑𝑊 ∈ Word 𝑆)       (𝜑𝑊:(0..^𝑁)⟶𝑆)
 
Theoremwrdres 30786 Condition for the restriction of a word to be a word itself. (Contributed by Thierry Arnoux, 5-Oct-2018.)
((𝑊 ∈ Word 𝑆𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 ↾ (0..^𝑁)) ∈ Word 𝑆)
 
Theoremwrdsplex 30787* Existence of a split of a word at a given index. (Contributed by Thierry Arnoux, 11-Oct-2018.) (Proof shortened by AV, 3-Nov-2022.)
((𝑊 ∈ Word 𝑆𝑁 ∈ (0...(♯‘𝑊))) → ∃𝑣 ∈ Word 𝑆𝑊 = ((𝑊 ↾ (0..^𝑁)) ++ 𝑣))
 
Theorempfx1s2 30788 The prefix of length 1 of a length 2 word. (Contributed by Thierry Arnoux, 19-Sep-2023.)
((𝐴𝑉𝐵𝑉) → (⟨“𝐴𝐵”⟩ prefix 1) = ⟨“𝐴”⟩)
 
Theorempfxrn2 30789 The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14136. (Contributed by Thierry Arnoux, 12-Dec-2023.)
((𝑊 ∈ Word 𝑆𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
 
Theorempfxrn3 30790 Express the range of a prefix of a word. Stronger version of pfxrn2 30789. (Contributed by Thierry Arnoux, 13-Dec-2023.)
((𝑊 ∈ Word 𝑆𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
 
Theorempfxf1 30791 Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝑊:dom 𝑊1-1𝑆)    &   (𝜑𝐿 ∈ (0...(♯‘𝑊)))       (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1𝑆)
 
Theorems1f1 30792 Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑𝐼𝐷)       (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1𝐷)
 
Theorems2rn 30793 Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)       (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
 
Theorems2f1 30794 Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)       (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
 
Theorems3rn 30795 Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)       (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
 
Theorems3f1 30796 Conditions for a length 3 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1𝐷)
 
Theorems3clhash 30797 Closure of the words of length 3 in a preimage using the hash function. (Contributed by Thierry Arnoux, 27-Sep-2023.)
⟨“𝐼𝐽𝐾”⟩ ∈ (♯ “ {3})
 
Theoremccatf1 30798 Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑𝑆𝑉)    &   (𝜑𝐴 ∈ Word 𝑆)    &   (𝜑𝐵 ∈ Word 𝑆)    &   (𝜑𝐴:dom 𝐴1-1𝑆)    &   (𝜑𝐵:dom 𝐵1-1𝑆)    &   (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)       (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1𝑆)
 
Theorempfxlsw2ccat 30799 Reconstruct a word from its prefix and its last two symbols. (Contributed by Thierry Arnoux, 26-Sep-2023.)
𝑁 = (♯‘𝑊)       ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ 𝑁) → 𝑊 = ((𝑊 prefix (𝑁 − 2)) ++ ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩))
 
Theoremwrdt2ind 30800* Perform an induction over the structure of a word of even length. (Contributed by Thierry Arnoux, 26-Sep-2023.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ++ ⟨“𝑖𝑗”⟩) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Word 𝐵𝑖𝐵𝑗𝐵) → (𝜒𝜃))       ((𝐴 ∈ Word 𝐵 ∧ 2 ∥ (♯‘𝐴)) → 𝜏)
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-45982
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