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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hashunif 30701* | The cardinality of a disjoint finite union of finite sets. Cf. hashuni 15274. (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 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) ·e (♯‘𝐵))) | ||
Theorem | hashgt1 30703 | 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))) | ||
Theorem | dvdszzq 30704 | Divisibility for an integer quotient. (Contributed by Thierry Arnoux, 17-Sep-2023.) |
⊢ 𝑁 = (𝐴 / 𝐵) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝑃 ∥ 𝐴) & ⊢ (𝜑 → ¬ 𝑃 ∥ 𝐵) ⇒ ⊢ (𝜑 → 𝑃 ∥ 𝑁) | ||
Theorem | prmdvdsbc 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)) | ||
Theorem | numdenneg 30706 | 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 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)))) | ||
Theorem | nnindf 30708* | 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 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 → (𝜓 ↔ 𝜒)) & ⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃)) & ⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏)) & ⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝜓) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏)) | ||
Theorem | subne0nn 30710 | 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 30711 | Subtracting an integer number from another number decreases it. See ltsubrpd 12546. (Contributed by Thierry Arnoux, 18-Apr-2017.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀) | ||
Theorem | fprodeq02 30712* | 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 30713 | The range of the indicator function is a subset of ℝ. (Contributed by Thierry Arnoux, 14-Aug-2017.) |
⊢ {0, 1} ⊆ ℝ | ||
Theorem | fprodex01 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 = 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 30715* | 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 30716* | 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 30717* | 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 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.) |
⊢ (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) | ||
Theorem | dfdec100 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) | ||
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 30720 | Constant used for decimal fraction constructor. See df-dp2 30721. |
class _𝐴𝐵 | ||
Definition | df-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)) | ||
Theorem | dp2eq1 30722 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶) | ||
Theorem | dp2eq2 30723 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵) | ||
Theorem | dp2eq1i 30724 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ _𝐴𝐶 = _𝐵𝐶 | ||
Theorem | dp2eq2i 30725 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ _𝐶𝐴 = _𝐶𝐵 | ||
Theorem | dp2eq12i 30726 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ _𝐴𝐶 = _𝐵𝐷 | ||
Theorem | dp20u 30727 | Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ _𝐴0 = 𝐴 | ||
Theorem | dp20h 30728 | Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℝ+ ⇒ ⊢ _0𝐴 = (𝐴 / ;10) | ||
Theorem | dp2cl 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) | ||
Theorem | dp2clq 30730 | Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℚ ⇒ ⊢ _𝐴𝐵 ∈ ℚ | ||
Theorem | rpdp2cl 30731 | 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 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+ | ||
Theorem | dp2lt10 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 | ||
Theorem | dp2lt 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 | ||
Theorem | dp2ltsuc 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 | ||
Theorem | dp2ltc 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 | ||
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. | ||
Syntax | cdp 30737 | Decimal point operator. See df-dp 30738. |
class . | ||
Definition | df-dp 30738* |
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 30739 | Define the value of the decimal point operator. See df-dp 30738. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | ||
Theorem | dpcl 30740 | Prove that the closure of the decimal point is ℝ as we have defined it. See df-dp 30738. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ) | ||
Theorem | dpfrac1 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 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) | ||
Theorem | dpval2 30742 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) | ||
Theorem | dpval3 30743 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = _𝐴𝐵 | ||
Theorem | dpmul10 30744 | Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵 | ||
Theorem | decdiv10 30745 | Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵) | ||
Theorem | dpmul100 30746 | 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 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) | ||
Theorem | dpmul1000 30748 | 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 30749 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ (𝐴.𝐵) = _𝐴𝐵 | ||
Theorem | dp0u 30750 | Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (𝐴.0) = 𝐴 | ||
Theorem | dp0h 30751 | Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℝ+ ⇒ ⊢ (0.𝐴) = (𝐴 / ;10) | ||
Theorem | rpdpcl 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+ | ||
Theorem | dplt 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) | ||
Theorem | dplti 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 | ||
Theorem | dpgti 30755 | 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 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) | ||
Theorem | dpexpp1 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)) | ||
Theorem | 0dp2dp 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) | ||
Theorem | dpadd2 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) | ||
Theorem | dpadd 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) | ||
Theorem | dpadd3 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) | ||
Theorem | dpmul 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) | ||
Theorem | dpmul4 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._𝐵_𝐶𝐷) · (𝐸._𝐹_𝐺𝐻)) < (𝑊._𝑋_𝑌𝑍) | ||
Theorem | threehalves 30764 | Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
⊢ (3 / 2) = (1.5) | ||
Theorem | 1mhdrd 30765 | Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
⊢ ((0._99) + (0._01)) = 1 | ||
Syntax | cxdiv 30766 | Extend class notation to include division of extended reals. |
class /𝑒 | ||
Definition | df-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)) | ||
Theorem | xdivval 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)) | ||
Theorem | xrecex 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) | ||
Theorem | xmulcand 30770 | 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 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) | ||
Theorem | xdivcld 30772 | 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 30773 | 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 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)) | ||
Theorem | rexdiv 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)) | ||
Theorem | xdivrec 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))) | ||
Theorem | xdivid 30777 | A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1) | ||
Theorem | xdiv0 30778 | Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0) | ||
Theorem | xdiv0rp 30779 | Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
⊢ (𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0) | ||
Theorem | eliccioo 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* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵))) | ||
Theorem | elxrge02 30781 | Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞)) | ||
Theorem | xdivpnfrp 30782 | Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞) | ||
Theorem | rpxdivcld 30783 | 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+) | ||
Theorem | xrpxdivcld 30784 | Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) | ||
Theorem | wrdfd 30785 | A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021.) |
⊢ (𝜑 → 𝑁 = (♯‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) ⇒ ⊢ (𝜑 → 𝑊:(0..^𝑁)⟶𝑆) | ||
Theorem | wrdres 30786 | Condition for the restriction of a word to be a word itself. (Contributed by Thierry Arnoux, 5-Oct-2018.) |
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 ↾ (0..^𝑁)) ∈ Word 𝑆) | ||
Theorem | wrdsplex 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 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ∃𝑣 ∈ Word 𝑆𝑊 = ((𝑊 ↾ (0..^𝑁)) ++ 𝑣)) | ||
Theorem | pfx1s2 30788 | The prefix of length 1 of a length 2 word. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (〈“𝐴𝐵”〉 prefix 1) = 〈“𝐴”〉) | ||
Theorem | pfxrn2 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 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊) | ||
Theorem | pfxrn3 30790 | Express the range of a prefix of a word. Stronger version of pfxrn2 30789. (Contributed by Thierry Arnoux, 13-Dec-2023.) |
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿))) | ||
Theorem | pfxf1 30791 | Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.) |
⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝑆) & ⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝑊))) ⇒ ⊢ (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆) | ||
Theorem | s1f1 30792 | Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝐷) ⇒ ⊢ (𝜑 → 〈“𝐼”〉:dom 〈“𝐼”〉–1-1→𝐷) | ||
Theorem | s2rn 30793 | Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) ⇒ ⊢ (𝜑 → ran 〈“𝐼𝐽”〉 = {𝐼, 𝐽}) | ||
Theorem | s2f1 30794 | Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) ⇒ ⊢ (𝜑 → 〈“𝐼𝐽”〉:dom 〈“𝐼𝐽”〉–1-1→𝐷) | ||
Theorem | s3rn 30795 | Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) ⇒ ⊢ (𝜑 → ran 〈“𝐼𝐽𝐾”〉 = {𝐼, 𝐽, 𝐾}) | ||
Theorem | s3f1 30796 | Conditions for a length 3 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉:dom 〈“𝐼𝐽𝐾”〉–1-1→𝐷) | ||
Theorem | s3clhash 30797 | Closure of the words of length 3 in a preimage using the hash function. (Contributed by Thierry Arnoux, 27-Sep-2023.) |
⊢ 〈“𝐼𝐽𝐾”〉 ∈ (◡♯ “ {3}) | ||
Theorem | ccatf1 30798 | Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.) |
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐵 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐴:dom 𝐴–1-1→𝑆) & ⊢ (𝜑 → 𝐵:dom 𝐵–1-1→𝑆) & ⊢ (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆) | ||
Theorem | pfxlsw2ccat 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))”〉)) | ||
Theorem | wrdt2ind 30800* | Perform an induction over the structure of a word of even length. (Contributed by Thierry Arnoux, 26-Sep-2023.) |
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 ++ 〈“𝑖𝑗”〉) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝐴 ∈ Word 𝐵 ∧ 2 ∥ (♯‘𝐴)) → 𝜏) |
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