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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodtp 29701* 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 29702* 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 29703* 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)
 
20.3.5.10  Decimal numbers
 
Theoremdfdec100 29704 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 29705 Constant used for decimal fraction constructor. See df-dp2 29706.
class 𝐴𝐵
 
Definitiondf-dp2 29706 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 11532. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
𝐴𝐵 = (𝐴 + (𝐵 / 10))
 
Theoremdfdp2OLD 29707 Obsolete version of df-dp2 29706 as of 9-Sep-2021. (Contributed by David A. Wheeler, 15-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 = (𝐴 + (𝐵 / 10))
 
Theoremdp2eq1 29708 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 
Theoremdp2eq2 29709 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 
Theoremdp2eq1i 29710 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶
 
Theoremdp2eq2i 29711 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵
 
Theoremdp2eq12i 29712 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷
 
Theoremdp20u 29713 Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       𝐴0 = 𝐴
 
Theoremdp20h 29714 Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       0𝐴 = (𝐴 / 10)
 
Theoremdp2cl 29715 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 29716 Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℚ       𝐴𝐵 ∈ ℚ
 
Theoremrpdp2cl 29717 Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴𝐵 ∈ ℝ+
 
Theoremrpdp2cl2 29718 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 29719 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 29720 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 29721 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 29722 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 29724 and df-dp2 29706 for more information; dpval2 29729 and dpfrac1 29727 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 11532.

 
Syntaxcdp 29723 Decimal point operator. See df-dp 29724.
class .
 
Definitiondf-dp 29724* 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 29725 Define the value of the decimal point operator. See df-dp 29724. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
 
Theoremdpcl 29726 Prove that the closure of the decimal point is as we have defined it. See df-dp 29724. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
 
Theoremdpfrac1 29727 Prove a simple equivalence involving the decimal point. See df-dp 29724 and dpcl 29726. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))
 
Theoremdpfrac1OLD 29728 Obsolete version of dpfrac1 29727 as of 9-Sep-2021. (Contributed by David A. Wheeler, 15-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))
 
Theoremdpval2 29729 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = (𝐴 + (𝐵 / 10))
 
Theoremdpval3 29730 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = 𝐴𝐵
 
Theoremdpmul10 29731 Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       ((𝐴.𝐵) · 10) = 𝐴𝐵
 
Theoremdecdiv10 29732 Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴𝐵 / 10) = (𝐴.𝐵)
 
Theoremdpmul100 29733 Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℝ       ((𝐴.𝐵𝐶) · 100) = 𝐴𝐵𝐶
 
Theoremdp3mul10 29734 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 29735 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 29736 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       (𝐴.𝐵) = 𝐴𝐵
 
Theoremdp0u 29737 Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       (𝐴.0) = 𝐴
 
Theoremdp0h 29738 Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       (0.𝐴) = (𝐴 / 10)
 
Theoremrpdpcl 29739 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 29740 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 29741 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 29742 Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴 < (𝐴.𝐵)
 
Theoremdpltc 29743 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 29744 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 29745 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 29746 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 29747 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 29748 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 29749 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 29750 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 29751 Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
(3 / 2) = (1.5)
 
Theorem1mhdrd 29752 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 29753 Extend class notation to include division of extended reals.
class /𝑒
 
Definitiondf-xdiv 29754* 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 29755* 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 29756* 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 29757 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 29758* 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 29759 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 29760 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)
 
Theoremxdivmul 29761 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 29762 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 29763 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 29764 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
 
Theoremxdiv0 29765 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
 
Theoremxdiv0rp 29766 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
 
Theoremeliccioo 29767 Membership in a closed interval of extended reals vs. 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 29768 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+𝐴 = +∞))
 
Theoremxdivpnfrp 29769 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
 
Theoremrpxdivcld 29770 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 29771 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  Prime Number Theory
 
20.3.7.1  Fermat's two square theorem
 
Theorembhmafibid1 29772 The Brahmagupta-Fibonacci identity. Express the product of two sums of two squares as a sum of two squares. First result. (Contributed by Thierry Arnoux, 1-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) − (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) + (𝐵 · 𝐶))↑2)))
 
Theorembhmafibid2 29773 The Brahmagupta-Fibonacci identity. Express the product of two sums of two squares as a sum of two squares. Second result. (Contributed by Thierry Arnoux, 1-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) + (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) − (𝐵 · 𝐶))↑2)))
 
Theorem2sqn0 29774 If the sum of two squares is prime, none of the original number is zero. (Contributed by Thierry Arnoux, 4-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)       (𝜑𝐴 ≠ 0)
 
Theorem2sqcoprm 29775 If the sum of two squares is prime, the two original numbers are coprime. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)       (𝜑 → (𝐴 gcd 𝐵) = 1)
 
Theorem2sqmod 29776 Given two decompositions of a prime as a sum of two squares, show that they are equal. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)    &   (𝜑 → ((𝐶↑2) + (𝐷↑2)) = 𝑃)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theorem2sqmo 29777* There exists at most one decomposition of a prime as a sum of two squares. See 2sqb 25202 for the existence of such a decomposition. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
 
20.3.8  Extensible Structures
 
20.3.8.1  Structure restriction operator
 
Theoremressplusf 29778 The group operation function +𝑓 of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
𝐵 = (Base‘𝐺)    &   𝐻 = (𝐺s 𝐴)    &    = (+g𝐺)    &    Fn (𝐵 × 𝐵)    &   𝐴𝐵       (+𝑓𝐻) = ( ↾ (𝐴 × 𝐴))
 
Theoremressnm 29779 The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑁 = (norm‘𝐺)       ((𝐺 ∈ Mnd ∧ 0𝐴𝐴𝐵) → (𝑁𝐴) = (norm‘𝐻))
 
Theoremabvpropd2 29780 Weaker version of abvpropd 18890. (Contributed by Thierry Arnoux, 8-Nov-2017.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (+g𝐾) = (+g𝐿))    &   (𝜑 → (.r𝐾) = (.r𝐿))       (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
 
20.3.8.2  The opposite group
 
Theoremoppgle 29781 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppg𝑅)    &    = (le‘𝑅)        = (le‘𝑂)
 
Theoremoppglt 29782 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppg𝑅)    &    < = (lt‘𝑅)       (𝑅𝑉< = (lt‘𝑂))
 
20.3.8.3  Posets
 
Theoremressprs 29783 The restriction of a preordered set is still a preordered set. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐵 = (Base‘𝐾)       ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Preset )
 
Theoremoduprs 29784 Being a preset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Preset → 𝐷 ∈ Preset )
 
Theoremposrasymb 29785 A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
 
Theoremtospos 29786 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ Toset → 𝐹 ∈ Poset)
 
Theoremresspos 29787 The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)
 
Theoremresstos 29788 The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Toset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Toset)
 
Theoremtleile 29789 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
 
Theoremtltnle 29790 In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 17013. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))
 
Theoremodutos 29791 Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Toset → 𝐷 ∈ Toset)
 
Theoremtlt2 29792 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 < 𝑋))
 
Theoremtlt3 29793 In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋))
 
Theoremtrleile 29794 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
 
Theoremtoslublem 29795* Lemma for toslub 29796 and xrsclat 29808. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &    = (le‘𝐾)       ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑏 𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏 𝑐𝑎 𝑐)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
 
Theoremtoslub 29796 In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐵, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)       (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
 
Theoremtosglblem 29797* Lemma for tosglb 29798 and xrsclat 29808. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &    = (le‘𝐾)       ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑎 𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐 𝑏𝑐 𝑎)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
 
Theoremtosglb 29798 Same theorem as toslub 29796, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)       (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))
 
20.3.8.4  Complete lattices
 
Theoremclatp0cl 29799 The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0.‘𝑊)       (𝑊 ∈ CLat → 0𝐵)
 
Theoremclatp1cl 29800 The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
𝐵 = (Base‘𝑊)    &    1 = (1.‘𝑊)       (𝑊 ∈ CLat → 1𝐵)
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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 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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 42801-42879
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