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Theorem List for Metamath Proof Explorer - 44601-44700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelbigofrcl 44601 Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)
(𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))

Theoremelbigo 44602* Properties of a function of order G(x). (Contributed by AV, 16-May-2020.)
(𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))

Theoremelbigo2 44603* Properties of a function of order G(x) under certain assumptions. (Contributed by AV, 17-May-2020.)
(((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵𝐴)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (𝐹𝑦) ≤ (𝑚 · (𝐺𝑦)))))

Theoremelbigo2r 44604* Sufficient condition for a function to be of order G(x). (Contributed by AV, 18-May-2020.)
(((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥𝐵 (𝐶𝑥 → (𝐹𝑥) ≤ (𝑀 · (𝐺𝑥))))) → 𝐹 ∈ (Ο‘𝐺))

Theoremelbigof 44605 A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)
(𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)

Theoremelbigodm 44606 The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020.)
(𝐹 ∈ (Ο‘𝐺) → dom 𝐹 ⊆ ℝ)

Theoremelbigoimp 44607* The defining property of a function of order G(x). (Contributed by AV, 18-May-2020.)
((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))

Theoremelbigolo1 44608 A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.) (Proof shortened by II, 16-Feb-2023.)
((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ+𝐹:𝐴⟶ℝ+) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 /f 𝐺) ∈ ≤𝑂(1)))

20.41.22.7  Logarithm to an arbitrary base (extension)

Theoremrege1logbrege0 44609 The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋))

Theoremrege1logbzge0 44610 The general logarithm, with an integer base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋))

Theoremfllogbd 44611 A real number is between the base of a logarithm to the power of the floor of the logarithm of the number and the base of the logarithm to the power of the floor of the logarithm of the number plus one. (Contributed by AV, 23-May-2020.)
(𝜑𝐵 ∈ (ℤ‘2))    &   (𝜑𝑋 ∈ ℝ+)    &   𝐸 = (⌊‘(𝐵 logb 𝑋))       (𝜑 → ((𝐵𝐸) ≤ 𝑋𝑋 < (𝐵↑(𝐸 + 1))))

Theoremrelogbmulbexp 44612 The logarithm of the product of a positive real number and the base to the power of a real number is the logarithm of the positive real number plus the real number. (Contributed by AV, 29-May-2020.)
((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+𝐶 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐵𝑐𝐶))) = ((𝐵 logb 𝐴) + 𝐶))

Theoremrelogbdivb 44613 The logarithm of the quotient of a positive real number and the base is the logarithm of the number minus 1. (Contributed by AV, 29-May-2020.)
((𝐵 ∈ (ℝ+ ∖ {1}) ∧ 𝐴 ∈ ℝ+) → (𝐵 logb (𝐴 / 𝐵)) = ((𝐵 logb 𝐴) − 1))

Theoremlogbge0b 44614 The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+) → (0 ≤ (𝐵 logb 𝑋) ↔ 1 ≤ 𝑋))

Theoremlogblt1b 44615 The logarithm of a number is less than 1 iff the number is less than the base of the logarithm. (Contributed by AV, 30-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+) → ((𝐵 logb 𝑋) < 1 ↔ 𝑋 < 𝐵))

20.41.22.8  The binary logarithm

If the binary logarithm is used more often, a separate symbol/definition could be provided for it, e.g. log2 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (2 logb 𝑋)). Then we can write "( log2 ` x )" (analogous to (log𝑥) for the natural logarithm) instead of (2 logb 𝑥).

Theoremfldivexpfllog2 44616 The floor of a positive real number divided by 2 to the power of the floor of the logarithm to base 2 of the number is 1. (Contributed by AV, 26-May-2020.)
(𝑋 ∈ ℝ+ → (⌊‘(𝑋 / (2↑(⌊‘(2 logb 𝑋))))) = 1)

Theoremnnlog2ge0lt1 44617 A positive integer is 1 iff its binary logarithm is between 0 and 1. (Contributed by AV, 30-May-2020.)
(𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)))

Theoremlogbpw2m1 44618 The floor of the binary logarithm of 2 to the power of a positive integer minus 1 is equal to the integer minus 1. (Contributed by AV, 31-May-2020.)
(𝐼 ∈ ℕ → (⌊‘(2 logb ((2↑𝐼) − 1))) = (𝐼 − 1))

Theoremfllog2 44619 The floor of the binary logarithm of 2 to the power of an element of a half-open integer interval bounded by powers of 2 is equal to the integer. (Contributed by AV, 31-May-2020.)
((𝐼 ∈ ℕ0𝑁 ∈ ((2↑𝐼)..^(2↑(𝐼 + 1)))) → (⌊‘(2 logb 𝑁)) = 𝐼)

20.41.22.9  Binary length

Syntaxcblen 44620 Extend class notation with the class of the binary length function.
class #b

Definitiondf-blen 44621 Define the binary length of an integer. Definition in section 1.3 of [AhoHopUll] p. 12. Although not restricted to integers, this definition is only meaningful for 𝑛 ∈ ℤ or even for 𝑛 ∈ ℂ. (Contributed by AV, 16-May-2020.)
#b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))

Theoremblenval 44622 The binary length of an integer. (Contributed by AV, 20-May-2020.)
(𝑁𝑉 → (#b𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))

Theoremblen0 44623 The binary length of 0. (Contributed by AV, 20-May-2020.)
(#b‘0) = 1

Theoremblenn0 44624 The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020.)
((𝑁𝑉𝑁 ≠ 0) → (#b𝑁) = ((⌊‘(2 logb (abs‘𝑁))) + 1))

Theoremblenre 44625 The binary length of a positive real number. (Contributed by AV, 20-May-2020.)
(𝑁 ∈ ℝ+ → (#b𝑁) = ((⌊‘(2 logb 𝑁)) + 1))

Theoremblennn 44626 The binary length of a positive integer. (Contributed by AV, 21-May-2020.)
(𝑁 ∈ ℕ → (#b𝑁) = ((⌊‘(2 logb 𝑁)) + 1))

Theoremblennnelnn 44627 The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.)
(𝑁 ∈ ℕ → (#b𝑁) ∈ ℕ)

Theoremblennn0elnn 44628 The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.)
(𝑁 ∈ ℕ0 → (#b𝑁) ∈ ℕ)

Theoremblenpw2 44629 The binary length of a power of 2 is the exponent plus 1. (Contributed by AV, 30-May-2020.)
(𝐼 ∈ ℕ0 → (#b‘(2↑𝐼)) = (𝐼 + 1))

Theoremblenpw2m1 44630 The binary length of a power of 2 minus 1 is the exponent. (Contributed by AV, 31-May-2020.)
(𝐼 ∈ ℕ → (#b‘((2↑𝐼) − 1)) = 𝐼)

Theoremnnpw2blen 44631 A positive integer is between 2 to the power of its binary length minus 1 and 2 to the power of its binary length. (Contributed by AV, 31-May-2020.)
(𝑁 ∈ ℕ → ((2↑((#b𝑁) − 1)) ≤ 𝑁𝑁 < (2↑(#b𝑁))))

Theoremnnpw2blenfzo 44632 A positive integer is between 2 to the power of the binary length of the integer minus 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.)
(𝑁 ∈ ℕ → 𝑁 ∈ ((2↑((#b𝑁) − 1))..^(2↑(#b𝑁))))

Theoremnnpw2blenfzo2 44633 A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.)
(𝑁 ∈ ℕ → (𝑁 = (2↑((#b𝑁) − 1)) ∨ 𝑁 ∈ (((2↑((#b𝑁) − 1)) + 1)..^(2↑(#b𝑁)))))

Theoremnnpw2pmod 44634 Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
(𝑁 ∈ ℕ → 𝑁 = ((2↑((#b𝑁) − 1)) + (𝑁 mod (2↑((#b𝑁) − 1)))))

Theoremblen1 44635 The binary length of 1. (Contributed by AV, 21-May-2020.)
(#b‘1) = 1

Theoremblen2 44636 The binary length of 2. (Contributed by AV, 21-May-2020.)
(#b‘2) = 2

Theoremnnpw2p 44637* Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
(𝑁 ∈ ℕ → ∃𝑖 ∈ ℕ0𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))

Theoremnnpw2pb 44638* A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
(𝑁 ∈ ℕ ↔ ∃𝑖 ∈ ℕ0𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))

Theoremblen1b 44639 The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.)
(𝑁 ∈ ℕ0 → ((#b𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1)))

Theoremblennnt2 44640 The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, 30-May-2010.)
(𝑁 ∈ ℕ → (#b‘(2 · 𝑁)) = ((#b𝑁) + 1))

Theoremnnolog2flm1 44641 The floor of the binary logarithm of an odd integer greater than 1 is the floor of the binary logarithm of the integer decreased by 1. (Contributed by AV, 2-Jun-2020.)
((𝑁 ∈ (ℤ‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ) → (⌊‘(2 logb 𝑁)) = (⌊‘(2 logb (𝑁 − 1))))

Theoremblennn0em1 44642 The binary length of the half of an even positive integer is the binary length of the integer minus 1. (Contributed by AV, 30-May-2010.)
((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b‘(𝑁 / 2)) = ((#b𝑁) − 1))

Theoremblennngt2o2 44643 The binary length of an odd integer greater than 1 is the binary length of the half of the integer decreased by 1, increased by 1. (Contributed by AV, 3-Jun-2020.)
((𝑁 ∈ (ℤ‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (#b𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1))

Theoremblengt1fldiv2p1 44644 The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.)
(𝑁 ∈ (ℤ‘2) → (#b𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))

Theoremblennn0e2 44645 The binary length of an even positive integer is the binary length of the half of the integer, increased by 1. (Contributed by AV, 29-May-2020.)
((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b𝑁) = ((#b‘(𝑁 / 2)) + 1))

20.41.22.10  Digits

Generalization of df-bits 15763. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 44665: ((𝐾(digit 2 ) N ) = 1 <-> K e. ( bits 𝑁)).

Syntaxcdig 44646 Extend class notation with the class of the digit extraction operation.
class digit

Definitiondf-dig 44647* Definition of an operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝑏. 𝑘 = − 1 corresponds to the first digit of the fractional part (for 𝑏 = 10 the first digit after the decimal point), 𝑘 = 0 corresponds to the last digit of the integer part (for 𝑏 = 10 the first digit before the decimal point). See also digit1 13590. Examples (not formal): ( 234.567 ( digit ` 10 ) 0 ) = 4; ( 2.567 ( digit ` 10 ) -2 ) = 6; ( 2345.67 ( digit ` 10 ) 2 ) = 3. (Contributed by AV, 16-May-2020.)
digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))

Theoremdigfval 44648* Operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
(𝐵 ∈ ℕ → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵)))

Theoremdigval 44649 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵))

Theoremdigvalnn0 44650 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵 is a nonnegative integer. (Contributed by AV, 28-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) ∈ ℕ0)

Theoremnn0digval 44651 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ0𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵𝐾))) mod 𝐵))

Theoremdignn0fr 44652 The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ (ℤ ∖ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)𝑁) = 0)

Theoremdignn0ldlem 44653 Lemma for dignnld 44654. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵𝐾))

Theoremdignnld 44654 The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → (𝐾(digit‘𝐵)𝑁) = 0)

Theoremdig2nn0ld 44655 The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘(#b𝑁))) → (𝐾(digit‘2)𝑁) = 0)

Theoremdig2nn1st 44656 The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
(𝑁 ∈ ℕ → (((#b𝑁) − 1)(digit‘2)𝑁) = 1)

Theoremdig0 44657 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)

Theoremdigexp 44658 The 𝐾 th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵𝑁)) = if(𝐾 = 𝑁, 1, 0))

Theoremdig1 44659 All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)1) = if(𝐾 = 0, 1, 0))

Theorem0dig1 44660 The 0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.)
(𝐵 ∈ (ℤ‘2) → (0(digit‘𝐵)1) = 1)

Theorem0dig2pr01 44661 The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
(𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁)

Theoremdig2nn0 44662 A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1})

Theorem0dig2nn0e 44663 The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 0)

Theorem0dig2nn0o 44664 The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 1)

Theoremdig2bits 44665 The 𝐾 th digit of a nonnegative integer 𝑁 in a binary system is its 𝐾 th bit. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)))

20.41.22.11  Nonnegative integer as sum of its shifted digits

Theoremdignn0flhalflem1 44666 Lemma 1 for dignn0flhalf 44669. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))

Theoremdignn0flhalflem2 44667 Lemma 2 for dignn0flhalf 44669. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))

Theoremdignn0ehalf 44668 The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)
(((𝐴 / 2) ∈ ℕ0𝐴 ∈ ℕ0𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))

Theoremdignn0flhalf 44669 The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)
((𝐴 ∈ (ℤ‘2) ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(⌊‘(𝐴 / 2))))

Theoremnn0sumshdiglemA 44670* Lemma for nn0sumshdig 44674 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)
(((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglemB 44671* Lemma for nn0sumshdig 44674 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.)
(((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglem1 44672* Lemma 1 for nn0sumshdig 44674 (induction step). (Contributed by AV, 7-Jun-2020.)
(𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝑦𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglem2 44673* Lemma 2 for nn0sumshdig 44674. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))

Theoremnn0sumshdig 44674* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
(𝐴 ∈ ℕ0𝐴 = Σ𝑘 ∈ (0..^(#b𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘)))

20.41.22.12  Algorithms for the multiplication of nonnegative integers

Theoremnn0mulfsum 44675* Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding up 𝑏 𝑎 times. (Contributed by AV, 17-May-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)

Theoremnn0mullong 44676* Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers 𝑎 and 𝑏 by multiplying the multiplicand 𝑏 by each digit of the multiplier 𝑎 and then add up all the properly shifted results. Here, the binary representation of the multiplier 𝑎 is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 15834. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))

20.41.23  Elementary geometry (extension)

20.41.23.1  Auxiliary theorems

Theoremfv1prop 44677 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐴𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘1) = 𝐴)

Theoremfv2prop 44678 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐵𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘2) = 𝐵)

Theoremsubmuladdmuld 44679 Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → (((𝐴𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷𝐶))))

Theoremaffinecomb1 44680* Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)    &   𝑆 = ((𝐺𝐹) / (𝐶𝐵))       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴𝐵)) + 𝐹)))

Theoremaffinecomb2 44681* Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶𝐵) · 𝐸) = (((𝐺𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺)))))

Theoremaffineid 44682 Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)

Theorem1subrec1sub 44683 Subtract the reciprocal of 1 minus a number from 1 results in the number divided by the number minus 1. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 1) → (1 − (1 / (1 − 𝐴))) = (𝐴 / (𝐴 − 1)))

Theoremresum2sqcl 44684 The sum of two squares of real numbers is a real number. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ)

Theoremresum2sqgt0 44685 The sum of the square of a nonzero real number and the square of another real number is greater than zero. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 0 < 𝑄)

Theoremresum2sqrp 44686 The sum of the square of a nonzero real number and the square of another real number is a positive real number. (Contributed by AV, 2-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ+)

Theoremresum2sqorgt0 44687 The sum of the square of two real numbers is greater than zero if at least one of the real numbers is nonzero. (Contributed by AV, 26-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) → 0 < 𝑄)

Theoremreorelicc 44688 Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶))

20.41.23.2  Real euclidean space of dimension 2

Theoremrrx2pxel 44689 The x-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (𝑋𝑃 → (𝑋‘1) ∈ ℝ)

Theoremrrx2pyel 44690 The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (𝑋𝑃 → (𝑋‘2) ∈ ℝ)

Theoremprelrrx2 44691 An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2. (Contributed by AV, 4-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {⟨1, 𝐴⟩, ⟨2, 𝐵⟩} ∈ 𝑃)

Theoremprelrrx2b 44692 An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2, determined by its coordinates. (Contributed by AV, 7-May-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((𝑍𝑃 ∧ (((𝑍‘1) = 𝐴 ∧ (𝑍‘2) = 𝐵) ∨ ((𝑍‘1) = 𝑋 ∧ (𝑍‘2) = 𝑌))) ↔ 𝑍 ∈ {{⟨1, 𝐴⟩, ⟨2, 𝐵⟩}, {⟨1, 𝑋⟩, ⟨2, 𝑌⟩}}))

Theoremrrx2pnecoorneor 44693 If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then they are different at least at one coordinate. (Contributed by AV, 26-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       ((𝑋𝑃𝑌𝑃𝑋𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))

Theoremrrx2pnedifcoorneor 44694 If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐴 = ((𝑌‘1) − (𝑋‘1))    &   𝐵 = ((𝑌‘2) − (𝑋‘2))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))

Theoremrrx2pnedifcoorneorr 44695 If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐴 = ((𝑌‘1) − (𝑋‘1))    &   𝐵 = ((𝑋‘2) − (𝑌‘2))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))

Theoremrrx2xpref1o 44696* There is a bijection between the set of ordered pairs of real numbers (the cartesian product of the real numbers) and the set of points in the two dimensional Euclidean plane (represented as mappings from {1, 2} to the real numbers). (Contributed by AV, 12-Mar-2023.)
𝑅 = (ℝ ↑m {1, 2})    &   𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})       𝐹:(ℝ × ℝ)–1-1-onto𝑅

Theoremrrx2xpreen 44697 The set of points in the two dimensional Euclidean plane and the set of ordered pairs of real numbers (the cartesian product of the real numbers) are equinumerous. (Contributed by AV, 12-Mar-2023.)
𝑅 = (ℝ ↑m {1, 2})       𝑅 ≈ (ℝ × ℝ)

Theoremrrx2plord 44698* The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point iff its first coordinate is less than the first coordinate of the other point, or the first coordinates of both points are equal and the second coordinate of the first point is less than the second coordinate of the other point: 𝑎, 𝑏⟩ ≤ ⟨𝑥, 𝑦 iff (𝑎 < 𝑥 ∨ (𝑎 = 𝑥𝑏𝑦)). (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}       ((𝑋𝑅𝑌𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))

Theoremrrx2plord1 44699* The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point if its first coordinate is less than the first coordinate of the other point. (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}       ((𝑋𝑅𝑌𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → 𝑋𝑂𝑌)

Theoremrrx2plord2 44700* The lexicographical ordering for points in the two dimensional Euclidean plane: if the first coordinates of two points are equal, a point is less than another point iff the second coordinate of the point is less than the second coordinate of the other point. (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   𝑅 = (ℝ ↑m {1, 2})       ((𝑋𝑅𝑌𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ (𝑋‘2) < (𝑌‘2)))

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