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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdpexpp1 29601 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 29602 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 29603 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)

𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   (𝐴𝐵 + 𝐶𝐷) = 𝐸𝐹       ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)

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 29606 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 29607 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 29608 Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
(3 / 2) = (1.5)

Theorem1mhdrd 29609 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 29610 Extend class notation to include division of extended reals.
class /𝑒

Definitiondf-xdiv 29611* 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 29612* 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 29613* 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 29614 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 29615* 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 29616 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 29617 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)

Theoremxdivmul 29618 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 29619 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 29620 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 29621 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)

Theoremxdiv0 29622 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)

Theoremxdiv0rp 29623 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)

Theoremeliccioo 29624 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 29625 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+𝐴 = +∞))

Theoremxdivpnfrp 29626 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)

Theoremrpxdivcld 29627 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 29628 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 29629 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 29630 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 29631 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 29632 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 29633 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 29634* There exists at most one decomposition of a prime as a sum of two squares. See 2sqb 25151 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 29635 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 29636 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 29637 Weaker version of abvpropd 18836. (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 29638 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppg𝑅)    &    = (le‘𝑅)        = (le‘𝑂)

Theoremoppglt 29639 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 29640 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 29641 Being a preset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Preset → 𝐷 ∈ Preset )

Theoremposrasymb 29642 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 29643 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ Toset → 𝐹 ∈ Poset)

Theoremresspos 29644 The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)

Theoremresstos 29645 The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Toset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Toset)

Theoremtleile 29646 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 29647 In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 16960. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))

Theoremodutos 29648 Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Toset → 𝐷 ∈ Toset)

Theoremtlt2 29649 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 29650 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 29651 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 29652* Lemma for toslub 29653 and xrsclat 29665. (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 29653 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 29654* Lemma for tosglb 29655 and xrsclat 29665. (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 29655 Same theorem as toslub 29653, 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 29656 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 29657 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)

20.3.8.5  Extended reals Structure - misc additions

Axiomax-xrssca 29658 Assume the scalar component of the extended real structure is the field of the real numbers (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
fld = (Scalar‘ℝ*𝑠)

Axiomax-xrsvsca 29659 Assume the scalar product of the extended real structure is the extended real number multiplication operation (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
·e = ( ·𝑠 ‘ℝ*𝑠)

Theoremxrs0 29660 The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 12076 and df-xrs 16156), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)
0 = (0g‘ℝ*𝑠)

Theoremxrslt 29661 The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.)
< = (lt‘ℝ*𝑠)

Theoremxrsinvgval 29662 The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 12076 and df-xrs 16156), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)
(𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵)

Theoremxrsmulgzz 29663 The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))

Theoremxrstos 29664 The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
*𝑠 ∈ Toset

Theoremxrsclat 29665 The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)
*𝑠 ∈ CLat

Theoremxrsp0 29666 The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Proof shortened by AV, 28-Sep-2020.)
-∞ = (0.‘ℝ*𝑠)

Theoremxrsp1 29667 The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
+∞ = (1.‘ℝ*𝑠)

Theoremressmulgnn 29668 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐴 ⊆ (Base‘𝐺)    &    = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝑁 ∈ ℕ ∧ 𝑋𝐴) → (𝑁(.g𝐻)𝑋) = (𝑁 𝑋))

Theoremressmulgnn0 29669 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐴 ⊆ (Base‘𝐺)    &    = (.g𝐺)    &   𝐼 = (invg𝐺)    &   (0g𝐺) = (0g𝐻)       ((𝑁 ∈ ℕ0𝑋𝐴) → (𝑁(.g𝐻)𝑋) = (𝑁 𝑋))

20.3.8.6  The extended nonnegative real numbers commutative monoid

Theoremxrge0base 29670 The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(0[,]+∞) = (Base‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge00 29671 The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
0 = (0g‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge0plusg 29672 The additive law of the extended nonnegative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
+𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge0le 29673 The lower-or-equal relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.)
≤ = (le‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge0mulgnn0 29674 The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
((𝐴 ∈ ℕ0𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵))

Theoremxrge0addass 29675 Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶)))

Theoremxrge0addgt0 29676 The sum of nonnegative and positive numbers is positive. See addgtge0 10513. (Contributed by Thierry Arnoux, 6-Jul-2017.)
(((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵))

Theoremxrge0adddir 29677 Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶)))

Theoremxrge0adddi 29678 Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ·e (𝐴 +𝑒 𝐵)) = ((𝐶 ·e 𝐴) +𝑒 (𝐶 ·e 𝐵)))

Theoremxrge0npcan 29679 Extended nonnegative real version of npcan 10287. (Contributed by Thierry Arnoux, 9-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)

Theoremfsumrp0cl 29680* Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → Σ𝑘𝐴 𝐵 ∈ (0[,)+∞))

20.3.9  Algebra

20.3.9.1  Monoids Homomorphisms

Theoremabliso 29681 The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.)
((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

20.3.9.2  Ordered monoids and groups

Syntaxcomnd 29682 Extend class notation with the class of all right ordered monoids.
class oMnd

Syntaxcogrp 29683 Extend class notation with the class of all right ordered groups.
class oGrp

Definitiondf-omnd 29684* Define class of all right ordered monoids. An ordered monoid is a monoid with a total ordering compatible with its operation. It is possible to use this definition also for left ordered monoids, by applying it to (oppg𝑀). (Contributed by Thierry Arnoux, 13-Mar-2018.)
oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+g𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}

Definitiondf-ogrp 29685 Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 13-Mar-2018.)
oGrp = (Grp ∩ oMnd)

Theoremisomnd 29686* A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (le‘𝑀)       (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))

Theoremisogrp 29687 A (left) ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Theoremogrpgrp 29688 An left ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
(𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Theoremomndmnd 29689 A left ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd → 𝑀 ∈ Mnd)

Theoremomndtos 29690 A left ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Theoremomndadd 29691 In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 + 𝑍) (𝑌 + 𝑍))

Theoremomndaddr 29692 In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)       (((oppg𝑀) ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑍 + 𝑋) (𝑍 + 𝑌))

Theoremomndadd2d 29693 In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑𝑀 ∈ CMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))

Theoremomndadd2rd 29694 In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑 → (oppg𝑀) ∈ oMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))

Theoremsubmomnd 29695 A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)

Theoremxrge0omnd 29696 The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.)
(ℝ*𝑠s (0[,]+∞)) ∈ oMnd

Theoremomndmul2 29697 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))

Theoremomndmul3 29698 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℕ0)    &   (𝜑𝑁𝑃)    &   (𝜑𝑋𝐵)    &   (𝜑0 𝑋)       (𝜑 → (𝑁 · 𝑋) (𝑃 · 𝑋))

Theoremomndmul 29699 In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 𝑌)       (𝜑 → (𝑁 · 𝑋) (𝑁 · 𝑌))

TheoremogrpinvOLD 29700 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 30-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) 0 )

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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 42301-42322
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