Home Metamath Proof ExplorerTheorem List (p. 196 of 449) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28689) Hilbert Space Explorer (28690-30212) Users' Mathboxes (30213-44899)

Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdrngunit 19501 Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ DivRing → (𝑋𝑈 ↔ (𝑋𝐵𝑋0 )))

Theoremdrngui 19502 The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑅 ∈ DivRing       (𝐵 ∖ { 0 }) = (Unit‘𝑅)

Theoremdrngring 19503 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing → 𝑅 ∈ Ring)

Theoremdrnggrp 19504 A division ring is a group. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Theoremisfld 19505 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))

Theoremisdrng2 19506 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))       (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))

Theoremdrngprop 19507 If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)    &   (.r𝐾) = (.r𝐿)       (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)

Theoremdrngmgp 19508 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))       (𝑅 ∈ DivRing → 𝐺 ∈ Grp)

Theoremdrngmcl 19509 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 }))

Theoremdrngid 19510 A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))       (𝑅 ∈ DivRing → 1 = (0g𝐺))

Theoremdrngunz 19511 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)
0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ DivRing → 10 )

Theoremdrngid2 19512 Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ DivRing → ((𝐼𝐵𝐼0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼))

Theoremdrnginvrcl 19513 Closure of the multiplicative inverse in a division ring. (reccl 11299 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝐼𝑋) ∈ 𝐵)

Theoremdrnginvrn0 19514 The multiplicative inverse in a division ring is nonzero. (recne0 11305 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝐼𝑋) ≠ 0 )

Theoremdrnginvrl 19515 Property of the multiplicative inverse in a division ring. (recid2 11307 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → ((𝐼𝑋) · 𝑋) = 1 )

Theoremdrnginvrr 19516 Property of the multiplicative inverse in a division ring. (recid 11306 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝑋 · (𝐼𝑋)) = 1 )

Theoremdrngmul0or 19517 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0𝑌 = 0 )))

Theoremdrngmulne0 19518 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋0𝑌0 )))

Theoremdrngmuleq0 19519 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑌0 )       (𝜑 → ((𝑋 · 𝑌) = 0𝑋 = 0 ))

Theoremopprdrng 19520 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
𝑂 = (oppr𝑅)       (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)

Theoremisdrngd 19521* Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a left-inverse 𝐼(𝑥). See isdrngd 19521 for the characterization using right-inverses. (Contributed by NM, 2-Aug-2013.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑0 = (0g𝑅))    &   (𝜑1 = (1r𝑅))    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝑥 · 𝑦) ≠ 0 )    &   (𝜑10 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼0 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → (𝐼 · 𝑥) = 1 )       (𝜑𝑅 ∈ DivRing)

Theoremisdrngrd 19522* Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a right-inverse 𝐼(𝑥). See isdrngd 19521 for the characterization using left-inverses. (Contributed by NM, 10-Aug-2013.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑0 = (0g𝑅))    &   (𝜑1 = (1r𝑅))    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝑥 · 𝑦) ≠ 0 )    &   (𝜑10 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼0 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → (𝑥 · 𝐼) = 1 )       (𝜑𝑅 ∈ DivRing)

Theoremdrngpropd 19523* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing))

Theoremfldpropd 19524* If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ Field ↔ 𝐿 ∈ Field))

10.4.2  Subrings of a ring

Syntaxcsubrg 19525 Extend class notation with all subrings of a ring.
class SubRing

Syntaxcrgspn 19526 Extend class notation with span of a set of elements over a ring.
class RingSpan

Definitiondf-subrg 19527* Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity.

The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset (ℤ × {0}) of (ℤ × ℤ) (where multiplication is componentwise) contains the false identity ⟨1, 0⟩ which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})

Definitiondf-rgspn 19528* The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡}))

Theoremissubrg 19529 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))

Theoremsubrgss 19530 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐵 = (Base‘𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 𝐴𝐵)

Theoremsubrgid 19531 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))

Theoremsubrgring 19532 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Theoremsubrgcrng 19533 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑆 = (𝑅s 𝐴)       ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ CRing)

Theoremsubrgrcl 19534 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
(𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)

Theoremsubrgsubg 19535 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
(𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))

Theoremsubrg0 19536 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)    &    0 = (0g𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g𝑆))

Theoremsubrg1cl 19537 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)

Theoremsubrgbas 19538 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))

Theoremsubrg1 19539 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)    &    1 = (1r𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r𝑆))

Theoremsubrgacl 19540 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
+ = (+g𝑅)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ∈ 𝐴)

Theoremsubrgmcl 19541 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
· = (.r𝑅)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝐴) → (𝑋 · 𝑌) ∈ 𝐴)

Theoremsubrgsubm 19542 A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑀 = (mulGrp‘𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀))

Theoremsubrgdvds 19543 If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &    = (∥r𝑅)    &   𝐸 = (∥r𝑆)       (𝐴 ∈ (SubRing‘𝑅) → 𝐸 )

Theoremsubrguss 19544 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (Unit‘𝑅)    &   𝑉 = (Unit‘𝑆)       (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)

Theoremsubrginv 19545 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &   𝐼 = (invr𝑅)    &   𝑈 = (Unit‘𝑆)    &   𝐽 = (invr𝑆)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) = (𝐽𝑋))

Theoremsubrgdv 19546 A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &    / = (/r𝑅)    &   𝑈 = (Unit‘𝑆)    &   𝐸 = (/r𝑆)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌))

Theoremsubrgunit 19547 An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (Unit‘𝑅)    &   𝑉 = (Unit‘𝑆)    &   𝐼 = (invr𝑅)       (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))

Theoremsubrgugrp 19548 The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (Unit‘𝑅)    &   𝑉 = (Unit‘𝑆)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Theoremissubrg2 19549* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Theoremopprsubrg 19550 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (SubRing‘𝑅) = (SubRing‘𝑂)

Theoremsubrgint 19551 The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubRing‘𝑅))

Theoremsubrgin 19552 The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑅)) → (𝐴𝐵) ∈ (SubRing‘𝑅))

Theoremsubrgmre 19553 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (Moore‘𝐵))

Theoremissubdrg 19554* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴))

Theoremsubsubrg 19555 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)))

Theoremsubsubrg2 19556 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → (SubRing‘𝑆) = ((SubRing‘𝑅) ∩ 𝒫 𝐴))

Theoremissubrg3 19557 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝑀 = (mulGrp‘𝑅)       (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀))))

Theoremresrhm 19558 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
𝑈 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (𝑈 RingHom 𝑇))

Theoremrhmeql 19559 The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹𝐺) ∈ (SubRing‘𝑆))

Theoremrhmima 19560 The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹𝑋) ∈ (SubRing‘𝑁))

Theoremrnrhmsubrg 19561 The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
(𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))

Theoremcntzsubr 19562 Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (Cntz‘𝑀)       ((𝑅 ∈ Ring ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubRing‘𝑅))

Theorempwsdiagrhm 19563* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐹 ∈ (𝑅 RingHom 𝑌))

Theoremsubrgpropd 19564* If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))

Theoremrhmpropd 19565* Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐽)𝑦) = (𝑥(.r𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝑀)𝑦))       (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀))

10.4.2.1  Sub-division rings

Syntaxcsdrg 19566 Syntax for subfields (sub-division-rings).
class SubDRing

Definitiondf-sdrg 19567* A sub-division-ring is a subset of a division ring's set which is a division ring under the induced operation. If the overring is commutative this is a field; no special consideration is made of the fields in the center of a skew field. (Contributed by Stefan O'Rear, 3-Oct-2015.)
SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})

Theoremissdrg 19568 Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
(𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))

Theoremsdrgid 19569 Every division ring is a division subring of itself. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝑅))

Theoremsdrgss 19570 A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝐵 = (Base‘𝑅)       (𝑆 ∈ (SubDRing‘𝑅) → 𝑆𝐵)

Theoremissdrg2 19571* Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐼 = (invr𝑅)    &    0 = (0g𝑅)       (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼𝑥) ∈ 𝑆))

Theoremacsfn1p 19572* Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎} ∈ (ACS‘𝑋))

Theoremsubrgacs 19573 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵))

Theoremsdrgacs 19574 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵))

Theoremcntzsdrg 19575 Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (Cntz‘𝑀)       ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubDRing‘𝑅))

Theoremsubdrgint 19576* The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝐿 = (𝑅s 𝑆)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑆 ⊆ (SubRing‘𝑅))    &   (𝜑𝑆 ≠ ∅)    &   ((𝜑𝑠𝑆) → (𝑅s 𝑠) ∈ DivRing)       (𝜑𝐿 ∈ DivRing)

Theoremsdrgint 19577 The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
((𝑅 ∈ DivRing ∧ 𝑆 ⊆ (SubDRing‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubDRing‘𝑅))

Theoremprimefld 19578 The smallest sub division ring of a division ring, here named 𝑃, is a field, called the Prime Field of 𝑅. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝑃 = (𝑅s (SubDRing‘𝑅))       (𝑅 ∈ DivRing → 𝑃 ∈ Field)

Theoremprimefld0cl 19579 The prime field contains the neutral element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
0 = (0g𝑅)       (𝑅 ∈ DivRing → 0 (SubDRing‘𝑅))

Theoremprimefld1cl 19580 The prime field contains the multiplicative neutral element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
1 = (1r𝑅)       (𝑅 ∈ DivRing → 1 (SubDRing‘𝑅))

10.4.3  Absolute value (abstract algebra)

Syntaxcabv 19581 The set of absolute values on a ring.
class AbsVal

Definitiondf-abv 19582* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 14589 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})

Theoremabvfval 19583* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})

Theoremisabv 19584* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))

Theoremisabvd 19585* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
(𝜑𝐴 = (AbsVal‘𝑅))    &   (𝜑𝐵 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑0 = (0g𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹:𝐵⟶ℝ)    &   (𝜑 → (𝐹0 ) = 0)    &   ((𝜑𝑥𝐵𝑥0 ) → 0 < (𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))       (𝜑𝐹𝐴)

Theoremabvrcl 19586 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)       (𝐹𝐴𝑅 ∈ Ring)

Theoremabvfge0 19587 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝐹𝐴𝐹:𝐵⟶(0[,)+∞))

Theoremabvf 19588 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝐹𝐴𝐹:𝐵⟶ℝ)

Theoremabvcl 19589 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)       ((𝐹𝐴𝑋𝐵) → (𝐹𝑋) ∈ ℝ)

Theoremabvge0 19590 The absolute value of a number is greater than or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)       ((𝐹𝐴𝑋𝐵) → 0 ≤ (𝐹𝑋))

Theoremabveq0 19591 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝐹𝐴𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))

Theoremabvne0 19592 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝐹𝐴𝑋𝐵𝑋0 ) → (𝐹𝑋) ≠ 0)

Theoremabvgt0 19593 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝐹𝐴𝑋𝐵𝑋0 ) → 0 < (𝐹𝑋))

Theoremabvmul 19594 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌)))

Theoremabvtri 19595 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))

Theoremabv0 19596 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &    0 = (0g𝑅)       (𝐹𝐴 → (𝐹0 ) = 0)

Theoremabv1z 19597 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝐹𝐴10 ) → (𝐹1 ) = 1)

Theoremabv1 19598 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ DivRing ∧ 𝐹𝐴) → (𝐹1 ) = 1)

Theoremabvneg 19599 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑁 = (invg𝑅)       ((𝐹𝐴𝑋𝐵) → (𝐹‘(𝑁𝑋)) = (𝐹𝑋))

Theoremabvsubtri 19600 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    = (-g𝑅)       ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44899
 Copyright terms: Public domain < Previous  Next >