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Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopprdrng 18501 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
𝑂 = (oppr𝑅)       (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)
 
Theoremisdrngd 18502* Properties that determine a division ring. 𝐼 (reciprocal) is normally dependent on 𝑥 i.e. read it as 𝐼(𝑥)." (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 18503* Properties that determine a division ring. 𝐼 (reciprocal) is normally dependent on 𝑥 i.e. read it as 𝐼(𝑥)." This version of isdrngd 18502 requires a right reciprocal instead of left. (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 18504* 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 18505* 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))
 
10.5.2  Subrings of a ring
 
Syntaxcsubrg 18506 Extend class notation with all subrings of a ring.
class SubRing
 
Syntaxcrgspn 18507 Extend class notation with span of a set of elements over a ring.
class RingSpan
 
Definitiondf-subrg 18508* 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 component-wise) 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 18509* 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 18510 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 18511 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐵 = (Base‘𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 𝐴𝐵)
 
Theoremsubrgid 18512 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
 
Theoremsubrgring 18513 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
 
Theoremsubrgcrng 18514 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 18515 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
(𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
 
Theoremsubrgsubg 18516 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
(𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
 
Theoremsubrg0 18517 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 18518 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
 
Theoremsubrgbas 18519 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
 
Theoremsubrg1 18520 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 18521 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
+ = (+g𝑅)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
 
Theoremsubrgmcl 18522 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
· = (.r𝑅)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
Theoremsubrgsubm 18523 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 18524 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 18525 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 18526 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 18527 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 18528 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 18529 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 18530* 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 18531 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (SubRing‘𝑅) = (SubRing‘𝑂)
 
Theoremsubrgint 18532 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 18533 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 18534 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 18535* 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 18536 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 18537 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 18538 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 18539 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 18540 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 18541 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))
 
Theoremcntzsubr 18542 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 18543* 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 18544* 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 18545* 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))
 
10.5.3  Absolute value (abstract algebra)
 
Syntaxcabv 18546 The set of absolute values on a ring.
class AbsVal
 
Definitiondf-abv 18547* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 13683 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 18548* 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 18549* 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 18550* 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 18551 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)       (𝐹𝐴𝑅 ∈ Ring)
 
Theoremabvfge0 18552 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 18553 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 18554 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 18555 The absolute value of a number is greater or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)       ((𝐹𝐴𝑋𝐵) → 0 ≤ (𝐹𝑋))
 
Theoremabveq0 18556 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 18557 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 18558 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 18559 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 18560 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 18561 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &    0 = (0g𝑅)       (𝐹𝐴 → (𝐹0 ) = 0)
 
Theoremabv1z 18562 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 18563 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 18564 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 18565 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)))
 
Theoremabvrec 18566 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       (((𝑅 ∈ DivRing ∧ 𝐹𝐴) ∧ (𝑋𝐵𝑋0 )) → (𝐹‘(𝐼𝑋)) = (1 / (𝐹𝑋)))
 
Theoremabvdiv 18567 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    / = (/r𝑅)       (((𝑅 ∈ DivRing ∧ 𝐹𝐴) ∧ (𝑋𝐵𝑌𝐵𝑌0 )) → (𝐹‘(𝑋 / 𝑌)) = ((𝐹𝑋) / (𝐹𝑌)))
 
Theoremabvdom 18568 Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)       ((𝐹𝐴 ∧ (𝑋𝐵𝑋0 ) ∧ (𝑌𝐵𝑌0 )) → (𝑋 · 𝑌) ≠ 0 )
 
Theoremabvres 18569 The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝑆 = (𝑅s 𝐶)    &   𝐵 = (AbsVal‘𝑆)       ((𝐹𝐴𝐶 ∈ (SubRing‘𝑅)) → (𝐹𝐶) ∈ 𝐵)
 
Theoremabvtrivd 18570* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐹 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, 1))    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑 ∧ (𝑦𝐵𝑦0 ) ∧ (𝑧𝐵𝑧0 )) → (𝑦 · 𝑧) ≠ 0 )       (𝜑𝐹𝐴)
 
Theoremabvtriv 18571* The trivial absolute value. (This theorem is true as long as 𝑅 is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 18568 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐹 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, 1))       (𝑅 ∈ DivRing → 𝐹𝐴)
 
Theoremabvpropd 18572* If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
 
10.5.4  Star rings
 
Syntaxcstf 18573 Extend class notation with the functionalization of the *-ring involution.
class *rf
 
Syntaxcsr 18574 Extend class notation with class of all *-rings.
class *-Ring
 
Definitiondf-staf 18575* Define the functionalization of the involution in a star ring. This is not strictly necessary but by having *𝑟 as an actual function we can state the principal properties of an involution much more cleanly. (Contributed by Mario Carneiro, 6-Oct-2015.)
*rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)))
 
Definitiondf-srng 18576* Define class of all star rings. A star ring is a ring with an involution (conjugation) function. Involution (unlike say the ring zero) is not unique and therefore must be added as a new component to the ring. For example, two possible involutions for complex numbers are the identity function and complex conjugation. Definition of involution in [Holland95] p. 204. (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
*-Ring = {𝑓[(*rf𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr𝑓)) ∧ 𝑖 = 𝑖)}
 
Theoremstaffval 18577* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐵 = (Base‘𝑅)    &    = (*𝑟𝑅)    &    = (*rf𝑅)        = (𝑥𝐵 ↦ ( 𝑥))
 
Theoremstafval 18578 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐵 = (Base‘𝑅)    &    = (*𝑟𝑅)    &    = (*rf𝑅)       (𝐴𝐵 → ( 𝐴) = ( 𝐴))
 
Theoremstaffn 18579 The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐵 = (Base‘𝑅)    &    = (*𝑟𝑅)    &    = (*rf𝑅)       ( Fn 𝐵 = )
 
Theoremissrng 18580 The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
𝑂 = (oppr𝑅)    &    = (*rf𝑅)       (𝑅 ∈ *-Ring ↔ ( ∈ (𝑅 RingHom 𝑂) ∧ = ))
 
Theoremsrngrhm 18581 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑂 = (oppr𝑅)    &    = (*rf𝑅)       (𝑅 ∈ *-Ring → ∈ (𝑅 RingHom 𝑂))
 
Theoremsrngring 18582 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑅 ∈ *-Ring → 𝑅 ∈ Ring)
 
Theoremsrngcnv 18583 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*rf𝑅)       (𝑅 ∈ *-Ring → = )
 
Theoremsrngf1o 18584 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*rf𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ *-Ring → :𝐵1-1-onto𝐵)
 
Theoremsrngcl 18585 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
 
Theoremsrngnvl 18586 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
 
Theoremsrngadd 18587 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 + 𝑌)) = (( 𝑋) + ( 𝑌)))
 
Theoremsrngmul 18588 The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 · 𝑌)) = (( 𝑌) · ( 𝑋)))
 
Theoremsrng1 18589 The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined *𝑟 to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 18591.) (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ *-Ring → ( 1 ) = 1 )
 
Theoremsrng0 18590 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)
= (*𝑟𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ *-Ring → ( 0 ) = 0 )
 
Theoremissrngd 18591* Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)
(𝜑𝐾 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑 = (*𝑟𝑅))    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑥𝐾) → ( 𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)))    &   ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)))    &   ((𝜑𝑥𝐾) → ( ‘( 𝑥)) = 𝑥)       (𝜑𝑅 ∈ *-Ring)
 
Theoremidsrngd 18592* A commutative ring is a star ring when the conjugate operation is the identity. (Contributed by Thierry Arnoux, 19-Apr-2019.)
𝐵 = (Base‘𝑅)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)       (𝜑𝑅 ∈ *-Ring)
 
10.6  Left modules
 
10.6.1  Definition and basic properties
 
Syntaxclmod 18593 Extend class notation with class of all left modules.
class LMod
 
Syntaxcscaf 18594 The functionalization of the scalar multiplication operation.
class ·sf
 
Definitiondf-lmod 18595* Define the class of all left modules, which are generalizations of left vector spaces. A left module over a ring is an (Abelian) group (vectors) together with a ring (scalars) and a left scalar product connecting them. (Contributed by NM, 4-Nov-2013.)
LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+g𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞𝑘𝑟𝑘𝑥𝑣𝑤𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r𝑓)𝑠𝑤) = 𝑤)))}
 
Definitiondf-scaf 18596* Define the functionalization of the ·𝑠 operator. This restricts the value of ·𝑠 to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
 
Theoremislmod 18597* The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)       (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
 
Theoremlmodlema 18598 Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)       ((𝑊 ∈ LMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑅 · 𝑌) ∈ 𝑉 ∧ (𝑅 · (𝑌 + 𝑋)) = ((𝑅 · 𝑌) + (𝑅 · 𝑋)) ∧ ((𝑄 𝑅) · 𝑌) = ((𝑄 · 𝑌) + (𝑅 · 𝑌))) ∧ (((𝑄 × 𝑅) · 𝑌) = (𝑄 · (𝑅 · 𝑌)) ∧ ( 1 · 𝑌) = 𝑌)))
 
Theoremislmodd 18599* Properties that determine a left module. See note in isgrpd2 17157 regarding the 𝜑 on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
(𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝐵 = (Base‘𝐹))    &   (𝜑 = (+g𝐹))    &   (𝜑× = (.r𝐹))    &   (𝜑1 = (1r𝐹))    &   (𝜑𝐹 ∈ Ring)    &   (𝜑𝑊 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝑉) → (𝑥 · 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝑉𝑧𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝑉)) → ((𝑥 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝑉)) → ((𝑥 × 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑𝑥𝑉) → ( 1 · 𝑥) = 𝑥)       (𝜑𝑊 ∈ LMod)
 
Theoremlmodgrp 18600 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
(𝑊 ∈ LMod → 𝑊 ∈ Grp)
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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 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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 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 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