Home Metamath Proof ExplorerTheorem List (p. 395 of 454) < 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 (1-28701) Hilbert Space Explorer (28702-30224) Users' Mathboxes (30225-45331)

Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsn-dtru 39401* dtru 5236 without ax-8 2113 or ax-12 2175. (Contributed by SN, 21-Sep-2023.)
¬ ∀𝑥 𝑥 = 𝑦

Theoremdifexd 39402 Existence of a difference. (Contributed by SN, 16-Jul-2024.)
(𝜑𝐴𝑉)       (𝜑 → (𝐴𝐵) ∈ V)

Theoremunexd 39403 The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝐵) ∈ V)

Theorempssexg 39404 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theorempssn0 39405 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)

Theorempsspwb 39406 Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

Theoremxppss12 39407 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))

Theoremelpwbi 39408 Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ 𝒫 𝐵)

Theoremopelxpii 39409 Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)
𝐴𝐶    &   𝐵𝐷       𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Theoremiunsn 39410* Indexed union of a singleton. Compare dfiun2 4920 and rnmpt 5791. (Contributed by Steven Nguyen, 7-Jun-2023.)
𝑥𝐴 {𝐵} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremimaopab 39411* The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}

Theoremfnsnbt 39412 A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.)
(𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))

Theoremfnimasnd 39413 The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝑆𝐴)       (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})

Theoremofun 39414 A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.)
(𝜑𝐴 Fn 𝑀)    &   (𝜑𝐵 Fn 𝑀)    &   (𝜑𝐶 Fn 𝑁)    &   (𝜑𝐷 Fn 𝑁)    &   (𝜑𝑀𝑉)    &   (𝜑𝑁𝑊)    &   (𝜑 → (𝑀𝑁) = ∅)       (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) = ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)))

Theoremdfqs2 39415* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)

Theoremdfqs3 39416* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}

Theoremqseq12d 39417 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Theoremqsalrel 39418* The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)
((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)    &   (𝜑 Er 𝐴)    &   (𝜑𝑁𝐴)       (𝜑 → (𝐴 / ) = {𝐴})

Theoremfzosumm1 39419* Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑 → (𝑁 − 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵))

Theoremccatcan2d 39420 Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.)
(𝜑𝐴 ∈ Word 𝑉)    &   (𝜑𝐵 ∈ Word 𝑉)    &   (𝜑𝐶 ∈ Word 𝑉)       (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))

Theoremnelsubginvcld 39421 The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝜑 → (𝑁𝑋) ∈ (𝐵𝑆))

Theoremnelsubgcld 39422 A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    + = (+g𝐺)       (𝜑 → (𝑋 + 𝑌) ∈ (𝐵𝑆))

Theoremnelsubgsubcld 39423 A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    = (-g𝐺)       (𝜑 → (𝑋 𝑌) ∈ (𝐵𝑆))

Theoremrnasclg 39424 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &    1 = (1r𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 }))

Theoremselvval2lem1 39425 𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼𝐽) and we have 𝐽𝑊 instead of 𝐽𝐼. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑇 ∈ AssAlg)

Theoremselvval2lem2 39426 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝐷 ∈ (𝑅 RingHom 𝑇))

Theoremselvval2lem3 39427 The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))

Theoremselvval2lemn 39428 A lemma to illustrate the purpose of selvval2lem3 39427 and the value of 𝑄. Will be renamed in the future when this section is moved to main. (Contributed by SN, 5-Nov-2023.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑆 = (𝑇s ran 𝐷)    &   𝑋 = (𝑇s (𝐵m 𝐼))    &   𝐵 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝑄 ∈ (𝑊 RingHom 𝑋))

Theoremselvval2lem4 39429 The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑆 = (𝑇s ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑋 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷𝐹) ∈ 𝑋)

Theoremselvval2lem5 39430* The fifth argument passed to evalSub is in the domain (a function 𝐼𝐸). (Contributed by SN, 22-Feb-2024.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐸 = (Base‘𝑇)    &   𝐹 = (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝐹 ∈ (𝐸m 𝐼))

Theoremselvcl 39431 Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐸 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸)

Theoremfrlmfielbas 39432 The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑋𝐵𝑋:𝐼𝑁))

Theoremfrlmfzwrd 39433 A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)

Theoremfrlmfzowrd 39434 A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)

Theoremfrlmfzolen 39435 The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝑁 ∈ ℕ0𝑋𝐵) → (♯‘𝑋) = 𝑁)

Theoremfrlmfzowrdb 39436 The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝐾𝑉𝑁 ∈ ℕ0) → (𝑋𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁)))

Theoremfrlmfzoccat 39437 The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾 ∈ Ring)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)       (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)

Theoremfrlmvscadiccat 39438 Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾 ∈ Ring)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)    &   𝑂 = ( ·𝑠𝑊)    &    = ( ·𝑠𝑋)    &    · = ( ·𝑠𝑌)    &   𝑆 = (Base‘𝐾)    &   (𝜑𝐴𝑆)       (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 𝑈) ++ (𝐴 · 𝑉)))

Theoremgrpmndd 39439 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝐺 ∈ Grp)       (𝜑𝐺 ∈ Mnd)

Theoremablcmnd 39440 An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝐺 ∈ Abel)       (𝜑𝐺 ∈ CMnd)

Theoremringabld 39441 A ring is an Abelian group. EDITORIAL: This cannot be used to shorten ringgrpd 19299 because ringabl 19326 depends on ringgrp 19295. (Contributed by SN, 1-Jun-2024.)
(𝜑𝑅 ∈ Ring)       (𝜑𝑅 ∈ Abel)

Theoremringcmnd 39442 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝑅 ∈ Ring)       (𝜑𝑅 ∈ CMnd)

Theoremdrngringd 39443 A division ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ DivRing)       (𝜑𝑅 ∈ Ring)

Theoremdrnggrpd 39444 A division ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ DivRing)       (𝜑𝑅 ∈ Grp)

Theoremlmodgrpd 39445 A left module is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LMod)       (𝜑𝑊 ∈ Grp)

Theoremlvecgrp 39446 A vector space is a group. (Contributed by SN, 28-May-2023.)
(𝑊 ∈ LVec → 𝑊 ∈ Grp)

Theoremlveclmodd 39447 A vector space is a left module. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LVec)       (𝜑𝑊 ∈ LMod)

Theoremlvecgrpd 39448 A vector space is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LVec)       (𝜑𝑊 ∈ Grp)

Theoremlvecring 39449 The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Theoremlmhmlvec 39450 The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec))

Theoremfrlmsnic 39451* Given a free module with a singleton as the index set, that is, a free module of one-dimensional vectors, the function that maps each vector to its coordinate is a module isomorphism from that module to its ring of scalars seen as a module. (Contributed by Steven Nguyen, 18-Aug-2023.)
𝑊 = (𝐾 freeLMod {𝐼})    &   𝐹 = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥𝐼))       ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))

Theoremuvccl 39452 A unit vector is a vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐼𝑊𝐽𝐼) → (𝑈𝐽) ∈ 𝐵)

Theoremuvcn0 39453 A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑌)       ((𝑅 ∈ NzRing ∧ 𝐼𝑊𝐽𝐼) → (𝑈𝐽) ≠ 0 )

Theoremfsuppind 39454* Induction on functions 𝐹:𝐴𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 20445 and frlmplusgval 20453). This theorem is structurally general for polynomial proof usage (see mplelbas 20668 and mpladd 20680). (Contributed by SN, 18-May-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ((𝜑 ∧ (𝑎𝐼𝑏𝐵)) → (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)       ((𝜑 ∧ (𝑋:𝐼𝐵𝑋 finSupp 0 )) → 𝑋𝐻)

Theoremfsuppssindlem1 39455* Lemma for fsuppssind 39457. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)
(𝜑0𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)       (𝜑𝐹 = (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )))

Theoremfsuppssindlem2 39456* Lemma for fsuppssind 39457. Write a function as a union. (Contributed by SN, 15-Jul-2024.)
(𝜑𝐵𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝐼)       (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))

Theoremfsuppssind 39457* Induction on functions 𝐹:𝐴𝐵 with finite support (see fsuppind 39454) whose supports are subsets of 𝑆. (Contributed by SN, 15-Jun-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝐼)    &   (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)    &   (𝜑𝑋:𝐼𝐵)    &   (𝜑𝑋 finSupp 0 )    &   (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)       (𝜑𝑋𝐻)

20.26.2  Arithmetic theorems

Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results.

For example, ax-1rid 10596 is used in mulid1 10628 related theorems, so one could trade off the extra axioms in mulid1 10628 for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ax-resscn 10583; in the other direction, real number closure laws can be avoided by using ax-resscn 10583 and then the complex number closure laws. (This only works if the result of (𝐴 + 𝐵) only needs to be a complex number).

The natural numbers are especially amenable to axiom reductions, as the set is the recursive set {1, (1 + 1), ((1 + 1) + 1)}, etc., i.e. the set of numbers formed by only additions of 1. The digits 2 through 9 are defined so that they expand into additions of 1. This makes adding natural numbers conveniently only require the rearrangement of parentheses, as shown with the following:

(4 + 3) = 7

((3 + 1) + (2 + 1)) = (6 + 1)

((((1 + 1) + 1) + 1) + ((1 + 1) + 1)) =

((((((1 + 1) + 1) + 1) + 1) + 1) + 1)

This only requires ax-addass 10591, ax-1cn 10584, and ax-addcl 10586. (And in practice, the expression isn't completely expanded into ones.)

Multiplication by 1 requires either mulid2i 10635 or (ax-1rid 10596 and 1re 10630) as seen in 1t1e1 11787 and 1t1e1ALT 39461. Multiplying with greater natural numbers uses ax-distr 10593. Still, this takes fewer axioms than adding zero, which is often implicit in theorems such as (9 + 1) = 10. Adding zero uses almost every complex number axiom, though notably not ax-mulcom 10590 (see readdid1 39545 and readdid2 39539).

Theoremc0exALT 39458 Alternate proof of c0ex 10624 using more set theory axioms but fewer complex number axioms (add ax-10 2142, ax-11 2158, ax-13 2379, ax-nul 5174, and remove ax-1cn 10584, ax-icn 10585, ax-addcl 10586, and ax-mulcl 10588). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ V

Theorem0cnALT3 39459 Alternate proof of 0cn 10622 using ax-resscn 10583, ax-addrcl 10587, ax-rnegex 10597, ax-cnre 10599 instead of ax-icn 10585, ax-addcl 10586, ax-mulcl 10588, ax-i2m1 10594. Version of 0cnALT 10863 using ax-1cn 10584 instead of ax-icn 10585. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ

Theoremelre0re 39460 Specialized version of 0red 10633 without using ax-1cn 10584 and ax-cnre 10599. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝐴 ∈ ℝ → 0 ∈ ℝ)

Theorem1t1e1ALT 39461 Alternate proof of 1t1e1 11787 using a different set of axioms (add ax-mulrcl 10589, ax-i2m1 10594, ax-1ne0 10595, ax-rrecex 10598 and remove ax-resscn 10583, ax-mulcom 10590, ax-mulass 10592, ax-distr 10593). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 · 1) = 1

Theoremremulcan2d 39462 mulcan2d 11263 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))

Theoremreaddid1addid2d 39463 Given some real number 𝐵 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 10803, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐵 + 𝐴) = 𝐵)       ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)

Theoremsn-1ne2 39464 A proof of 1ne2 11833 without using ax-mulcom 10590, ax-mulass 10592, ax-pre-mulgt0 10603. Based on mul02lem2 10806. (Contributed by SN, 13-Dec-2023.)
1 ≠ 2

Theoremnnn1suc 39465* A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.)
((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)

Theoremnnadd1com 39466 Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
(𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Theoremnnaddcom 39467 Addition is commutative for natural numbers. Uses fewer axioms than addcom 10815. (Contributed by Steven Nguyen, 9-Dec-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremnnaddcomli 39468 Version of addcomli 10821 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶

Theoremnnadddir 39469 Right-distributivity for natural numbers without ax-mulcom 10590. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theoremnnmul1com 39470 Multiplication with 1 is commutative for natural numbers, without ax-mulcom 10590. Since (𝐴 · 1) is 𝐴 by ax-1rid 10596, this is equivalent to remulid2 39568 for natural numbers, but using fewer axioms (avoiding ax-resscn 10583, ax-addass 10591, ax-mulass 10592, ax-rnegex 10597, ax-pre-lttri 10600, ax-pre-lttrn 10601, ax-pre-ltadd 10602). (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))

Theoremnnmulcom 39471 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

Theoremaddsubeq4com 39472 Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴𝐶) = (𝐷𝐵)))

Theoremsqsumi 39473 A sum squared. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + (𝐵 · 𝐵)) + (2 · (𝐴 · 𝐵)))

Theoremnegn0nposznnd 39474 Lemma for dffltz 39613. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴 ≠ 0)    &   (𝜑 → ¬ 0 < 𝐴)    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℕ)

Theoremsqmid3api 39475 Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022.)
𝐴 ∈ ℂ    &   𝑁 ∈ ℂ    &   (𝐴 + 𝑁) = 𝐵    &   (𝐵 + 𝑁) = 𝐶       (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁))

Theoremdecaddcom 39476 Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (𝐴𝐵 + 𝐶) = (𝐴𝐶 + 𝐵)

Theoremsqn5i 39477 The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ ℕ0       (𝐴5 · 𝐴5) = (𝐴 · (𝐴 + 1))25

Theoremsqn5ii 39478 The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ ℕ0    &   (𝐴 + 1) = 𝐵    &   (𝐴 · 𝐵) = 𝐶       (𝐴5 · 𝐴5) = 𝐶25

Theoremdecpmulnc 39479 Partial products algorithm for two digit multiplication, no carry. Compare muladdi 11080. (Contributed by Steven Nguyen, 9-Dec-2022.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝐸    &   ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹    &   (𝐵 · 𝐷) = 𝐺       (𝐴𝐵 · 𝐶𝐷) = 𝐸𝐹𝐺

Theoremdecpmul 39480 Partial products algorithm for two digit multiplication. (Contributed by Steven Nguyen, 10-Dec-2022.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝐸    &   ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹    &   (𝐵 · 𝐷) = 𝐺𝐻    &   (𝐸𝐺 + 𝐹) = 𝐼    &   𝐺 ∈ ℕ0    &   𝐻 ∈ ℕ0       (𝐴𝐵 · 𝐶𝐷) = 𝐼𝐻

Theoremsqdeccom12 39481 The square of a number in terms of its digits switched. (Contributed by Steven Nguyen, 3-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       ((𝐴𝐵 · 𝐴𝐵) − (𝐵𝐴 · 𝐵𝐴)) = (99 · ((𝐴 · 𝐴) − (𝐵 · 𝐵)))

Theoremsq3deccom12 39482 Variant of sqdeccom12 39481 with a three digit square. (Contributed by Steven Nguyen, 3-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   (𝐴 + 𝐶) = 𝐷       ((𝐴𝐵𝐶 · 𝐴𝐵𝐶) − (𝐷𝐵 · 𝐷𝐵)) = (99 · ((𝐴𝐵 · 𝐴𝐵) − (𝐶 · 𝐶)))

Theorem235t711 39483 Calculate a product by long multiplication as a base comparison with other multiplication algorithms.

Conveniently, 711 has two ones which greatly simplifies calculations like 235 · 1. There isn't a higher level mulcomli 10639 saving the lower level uses of mulcomli 10639 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12195 are added then this proof would benefit more than ex-decpmul 39484.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11760 or 8t7e56 12206. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

(235 · 711) = 167085

Theoremex-decpmul 39484 Example usage of decpmul 39480. This proof is significantly longer than 235t711 39483. There is more unnecessary carrying compared to 235t711 39483. Although saving 5 visual steps, using mulcomli 10639 early on increases the compressed proof length. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(235 · 711) = 167085

20.26.3  Exponents

Theoremoexpreposd 39485 Lemma for dffltz 39613. (Contributed by Steven Nguyen, 4-Mar-2023.)
(𝜑𝑁 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)       (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁𝑀)))

Theoremcxpgt0d 39486 Exponentiation with a positive mantissa is positive. (Contributed by Steven Nguyen, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℝ)       (𝜑 → 0 < (𝐴𝑐𝑁))

Theoremdvdsexpim 39487 dvdssqim 15894 generalized to nonnegative exponents. (Contributed by Steven Nguyen, 2-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝐵 → (𝐴𝑁) ∥ (𝐵𝑁)))

Theoremnn0rppwr 39488 If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵𝑁. rppwr 15898 extended to nonnegative integers. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))

Theoremexpgcd 39489 Exponentiation distributes over GCD. sqgcd 15899 extended to nonnegative exponents. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))

Theoremnn0expgcd 39490 Exponentiation distributes over GCD. nn0gcdsq 16082 extended to nonnegative exponents. expgcd 39489 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))

Theoremzexpgcd 39491 Exponentiation distributes over GCD. zgcdsq 16083 extended to nonnegative exponents. nn0expgcd 39490 extended to integer bases by symmetry. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))

Theoremnumdenexp 39492 numdensq 16084 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → ((numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁) ∧ (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁)))

Theoremnumexp 39493 numsq 16085 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁))

Theoremdenexp 39494 densq 16086 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁))

Theoremexp11d 39495 sq11d 13617 for positive real bases and nonzero exponents. (Contributed by Steven Nguyen, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑁 ≠ 0)    &   (𝜑 → (𝐴𝑁) = (𝐵𝑁))       (𝜑𝐴 = 𝐵)

Theoremltexp1d 39496 ltmul1d 12460 for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴𝑁) < (𝐵𝑁)))

Theoremltexp1dd 39497 Raising both sides of 'less than' to the same positive integer preserves ordering. (Contributed by Steven Nguyen, 24-Aug-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴𝑁) < (𝐵𝑁))

Theoremzrtelqelz 39498 zsqrtelqelz 16088 generalized to positive integer roots. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴𝑐(1 / 𝑁)) ∈ ℚ) → (𝐴𝑐(1 / 𝑁)) ∈ ℤ)

Theoremzrtdvds 39499 A positive integer root divides its integer. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴𝑐(1 / 𝑁)) ∈ ℕ) → (𝐴𝑐(1 / 𝑁)) ∥ 𝐴)

Theoremrtprmirr 39500 The root of a prime number is irrational. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (ℤ‘2)) → (𝑃𝑐(1 / 𝑁)) ∈ (ℝ ∖ ℚ))

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 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45331
 Copyright terms: Public domain < Previous  Next >