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Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnimasnd 39001 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)})
 
Theoremdfqs2 39002* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
 
Theoremdfqs3 39003* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
 
Theoremqseq12d 39004 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
 
Theoremqsalrel 39005* The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is a nonempty set. (Contributed by Steven Nguyen, 8-Jun-2023.)
((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)    &   (𝜑 Er 𝐴)    &   (𝜑𝑁𝐴)    &   (𝜑𝐴𝑉)       (𝜑 → (𝐴 / ) = {𝐴})
 
Theoremfzosumm1 39006* 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 39007 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 39008 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 39009 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 39010 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 39011 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 39012 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 39013 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 39014 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 39015 A lemma to illustrate the purpose of selvval2lem3 39014 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 39016 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 39017* 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 39018 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 39019 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 39020 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 39021 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 39022 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 39023 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 39024 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 39025 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)))
 
Theoremlvecgrp 39026 A left vector is a group. (Contributed by Steven Nguyen, 28-May-2023.)
(𝑊 ∈ LVec → 𝑊 ∈ Grp)
 
Theoremlvecring 39027 The scalar component of a left vector is a ring. (Contributed by Steven Nguyen, 28-May-2023.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → 𝐹 ∈ Ring)
 
Theoremlmhmlvec 39028 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 39029* 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 39030 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 39031 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 )
 
20.25.2  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 11788 and 1t1e1ALT 39035. Multiplying with greater natural numbers uses ax-distr 10593. Still, this takes fewer axioms than adding zero. When zero is involved in the decimal constructor, there's an implicit addition operation which causes such theorems (e.g. (9 + 1) = 10) to use almost every complex number axiom.

 
Theoremc0exALT 39032 Alternate proof of c0ex 10624 using more set theory axioms but fewer complex number axioms (add ax-10 2136, ax-11 2151, ax-13 2383, ax-nul 5202, 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 39033 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 39034 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 39035 Alternate proof of 1t1e1 11788 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 39036 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 39037 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 39038 A proof of 1ne2 11834 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 39039* 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 39040 Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
(𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
 
Theoremnnaddcom 39041 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 39042 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 39043 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 39044 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 39129 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 39045 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaddsubeq4com 39046 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 39047 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 39048 Lemma for dffltz 39151. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴 ≠ 0)    &   (𝜑 → ¬ 0 < 𝐴)    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℕ)
 
Theoremsqmid3api 39049 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 39050 Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (𝐴𝐵 + 𝐶) = (𝐴𝐶 + 𝐵)
 
Theoremsqn5i 39051 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 39052 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 39053 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 39054 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 39055 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 39056 Variant of sqdeccom12 39055 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 39057 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 12196 are added then this proof would benefit more than ex-decpmul 39058.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11761 or 8t7e56 12207. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

(235 · 711) = 167085
 
Theoremex-decpmul 39058 Example usage of decpmul 39054. This proof is significantly longer than 235t711 39057. There is more unnecessary carrying compared to 235t711 39057. 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
 
20.25.3  Exponents
 
Theoremoexpreposd 39059 Lemma for dffltz 39151. (Contributed by Steven Nguyen, 4-Mar-2023.)
(𝜑𝑁 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)       (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁𝑀)))
 
Theoremcxpgt0d 39060 Exponentiation with a positive mantissa is positive. (Contributed by Steven Nguyen, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℝ)       (𝜑 → 0 < (𝐴𝑐𝑁))
 
Theoremdvdsexpim 39061 dvdssqim 15894 generalized to nonnegative exponents. (Contributed by Steven Nguyen, 2-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝐵 → (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremnn0rppwr 39062 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 39063 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 39064 Exponentiation distributes over GCD. nn0gcdsq 16082 extended to nonnegative exponents. expgcd 39063 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremzexpgcd 39065 Exponentiation distributes over GCD. zgcdsq 16083 extended to nonnegative exponents. nn0expgcd 39064 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 39066 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 39067 numsq 16085 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁))
 
Theoremdenexp 39068 densq 16086 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁))
 
Theoremexp11d 39069 sq11d 13611 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 39070 ltmul1d 12462 for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴𝑁) < (𝐵𝑁)))
 
Theoremltexp1dd 39071 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 39072 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 39073 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 39074 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))
 
20.25.4  Real subtraction
 
Syntaxcresub 39075 Real number subtraction.
class
 
Definitiondf-resub 39076* Define subtraction between real numbers. This operator saves a few axioms over df-sub 10861 in certain situations. Theorem resubval 39077 shows its value, resubadd 39089 relates it to addition, and rersubcl 39088 proves its closure. Based on df-sub 10861. (Contributed by Steven Nguyen, 7-Jan-2022.)
= (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
 
Theoremresubval 39077* Value of real subtraction, which is the (unique) real 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by Steven Nguyen, 7-Jan-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
 
Theoremrenegeulemv 39078* Lemma for renegeu 39080 and similar. Derive existential uniqueness from existence. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)       (𝜑 → ∃!𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴)
 
Theoremrenegeulem 39079* Lemma for renegeu 39080 and similar. Remove a change in bound variables from renegeulemv 39078. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)       (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)
 
Theoremrenegeu 39080* Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Theoremrernegcl 39081 Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (0 − 𝐴) ∈ ℝ)
 
Theoremrenegadd 39082 Relationship between real negation and addition. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 − 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 0))
 
Theoremrenegid 39083 Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (𝐴 + (0 − 𝐴)) = 0)
 
Theoremreneg0addid2 39084 Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → ((0 − 0) + 𝐴) = 𝐴)
 
Theoremresubeulem1 39085 Lemma for resubeu 39087. A value which when added to zero, results in negative zero. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (0 + (0 − (0 + 0))) = (0 − 0))
 
Theoremresubeulem2 39086 Lemma for resubeu 39087. A value which when added to 𝐴, results in 𝐵. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + ((0 − 𝐴) + ((0 − (0 + 0)) + 𝐵))) = 𝐵)
 
Theoremresubeu 39087* Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵)
 
Theoremrersubcl 39088 Closure for real subtraction. Based on subcl 10874. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) ∈ ℝ)
 
Theoremresubadd 39089 Relation between real subtraction and addition. Based on subadd 10878. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremresubaddd 39090 Relationship between subtraction and addition. Based on subaddd 11004. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremresubf 39091 Real subtraction is an operation on the real numbers. Based on subf 10877. (Contributed by Steven Nguyen, 7-Jan-2023.)
:(ℝ × ℝ)⟶ℝ
 
Theoremrepncan2 39092 Addition and subtraction of equals. Compare pncan2 10882. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
 
Theoremrepncan3 39093 Addition and subtraction of equals. Based on pncan3 10883. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 𝐴)) = 𝐵)
 
Theoremreaddsub 39094 Law for addition and subtraction. (Contributed by Steven Nguyen, 28-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 𝐶) + 𝐵))
 
Theoremreladdrsub 39095 Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11040 with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶 𝐴))
 
Theoremreltsub1 39096 Subtraction from both sides of 'less than'. Compare ltsub1 11125. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 𝐶) < (𝐵 𝐶)))
 
Theoremreltsubadd2 39097 'Less than' relationship between addition and subtraction. Compare ltsubadd2 11100. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))
 
Theoremresubcan2 39098 Cancellation law for real subtraction. Compare subcan2 10900. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐶) = (𝐵 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremresubsub4 39099 Law for double subtraction. Compare subsub4 10908. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) − 𝐶) = (𝐴 (𝐵 + 𝐶)))
 
Theoremrennncan2 39100 Cancellation law for real subtraction. Compare nnncan2 10912. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐶) − (𝐵 𝐶)) = (𝐴 𝐵))
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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 44801-44804
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