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Theorem List for Metamath Proof Explorer - 42401-42430   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmvlrmuld 42401 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)       (𝜑𝐴 = (𝐶 / 𝐵))
 
Theoremmvlrmuli 42402 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0    &   (𝐴 · 𝐵) = 𝐶       𝐴 = (𝐶 / 𝐵)
 
20.35.11  Algebra helper examples

Examples using the algebra helpers.

 
Theoremi2linesi 42403 Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use inference form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝑋 ∈ ℂ    &   𝑌 = ((𝐴 · 𝑋) + 𝐵)    &   𝑌 = ((𝐶 · 𝑋) + 𝐷)    &   (𝐴𝐶) ≠ 0       𝑋 = ((𝐷𝐵) / (𝐴𝐶))
 
Theoremi2linesd 42404 Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use deduction form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 = ((𝐴 · 𝑋) + 𝐵))    &   (𝜑𝑌 = ((𝐶 · 𝑋) + 𝐷))    &   (𝜑 → (𝐴𝐶) ≠ 0)       (𝜑𝑋 = ((𝐷𝐵) / (𝐴𝐶)))
 
20.35.12  Formal methods "surprises"

Prove that some formal expressions using classical logic have meanings that might not be obvious to some lay readers. I find these are common mistakes and are worth pointing out to new people. In particular we prove alimp-surprise 42405, empty-surprise 42407, and eximp-surprise 42409.

 
Theoremalimp-surprise 42405 Demonstrate that when using "for all" and material implication the consequent can be both always true and always false if there is no case where the antecedent is true.

Those inexperienced with formal notations of classical logic can be surprised with what "for all" and material implication do together when the implication's antecedent is never true. This can happen, for example, when the antecedent is set membership but the set is the empty set (e.g., 𝑥𝑀 and 𝑀 = ∅).

This is perhaps best explained using an example. The sentence "All Martians are green" would typically be represented formally using the expression 𝑥(𝜑𝜓). In this expression 𝜑 is true iff 𝑥 is a Martian and 𝜓 is true iff 𝑥 is green. Similarly, "All Martians are not green" would typically be represented as 𝑥(𝜑 → ¬ 𝜓). However, if there are no Martians (¬ ∃𝑥𝜑), then both of those expressions are true. That is surprising to the inexperienced, because the two expressions seem to be the opposite of each other. The reason this occurs is because in classical logic the implication (𝜑𝜓) is equivalent to ¬ 𝜑𝜓 (as proven in imor 426). When 𝜑 is always false, ¬ 𝜑 is always true, and an or with true is always true.

Here are a few technical notes. In this notation, 𝜑 and 𝜓 are predicates that return a true or false value and may depend on 𝑥. We only say may because it actually doesn't matter for our proof. In metamath this simply means that we do not require that 𝜑, 𝜓, and 𝑥 be distinct (so 𝑥 can be part of 𝜑 or 𝜓).

In natural language the term "implies" often presumes that the antecedent can occur in at one least circumstance and that there is some sort of causality. However, exactly what causality means is complex and situation-dependent. Modern logic typically uses material implication instead; this has a rigorous definition, but it is important for new users of formal notation to precisely understand it. There are ways to solve this, e.g., expressly stating that the antecedent exists (see alimp-no-surprise 42406) or using the allsome quantifier (df-alsi 42413) .

For other "surprises" for new users of classical logic, see empty-surprise 42407 and eximp-surprise 42409. (Contributed by David A. Wheeler, 17-Oct-2018.)

¬ ∃𝑥𝜑       (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓))
 
Theoremalimp-no-surprise 42406 There is no "surprise" in a for-all with implication if there exists a value where the antecedent is true. This is one way to prevent for-all with implication from allowing anything. For a contrast, see alimp-surprise 42405. The allsome quantifier also counters this problem, see df-alsi 42413. (Contributed by David A. Wheeler, 27-Oct-2018.)
¬ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)
 
Theoremempty-surprise 42407 Demonstrate that when using restricted "for all" over a class the expression can be both always true and always false if the class is empty.

Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. It is important to note that 𝑥𝐴𝜑 is simply an abbreviation for 𝑥(𝑥𝐴𝜑) (per df-ral 2805). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 42405).

If you want the expression 𝑥𝐴𝜑 to not be vacuously true, you need to ensure that set 𝐴 is inhabited (e.g., 𝑥𝐴). (Technical note: You can also assert that 𝐴 ≠ ∅; this is an equivalent claim in classical logic as proven in n0 3793, but in intuitionistic logic the statement 𝐴 ≠ ∅ is a weaker claim than 𝑥𝐴.)

Some materials on logic (particularly those that discuss "syllogisms") are based on the much older work by Aristotle, but Aristotle expressly excluded empty sets from his system. Aristotle had a specific goal; he was trying to develop a "companion-logic" for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature... This is why he leaves no room for such non-existent entities in his logic." (Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/aris-log/). While this made sense for his purposes, it is less flexible than modern (classical) logic which does permit empty sets. If you wish to make claims that require a nonempty set, you must expressly include that requirement, e.g., by stating 𝑥𝜑. Examples of proofs that do this include barbari 2459, celaront 2460, and cesaro 2465.

For another "surprise" for new users of classical logic, see alimp-surprise 42405 and eximp-surprise 42409. (Contributed by David A. Wheeler, 20-Oct-2018.)

¬ ∃𝑥 𝑥𝐴       𝑥𝐴 𝜑
 
Theoremempty-surprise2 42408 "Prove" that false is true when using a restricted "for all" over the empty set, to demonstrate that the expression is always true if the value ranges over the empty set.

Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. We proved the general case in empty-surprise 42407. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1509); the problem is that restricted "for all" works in ways that might seem counterintuitive to the inexperienced when given an empty set. Solutions to this can include requiring that the set not be empty or by using the allsome quantifier df-alsc 42414. (Contributed by David A. Wheeler, 20-Oct-2018.)

¬ ∃𝑥 𝑥𝐴       𝑥𝐴
 
Theoremeximp-surprise 42409 Show what implication inside "there exists" really expands to (using implication directly inside "there exists" is usually a mistake).

Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. That is usually a mistake, because as proven using imor 426, such an expression can be rewritten using not with or - and that is often not what the author intended. New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". A stark example is shown in eximp-surprise2 42410. See also alimp-surprise 42405 and empty-surprise 42407. (Contributed by David A. Wheeler, 17-Oct-2018.)

(∃𝑥(𝜑𝜓) ↔ ∃𝑥𝜑𝜓))
 
Theoremeximp-surprise2 42410 Show that "there exists" with an implication is always true if there exists a situation where the antecedent is false.

Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. This is usually a mistake, because that combination does not mean what an inexperienced person might think it means. For example, if there is some object that does not meet the precondition 𝜑, then the expression 𝑥(𝜑𝜓) as a whole is always true, no matter what 𝜓 is (𝜓 could even be false, ). New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". See eximp-surprise 42409, which shows what implication really expands to. See also empty-surprise 42407. (Contributed by David A. Wheeler, 18-Oct-2018.)

𝑥 ¬ 𝜑       𝑥(𝜑𝜓)
 
20.35.13  Allsome quantifier

These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some").

In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like 𝑥𝜑𝜓 do not imply that 𝜑 is ever true, leading to vacuous truths. See alimp-surprise 42405 and empty-surprise 42407 as examples of the problem. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem.

The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines.

I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it.

The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ∀!𝑥(𝜑𝜓), and when restricted (applied to a class) we allow ∀!𝑥𝐴𝜑. The first symbol after the setvar variable must always be if it is the form applied to a class, and since cannot begin a wff, it is unambiguous. The looks like it would be a problem because 𝜑 or 𝜓 might include implications, but any implication arrow within any wff must be surrounded by parentheses, so only the implication arrow of ∀! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax.

For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome.

 
Syntaxwalsi 42411 Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff ∀!𝑥(𝜑𝜓)
 
Syntaxwalsc 42412 Extend wff definition to include "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff ∀!𝑥𝐴𝜑
 
Definitiondf-alsi 42413 Define "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
(∀!𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑))
 
Definitiondf-alsc 42414 Define "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴 and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.)
(∀!𝑥𝐴𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∃𝑥 𝑥𝐴))
 
Theoremalsconv 42415 There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
(∀!𝑥(𝑥𝐴𝜑) ↔ ∀!𝑥𝐴𝜑)
 
Theoremalsi1d 42416 Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(𝜑 → ∀!𝑥(𝜓𝜒))       (𝜑 → ∀𝑥(𝜓𝜒))
 
Theoremalsi2d 42417 Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(𝜑 → ∀!𝑥(𝜓𝜒))       (𝜑 → ∃𝑥𝜓)
 
Theoremalsc1d 42418 Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(𝜑 → ∀!𝑥𝐴𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremalsc2d 42419 Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(𝜑 → ∀!𝑥𝐴𝜓)       (𝜑 → ∃𝑥 𝑥𝐴)
 
Theoremalscn0d 42420* Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.)
(𝜑 → ∀!𝑥𝐴𝜓)       (𝜑𝐴 ≠ ∅)
 
Theoremalsi-no-surprise 42421 Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-alsi 42413; the proof itself builds on alimp-no-surprise 42406. For a contrast, see alimp-surprise 42405. (Contributed by David A. Wheeler, 27-Oct-2018.)
¬ (∀!𝑥(𝜑𝜓) ∧ ∀!𝑥(𝜑 → ¬ 𝜓))
 
20.35.14  Miscellaneous

Miscellaneous proofs.

 
Theorem5m4e1 42422 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
(5 − 4) = 1
 
Theorem2p2ne5 42423 Prove that 2 + 2 ≠ 5. In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase 2 + 2 = 5 has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)
(2 + 2) ≠ 5
 
Theoremresolution 42424 Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) → (𝜓𝜒))
 
Theoremtestable 42425 In classical logic all wffs are testable, that is, it is always true that 𝜑 ∨ ¬ ¬ 𝜑). This is not necessarily true in intuitionistic logic. In intuitionistic logic, if this statement is true for some 𝜑, then 𝜑 is testable. The proof is trivial because it's simply a special case of the law of the excluded middle, which is true in classical logic but not necessarily true in intuitionisic logic. (Contributed by David A. Wheeler, 5-Dec-2018.)
𝜑 ∨ ¬ ¬ 𝜑)
 
20.35.15  AA theorems
 
Theoremaacllem 42426* Lemma for other theorems about 𝔸. (Contributed by Brendan Leahy, 3-Jan-2020.) (Revised by Alexander van der Vekens and David A. Wheeler, 25-Apr-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)    &   ((𝜑𝑘 ∈ (0...𝑁)) → (𝐴𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))       (𝜑𝐴 ∈ 𝔸)
 
20.36  Mathbox for Kunhao Zheng
 
20.36.1  Weighted AM-GM Inequality
 
Theoremamgmwlem 42427 Weighted version of amgmlem 24407. (Contributed by Kunhao Zheng, 19-Jun-2021.)
𝑀 = (mulGrp‘ℂfld)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐹:𝐴⟶ℝ+)    &   (𝜑𝑊:𝐴⟶ℝ+)    &   (𝜑 → (ℂfld Σg 𝑊) = 1)       (𝜑 → (𝑀 Σg (𝐹𝑓𝑐𝑊)) ≤ (ℂfld Σg (𝐹𝑓 · 𝑊)))
 
TheoremamgmlemALT 42428 Alternative proof of amgmlem 24407 using amgmwlem 42427. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Kunhao Zheng, 20-Jun-2021.)
𝑀 = (mulGrp‘ℂfld)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐹:𝐴⟶ℝ+)       (𝜑 → ((𝑀 Σg 𝐹)↑𝑐(1 / (#‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (#‘𝐴)))
 
Theoremamgmw2d 42429 Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 37424). (Contributed by Kunhao Zheng, 20-Jun-2021.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑃 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑄 ∈ ℝ+)    &   (𝜑 → (𝑃 + 𝑄) = 1)       (𝜑 → ((𝐴𝑐𝑃) · (𝐵𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
 
Theoremyoung2d 42430 Young's inequality for 𝑛 = 2, a direct application of amgmw2d 42429. (Contributed by Kunhao Zheng, 6-Jul-2021.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑃 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑄 ∈ ℝ+)    &   (𝜑 → ((1 / 𝑃) + (1 / 𝑄)) = 1)       (𝜑 → (𝐴 · 𝐵) ≤ (((𝐴𝑐𝑃) / 𝑃) + ((𝐵𝑐𝑄) / 𝑄)))
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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 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 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