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Theorem List for Metamath Proof Explorer - 38101-38200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhlhilvsca 38101 The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &    · = ( ·𝑠𝐿)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑· = ( ·𝑠𝑈))
 
Theoremhlhilip 38102* Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐿)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &    , = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥))       (𝜑, = (·𝑖𝑈))
 
Theoremhlhilipval 38103 Value of inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐿)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &    , = (·𝑖𝑈)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 , 𝑌) = ((𝑆𝑌)‘𝑋))
 
Theoremhlhilnvl 38104 The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 = (*𝑟𝑅))
 
Theoremhlhillvec 38105 The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑈 ∈ LVec)
 
Theoremhlhildrng 38106 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑅 = (Scalar‘𝑈)       (𝜑𝑅 ∈ DivRing)
 
Theoremhlhilsrnglem 38107 Lemma for hlhilsrng 38108. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑅 = (Scalar‘𝑈)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐿)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    · = (.r𝑆)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)       (𝜑𝑅 ∈ *-Ring)
 
Theoremhlhilsrng 38108 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑅 = (Scalar‘𝑈)       (𝜑𝑅 ∈ *-Ring)
 
Theoremhlhil0 38109 The zero vector for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &    0 = (0g𝐿)       (𝜑0 = (0g𝑈))
 
Theoremhlhillsm 38110 The vector sum operation for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &    = (LSSum‘𝐿)       (𝜑 = (LSSum‘𝑈))
 
Theoremhlhilocv 38111 The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑉 = (Base‘𝐿)    &   𝑁 = ((ocH‘𝐾)‘𝑊)    &   𝑂 = (ocv‘𝑈)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑂𝑋) = (𝑁𝑋))
 
Theoremhlhillcs 38112 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 38090 is applied over and over to conclusion rather than applied once to antecedent - would compressed proof be shorter if applied once to antecedent? (Contributed by NM, 23-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝐶 = (ClSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐶 = ran 𝐼)
 
Theoremhlhilphllem 38113* Lemma for hlhil 23649. (Contributed by NM, 23-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐿)    &    + = (+g𝐿)    &    · = ( ·𝑠𝐿)    &   𝑅 = (Scalar‘𝐿)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &    × = (.r𝑅)    &   𝑄 = (0g𝑅)    &    0 = (0g𝐿)    &    , = (·𝑖𝑈)    &   𝐽 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   𝐸 = (𝑥𝑉, 𝑦𝑉 ↦ ((𝐽𝑦)‘𝑥))       (𝜑𝑈 ∈ PreHil)
 
Theoremhlhilhillem 38114* Lemma for hlhil 23649. (Contributed by NM, 23-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐿)    &    + = (+g𝐿)    &    · = ( ·𝑠𝐿)    &   𝑅 = (Scalar‘𝐿)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &    × = (.r𝑅)    &   𝑄 = (0g𝑅)    &    0 = (0g𝐿)    &    , = (·𝑖𝑈)    &   𝐽 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   𝐸 = (𝑥𝑉, 𝑦𝑉 ↦ ((𝐽𝑦)‘𝑥))    &   𝑂 = (ocv‘𝑈)    &   𝐶 = (ClSubSp‘𝑈)       (𝜑𝑈 ∈ Hil)
 
Theoremhlathil 38115 Construction of a Hilbert space (df-hil 20447) 𝑈 from a Hilbert lattice (df-hlat 35505) 𝐾, where 𝑊 is a fixed but arbitrary hyperplane (co-atom) in 𝐾.

The Hilbert space 𝑈 is identical to the vector space ((DVecH‘𝐾)‘𝑊) (see dvhlvec 37263) except that it is extended with involution and inner product components. The construction of these two components is provided by Theorem 3.6 in [Holland95] p. 13, whose proof we follow loosely.

An example of involution is the complex conjugate when the division ring is the field of complex numbers. The nature of the division ring we constructed is indeterminate, however, until we specialize the initial Hilbert lattice with additional conditions found by Maria Solèr in 1995 and refined by René Mayet in 1998 that result in a division ring isomorphic to . See additional discussion at http://us.metamath.org/qlegif/mmql.html#what.

𝑊 corresponds to the w in the proof of Theorem 13.4 of [Crawley] p. 111. Such a 𝑊 always exists since HL has lattice rank of at least 4 by df-hil 20447. It can be eliminated if we just want to show the existence of a Hilbert space, as is done in the literature. (Contributed by NM, 23-Jun-2015.)

𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑈 ∈ Hil)
 
20.24  Mathbox for Steven Nguyen
 
20.24.1  Utility theorems
 
Theoremioin9i8 38116 Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ¬ 𝜃)    &   (𝜓𝜃)       (𝜑 → (𝜓𝜃))
 
Theoremjaodd 38117 Double deduction form of jaoi 846. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜏𝜃)))       (𝜑 → (𝜓 → ((𝜒𝜏) → 𝜃)))
 
Theoremnsb 38118 Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.)
¬ 𝜑        ¬ [𝑥 / 𝑦]𝜑
 
Theoremsbtv 38119* Version of sbt 2496 with a disjoint variable condition, which does not use ax-5 1953, ax-7 2055, ax-12 2163, or ax-13 2334. (Contributed by Steven Nguyen, 25-Apr-2023.)
𝜑       [𝑥 / 𝑦]𝜑
 
Theorem3rspcedvd 38120* Triple application of rspcedvd 3518. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐵) → (𝜒𝜃))    &   ((𝜑𝑧 = 𝐶) → (𝜃𝜏))    &   (𝜑𝜏)       (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 𝜓)
 
Theoremcleljust2 38121* A version of cleljust 2115 where the second class is instead replaced with a class abstraction. This provides another part of the justification for df-clel 2774. (Contributed by Steven Nguyen, 19-May-2023.)
(𝑎 ∈ {𝑏𝜑} ↔ ∃𝑥(𝑥 = 𝑎𝑥 ∈ {𝑏𝜑}))
 
Theoremsn-vexwv 38122* A universal class. Proof of bj-vexwv 33426 without ax-5 1953, ax-7 2055 or ax-12 2163. (Contributed by Steven Nguyen, 25-Apr-2023.)
𝜑       𝑦 ∈ {𝑥𝜑}
 
Theorembrabg2a 38123* Generalized version of brabga 5226 when the abstraction is in the style of opelopab2a 5227. (Contributed by Steven Nguyen, 29-Apr-2023.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}       (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
 
Theorempssexg 38124 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorempssn0 38125 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)
 
Theorempsspwb 38126 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 38127 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))
 
Theoremelpwbi 38128 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 38129 Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)
𝐴𝐶    &   𝐵𝐷       𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)
 
Theoremqseq12d 38130 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
 
Theoremnelsubginvcld 38131 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 38132 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 38133 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))
 
Theoremlvecgrp 38134 A left vector is a group. (Contributed by Steven Nguyen, 28-May-2023.)
(𝑊 ∈ LVec → 𝑊 ∈ Grp)
 
Theoremlvecring 38135 The scalar component of a left vector is a ring. (Contributed by Steven Nguyen, 28-May-2023.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → 𝐹 ∈ Ring)
 
20.24.2  Russell's paradox
 
Theoremcbvabvw 38136* Version of cbvabv 2914 with a distinct variable condition on x and y, which removes dependencies on ax-11 2150 and ax-13 2334. (Contributed by Steven Nguyen, 4-Dec-2022.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}
 
Theoremcbvrabvw 38137* Version of cbvrabv 3396 with a distinct variable condition on x and y, which removes dependencies on ax-11 2150 and ax-13 2334. (Contributed by Steven Nguyen, 4-Dec-2022.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
 
Theoremrru 38138* Relative version of Russell's paradox ru 3651 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5064. (Contributed by Stefan O'Rear, 22-Feb-2015.) Remove usage of ax-13 2334. (Revised by Steven Nguyen, 20-Nov-2022.)

¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
 
20.24.3  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 10342 is used in mulid1 10374 related theorems, so one could trade off the extra axioms in mulid1 10374 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 10329; in the other direction, real number closure laws can be avoided by using ax-resscn 10329 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 10337, ax-1cn 10330, and ax-addcl 10332. (And in practice, the expression isn't completely expanded into ones.)

Multiplication by 1 requires either mulid2i 10382 or (ax-1rid 10342 and 1re 10376) as seen in 1t1e1 11544 and 1t1e1ALT 38142. Multiplying with greater natural numbers uses ax-distr 10339. 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 38139 Alternate proof of c0ex 10370 using more set theory axioms but fewer complex number axioms (add ax-10 2135, ax-11 2150, ax-13 2334, ax-nul 5025, and remove ax-1cn 10330, ax-icn 10331, ax-addcl 10332, and ax-mulcl 10334). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ V
 
Theorem0cnALT3 38140 Alternate proof of 0cn 10368 using ax-resscn 10329, ax-addrcl 10333, ax-rnegex 10343, ax-cnre 10345 instead of ax-icn 10331, ax-addcl 10332, ax-mulcl 10334, ax-i2m1 10340. Version of 0cnALT 10610 using ax-1cn 10330 instead of ax-icn 10331. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ
 
Theoremelre0re 38141 Specialized version of 0red 10380 without using ax-1cn 10330 and ax-cnre 10345. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝐴 ∈ ℝ → 0 ∈ ℝ)
 
Theorem1t1e1ALT 38142 Alternate proof of 1t1e1 11544 using a different set of axioms (add ax-mulrcl 10335, ax-i2m1 10340, ax-1ne0 10341, ax-rrecex 10344 and remove ax-resscn 10329, ax-mulcom 10336, ax-mulass 10338, ax-distr 10339). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 · 1) = 1
 
Theoremremulcan2d 38143 mulcan2d 11009 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))
 
Theoremnnadd1com 38144 Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
(𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
 
Theoremnnaddcom 38145 Addition is commutative for natural numbers. Uses fewer axioms than addcom 10562. (Contributed by Steven Nguyen, 9-Dec-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremaddsubeq4com 38146 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 38147 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 38148 Lemma for dffltz 38213. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴 ≠ 0)    &   (𝜑 → ¬ 0 < 𝐴)    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℕ)
 
Theoremsqmid3api 38149 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 38150 Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (𝐴𝐵 + 𝐶) = (𝐴𝐶 + 𝐵)
 
Theoremsqn5i 38151 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 38152 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 38153 Partial products algorithm for two digit multiplication, no carry. Compare muladdi 10826. (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 38154 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 38155 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 38156 Variant of sqdeccom12 38155 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 38157 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 10386 saving the lower level uses of mulcomli 10386 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 11956 are added then this proof would benefit more than ex-decpmul 38158.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11517 or 8t7e56 11967. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

(235 · 711) = 167085
 
Theoremex-decpmul 38158 Example usage of decpmul 38154. This proof is significantly longer than 235t711 38157. There is more unnecessary carrying compared to 235t711 38157. Although saving 5 visual steps, using mulcomli 10386 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
 
20.24.4  Exponents
 
Theoremoexpreposd 38159 Lemma for dffltz 38213. (Contributed by Steven Nguyen, 4-Mar-2023.)
(𝜑𝑁 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)       (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁𝑀)))
 
Theoremcxpgt0d 38160 Exponentiation with a positive mantissa is positive. (Contributed by Steven Nguyen, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℝ)       (𝜑 → 0 < (𝐴𝑐𝑁))
 
Theoremdvdsexpim 38161 dvdssqim 15679 generalized to nonnegative exponents. (Contributed by Steven Nguyen, 2-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝐵 → (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremnn0rppwr 38162 If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵𝑁. rppwr 15683 extended to nonnegative integers. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
 
Theoremexpgcd 38163 Exponentiation distributes over GCD. sqgcd 15684 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 38164 Exponentiation distributes over GCD. nn0gcdsq 15864 extended to nonnegative exponents. expgcd 38163 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremzexpgcd 38165 Exponentiation distributes over GCD. zgcdsq 15865 extended to nonnegative exponents. nn0expgcd 38164 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 38166 numdensq 15866 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 38167 numsq 15867 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁))
 
Theoremdenexp 38168 densq 15868 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁))
 
Theoremexp11d 38169 sq11d 13366 for positive real bases and non-zero exponents. (Contributed by Steven Nguyen, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑁 ≠ 0)    &   (𝜑 → (𝐴𝑁) = (𝐵𝑁))       (𝜑𝐴 = 𝐵)
 
Theoremzrtelqelz 38170 zsqrtelqelz 15870 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 38171 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 38172 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))
 
20.24.5  Real subtraction
 
Syntaxcresub 38173 Real number subtraction.
class
 
Definitiondf-resub 38174* Define subtraction between real numbers. This operator saves a few axioms over df-sub 10608 in certain situations. Theorem resubval 38175 shows its value, resubadd 38188 relates it to addition, and rersubcl 38187 proves its closure. Based on df-sub 10608. (Contributed by Steven Nguyen, 7-Jan-2022.)
= (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
 
Theoremresubval 38175* 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 38176* Lemma for renegeu 38178 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 38177* Lemma for renegeu 38178 and similar. Remove a change in bound variables from renegeulemv 38176. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)       (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)
 
Theoremrenegeu 38178* Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Theoremrernegcl 38179 Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (0 − 𝐴) ∈ ℝ)
 
Theoremrenegadd 38180 Relationship between real negation and addition. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 − 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 0))
 
Theoremrenegid 38181 Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (𝐴 + (0 − 𝐴)) = 0)
 
Theoremreaddid2addid1d 38182 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 10550, 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)
 
Theoremreneg0addid1 38183 Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → ((0 − 0) + 𝐴) = 𝐴)
 
Theoremresubeulem1 38184 Lemma for resubeu 38186. 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 38185 Lemma for resubeu 38186. 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 38186* Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵)
 
Theoremrersubcl 38187 Closure for real subtraction. Based on subcl 10621. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) ∈ ℝ)
 
Theoremresubadd 38188 Relation between real subtraction and addition. Based on subadd 10625. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremresubaddd 38189 Relationship between subtraction and addition. Based on subaddd 10752. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremresubf 38190 Real subtraction is an operation on the real numbers. Based on subf 10624. (Contributed by Steven Nguyen, 7-Jan-2023.)
:(ℝ × ℝ)⟶ℝ
 
Theoremrepncan2 38191 Addition and subtraction of equals. Compare pncan2 10629. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
 
Theoremrepncan3 38192 Addition and subtraction of equals. Based on pncan3 10630. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 𝐴)) = 𝐵)
 
Theoremreaddsub 38193 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 38194 Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 43621 with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶 𝐴))
 
Theoremresubcan2 38195 Cancellation law for real subtraction. Compare subcan2 10648. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐶) = (𝐵 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremresubsub4 38196 Law for double subtraction. Compare subsub4 10656. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) − 𝐶) = (𝐴 (𝐵 + 𝐶)))
 
Theoremrennncan2 38197 Cancellation law for real subtraction. Compare nnncan2 10660. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐶) − (𝐵 𝐶)) = (𝐴 𝐵))
 
Theoremrenpncan3 38198 Cancellation law for real subtraction. Compare npncan3 10661. (Contributed by Steven Nguyen, 28-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) + (𝐶 𝐴)) = (𝐶 𝐵))
 
Theoremrepnpcan 38199 Cancellation law for addition and real subtraction. Compare pnpcan 10662. (Contributed by Steven Nguyen, 19-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 𝐶))
 
Theoremresubidaddid1lem 38200 Lemma for resubidaddid1 38201. A special case of npncan 10644. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴 𝐵) = (𝐵 𝐶))       (𝜑 → ((𝐴 𝐵) + (𝐵 𝐶)) = (𝐴 𝐶))
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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 43601-43657
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