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Theorem List for Metamath Proof Explorer - 35301-35400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlduallmodlem 35301 Lemma for lduallmod 35302. (Contributed by NM, 22-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   𝑉 = (Base‘𝑊)    &    + = ∘𝑓 (+g𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    · = ( ·𝑠𝐷)       (𝜑𝐷 ∈ LMod)

Theoremlduallmod 35302 The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐷 ∈ LMod)

Theoremlduallvec 35303 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 19010; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LVec)       (𝜑𝐷 ∈ LVec)

Theoremldualvsub 35304 The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
𝑅 = (Scalar‘𝑊)    &   𝑁 = (invg𝑅)    &    1 = (1r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &    · = ( ·𝑠𝐷)    &    = (-g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺 𝐻) = (𝐺 + ((𝑁1 ) · 𝐻)))

Theoremldualvsubcl 35305 Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    = (-g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺 𝐻) ∈ 𝐹)

Theoremldualvsubval 35306 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 35304? (Requires 𝐷 to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝑆 = (-g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    = (-g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐺 𝐻)‘𝑋) = ((𝐺𝑋)𝑆(𝐻𝑋)))

Theoremldualssvscl 35307 Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   𝑆 = (LSubSp‘𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋 · 𝑌) ∈ 𝑈)

Theoremldualssvsubcl 35308 Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
𝐷 = (LDual‘𝑊)    &    = (-g𝐷)    &   𝑆 = (LSubSp‘𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋 𝑌) ∈ 𝑈)

Theoremldual0vs 35309 Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    0 = (0g𝑅)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   𝑂 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → ( 0 · 𝐺) = 𝑂)

Theoremlkr0f2 35310 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) = 𝑉𝐺 = 0 ))

Theoremlduallkr3 35311 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) ∈ 𝐻𝐺0 ))

TheoremlkrpssN 35312 Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ⊊ (𝐾𝐻) ↔ (𝐺0𝐻 = 0 )))

Theoremlkrin 35313 Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ∩ (𝐾𝐻)) ⊆ (𝐾‘(𝐺 + 𝐻)))

Theoremeqlkr4 35314* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑 → (𝐾𝐺) = (𝐾𝐻))       (𝜑 → ∃𝑟𝑅 𝐻 = (𝑟 · 𝐺))

Theoremldual1dim 35315* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &   𝑁 = (LSpan‘𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → (𝑁‘{𝐺}) = {𝑔𝐹 ∣ (𝐿𝐺) ⊆ (𝐿𝑔)})

Theoremldualkrsc 35316 The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝐾)    &   (𝜑𝑋0 )       (𝜑 → (𝐿‘(𝑋 · 𝐺)) = (𝐿𝐺))

Theoremlkrss 35317 The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝐾)       (𝜑 → (𝐿𝐺) ⊆ (𝐿‘(𝑋 · 𝐺)))

Theoremlkrss2N 35318* Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ⊆ (𝐾𝐻) ↔ ∃𝑟𝑅 𝐻 = (𝑟 · 𝐺)))

TheoremlkreqN 35319 Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴 ∈ (𝑅 ∖ { 0 }))    &   (𝜑𝐻𝐹)    &   (𝜑𝐺 = (𝐴 · 𝐻))       (𝜑 → (𝐾𝐺) = (𝐾𝐻))

TheoremlkrlspeqN 35320 Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   𝑁 = (LSpan‘𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐻𝐹)    &   (𝜑𝐺 ∈ ((𝑁‘{𝐻}) ∖ { 0 }))       (𝜑 → (𝐿𝐺) = (𝐿𝐻))

20.23.9  Ortholattices and orthomodular lattices

Syntaxcops 35321 Extend class notation with orthoposets.
class OP

SyntaxccmtN 35322 Extend class notation with the commutes relation.
class cm

Syntaxcol 35323 Extend class notation with orthlattices.
class OL

Syntaxcoml 35324 Extend class notation with orthomodular lattices.
class OML

Definitiondf-oposet 35325* Define the class of orthoposets, which are bounded posets with an orthocomplementation operation. Note that (Base p ) e. dom ( lub 𝑝) means there is an upper bound 1., and similarly for the 0. element. (Contributed by NM, 20-Oct-2011.) (Revised by NM, 13-Sep-2018.)
OP = {𝑝 ∈ Poset ∣ (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑜(𝑜 = (oc‘𝑝) ∧ ∀𝑎 ∈ (Base‘𝑝)∀𝑏 ∈ (Base‘𝑝)(((𝑜𝑎) ∈ (Base‘𝑝) ∧ (𝑜‘(𝑜𝑎)) = 𝑎 ∧ (𝑎(le‘𝑝)𝑏 → (𝑜𝑏)(le‘𝑝)(𝑜𝑎))) ∧ (𝑎(join‘𝑝)(𝑜𝑎)) = (1.‘𝑝) ∧ (𝑎(meet‘𝑝)(𝑜𝑎)) = (0.‘𝑝))))}

Definitiondf-cmtN 35326* Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
cm = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))})

Definitiondf-ol 35327 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
OL = (Lat ∩ OP)

Definitiondf-oml 35328* Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}

Theoremisopos 35329* The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))

Theoremopposet 35330 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
(𝐾 ∈ OP → 𝐾 ∈ Poset)

Theoremoposlem 35331 Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ))

Theoremop01dm 35332 Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))

Theoremop0cl 35333 An orthoposet has a zero element. (h0elch 28684 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ OP → 0𝐵)

Theoremop1cl 35334 An orthoposet has a unit element. (helch 28672 analog.) (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP → 1𝐵)

Theoremop0le 35335 Orthoposet zero is less than or equal to any element. (ch0le 28872 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → 0 𝑋)

Theoremople0 35336 An element less than or equal to zero equals zero. (chle0 28874 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 0𝑋 = 0 ))

Theoremopnlen0 35337 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2982 and op0le 35335 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑋 𝑌) → 𝑋0 )

Theoremlub0N 35338 The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)
1 = (lub‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Theoremopltn0 35339 A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 0 < 𝑋𝑋0 ))

Theoremople1 35340 Any element is less than the orthoposet unit. (chss 28658 analog.) (Contributed by NM, 23-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → 𝑋 1 )

Theoremop1le 35341 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 28874 analog.) (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 1 𝑋𝑋 = 1 ))

Theoremglb0N 35342 The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)
𝐺 = (glb‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Theoremopoccl 35343 Closure of orthocomplement operation. (choccl 28737 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)

Theoremopococ 35344 Double negative law for orthoposets. (ococ 28837 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)

Theoremopcon3b 35345 Contraposition law for orthoposets. (chcon3i 28897 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))

Theoremopcon2b 35346 Orthocomplement contraposition law. (negcon2 10676 analog.) (Contributed by NM, 16-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))

Theoremopcon1b 35347 Orthocomplement contraposition law. (negcon1 10675 analog.) (Contributed by NM, 24-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = 𝑌 ↔ ( 𝑌) = 𝑋))

Theoremoplecon3 35348 Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))

Theoremoplecon3b 35349 Contraposition law for orthoposets. (chsscon3 28931 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))

Theoremoplecon1b 35350 Contraposition law for strict ordering in orthoposets. (chsscon1 28932 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))

Theoremopoc1 35351 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OP → ( 1 ) = 0 )

Theoremopoc0 35352 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OP → ( 0 ) = 1 )

Theoremopltcon3b 35353 Contraposition law for strict ordering in orthoposets. (chpsscon3 28934 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ( 𝑌) < ( 𝑋)))

Theoremopltcon1b 35354 Contraposition law for strict ordering in orthoposets. (chpsscon1 28935 analog.) (Contributed by NM, 5-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) < 𝑌 ↔ ( 𝑌) < 𝑋))

Theoremopltcon2b 35355 Contraposition law for strict ordering in orthoposets. (chsscon2 28933 analog.) (Contributed by NM, 5-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < ( 𝑌) ↔ 𝑌 < ( 𝑋)))

Theoremopexmid 35356 Law of excluded middle for orthoposets. (chjo 28946 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 1 )

Theoremopnoncon 35357 Law of contradiction for orthoposets. (chocin 28926 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 0 )

TheoremriotaocN 35358* The orthocomplement of the unique poset element such that 𝜓. (riotaneg 11356 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   (𝑥 = ( 𝑦) → (𝜑𝜓))       ((𝐾 ∈ OP ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = ( ‘(𝑦𝐵 𝜓)))

TheoremcmtfvalN 35359* Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})

TheoremcmtvalN 35360 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 29015 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))

Theoremisolat 35361 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

Theoremollat 35362 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL → 𝐾 ∈ Lat)

Theoremolop 35363 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL → 𝐾 ∈ OP)

TheoremolposN 35364 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
(𝐾 ∈ OL → 𝐾 ∈ Poset)

TheoremisolatiN 35365 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
𝐾 ∈ Lat    &   𝐾 ∈ OP       𝐾 ∈ OL

Theoremoldmm1 35366 De Morgan's law for meet in an ortholattice. (chdmm1 28956 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))

Theoremoldmm2 35367 De Morgan's law for meet in an ortholattice. (chdmm2 28957 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) = (𝑋 ( 𝑌)))

Theoremoldmm3N 35368 De Morgan's law for meet in an ortholattice. (chdmm3 28958 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))

Theoremoldmm4 35369 De Morgan's law for meet in an ortholattice. (chdmm4 28959 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌))) = (𝑋 𝑌))

Theoremoldmj1 35370 De Morgan's law for join in an ortholattice. (chdmj1 28960 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))

Theoremoldmj2 35371 De Morgan's law for join in an ortholattice. (chdmj2 28961 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) = (𝑋 ( 𝑌)))

Theoremoldmj3 35372 De Morgan's law for join in an ortholattice. (chdmj3 28962 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))

Theoremoldmj4 35373 De Morgan's law for join in an ortholattice. (chdmj4 28963 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌))) = (𝑋 𝑌))

Theoremolj01 35374 An ortholattice element joined with zero equals itself. (chj0 28928 analog.) (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 0 ) = 𝑋)

Theoremolj02 35375 An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → ( 0 𝑋) = 𝑋)

Theoremolm11 35376 The meet of an ortholattice element with one equals itself. (chm1i 28887 analog.) (Contributed by NM, 22-May-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 1 ) = 𝑋)

Theoremolm12 35377 The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → ( 1 𝑋) = 𝑋)

TheoremlatmassOLD 35378 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4043 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Theoremlatm12 35379 A rearrangement of lattice meet. (in12 4044 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑌 (𝑋 𝑍)))

Theoremlatm32 35380 A rearrangement of lattice meet. (in12 4044 analog.) (Contributed by NM, 13-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Theoremlatmrot 35381 Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑍 𝑋) 𝑌))

Theoremlatm4 35382 Rearrangement of lattice meet of 4 classes. (in4 4049 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌) (𝑍 𝑊)) = ((𝑋 𝑍) (𝑌 𝑊)))

TheoremlatmmdiN 35383 Lattice meet distributes over itself. (inindi 4050 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Theoremlatmmdir 35384 Lattice meet distributes over itself. (inindir 4051 analog.) (Contributed by NM, 6-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) (𝑌 𝑍)))

Theoremolm01 35385 Meet with lattice zero is zero. (chm0 28922 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 0 ) = 0 )

Theoremolm02 35386 Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → ( 0 𝑋) = 0 )

Theoremisoml 35387* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))

TheoremisomliN 35388* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
𝐾 ∈ OL    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))       𝐾 ∈ OML

Theoremomlol 35389 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OL)

Theoremomlop 35390 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OP)

Theoremomllat 35391 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ Lat)

Theoremomllaw 35392 The orthomodular law. (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))

Theoremomllaw2N 35393 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 29016 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))

Theoremomllaw3 35394 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 28867 analog.) (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))

Theoremomllaw4 35395 Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋))

Theoremomllaw5N 35396 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 29044 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌))

TheoremcmtcomlemN 35397 Lemma for cmtcomN 35398. (cmcmlem 29022 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))

TheoremcmtcomN 35398 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 29023 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))

Theoremcmt2N 35399 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 29024 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶( 𝑌)))

Theoremcmt3N 35400 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 29026 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶𝑌))

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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 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