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Theorem List for Metamath Proof Explorer - 36901-37000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempmaple 36901 The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑀𝑋) ⊆ (𝑀𝑌)))

Theorempmap11 36902 The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) = (𝑀𝑌) ↔ 𝑋 = 𝑌))

Theorempmapat 36903 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑀𝑃) = {𝑃})

Theoremelpmapat 36904 Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑋 ∈ (𝑀𝑃) ↔ 𝑋 = 𝑃))

Theorempmap0 36905 Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
0 = (0.‘𝐾)    &   𝑀 = (pmap‘𝐾)       (𝐾 ∈ AtLat → (𝑀0 ) = ∅)

Theorempmapeq0 36906 A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) = ∅ ↔ 𝑋 = 0 ))

Theorempmap1N 36907 Value of the projective map of a Hilbert lattice at lattice unit. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)
1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       (𝐾 ∈ OP → (𝑀1 ) = 𝐴)

Theorempmapsub 36908 The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)

Theorempmapglbx 36909* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 36910, where we read 𝑆 as 𝑆(𝑖). Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))

Theorempmapglb 36910* The projective map of the GLB of a set of lattice elements 𝑆. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))

Theorempmapglb2N 36911* The projective map of the GLB of a set of lattice elements 𝑆. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows 𝑆 = ∅. (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑀‘(𝐺𝑆)) = (𝐴 𝑥𝑆 (𝑀𝑥)))

Theorempmapglb2xN 36912* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 36911, where we read 𝑆 as 𝑆(𝑖). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows 𝐼 = ∅. (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))

Theorempmapmeet 36913 The projective map of a meet. (Contributed by NM, 25-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑃 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑃‘(𝑋 𝑌)) = ((𝑃𝑋) ∩ (𝑃𝑌)))

Theoremisline2 36914* Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞)))))

Theoremlinepmap 36915 A line described with a projective map. (Contributed by NM, 3-Feb-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) ∈ 𝑁)

Theoremisline3 36916* Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑝 𝑞))))

Theoremisline4N 36917* Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))

Theoremlneq2at 36918 A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋 = (𝑃 𝑄))

TheoremlnatexN 36919* There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))

TheoremlnjatN 36920* Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → ∃𝑞𝐴 (𝑞𝑃𝑋 = (𝑃 𝑞)))

TheoremlncvrelatN 36921 A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)

Theoremlncvrat 36922 A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → 𝑃𝐶𝑋)

Theoremlncmp 36923 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))

Theorem2lnat 36924 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝐹 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)

Theorem2atm2atN 36925 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
= (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑅 𝑃) (𝑅 𝑄)) ≠ 0 )

Theorem2llnma1b 36926 Generalization of 2llnma1 36927. (Contributed by NM, 26-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) = 𝑃)

Theorem2llnma1 36927 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑄 𝑃) (𝑄 𝑅)) = 𝑄)

Theorem2llnma3r 36928 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑅) ≠ (𝑄 𝑅)) → ((𝑃 𝑅) (𝑄 𝑅)) = 𝑅)

Theorem2llnma2 36929 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑃) (𝑅 𝑄)) = 𝑅)

Theorem2llnma2rN 36930 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) (𝑄 𝑅)) = 𝑅)

20.24.13  Construction of a vector space from a Hilbert lattice

Theoremcdlema1N 36931 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝐹 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) = (𝑋 𝑌))

Theoremcdlema2N 36932 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑅 𝑋) = 0 )

Theoremcdlemblem 36933 Lemma for cdlemb 36934. (Contributed by NM, 8-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    < = (lt‘𝐾)    &    = (meet‘𝐾)    &   𝑉 = ((𝑃 𝑄) 𝑋)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))

Theoremcdlemb 36934* Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))

Syntaxcpadd 36935 Extend class notation with projective subspace sum.
class +𝑃

Definitiondf-padd 36936* Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)
+𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))

Theorempaddfval 36937* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (𝐾𝐵+ = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))

Theorempaddval 36938* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = ((𝑋𝑌) ∪ {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝 (𝑞 𝑟)}))

Theoremelpadd 36939* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑆 ∈ (𝑋 + 𝑌) ↔ ((𝑆𝑋𝑆𝑌) ∨ (𝑆𝐴 ∧ ∃𝑞𝑋𝑟𝑌 𝑆 (𝑞 𝑟)))))

Theoremelpaddn0 36940* Member of projective subspace sum of nonempty sets. (Contributed by NM, 3-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆𝐴 ∧ ∃𝑞𝑋𝑟𝑌 𝑆 (𝑞 𝑟))))

Theorempaddvaln0N 36941* Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝 (𝑞 𝑟)})

Theoremelpaddri 36942 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑄𝑋𝑅𝑌) ∧ (𝑆𝐴𝑆 (𝑄 𝑅))) → 𝑆 ∈ (𝑋 + 𝑌))

TheoremelpaddatriN 36943 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ (𝑅𝑋𝑆𝐴𝑆 (𝑅 𝑄))) → 𝑆 ∈ (𝑋 + {𝑄}))

Theoremelpaddat 36944* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆𝐴 ∧ ∃𝑝𝑋 𝑆 (𝑝 𝑄))))

TheoremelpaddatiN 36945* Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑅 ∈ (𝑋 + {𝑄}))) → ∃𝑝𝑋 𝑅 (𝑝 𝑄))

Theoremelpadd2at 36946 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑄𝐴𝑅𝐴) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ (𝑆𝐴𝑆 (𝑄 𝑅))))

Theoremelpadd2at2 36947 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ (𝑄𝐴𝑅𝐴𝑆𝐴)) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ 𝑆 (𝑄 𝑅)))

TheorempaddunssN 36948 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋𝑌) ⊆ (𝑋 + 𝑌))

Theoremelpadd0 36949 Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾𝐵𝑋𝐴𝑌𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆𝑋𝑆𝑌)))

Theorempaddval0 36950 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾𝐵𝑋𝐴𝑌𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = (𝑋𝑌))

Theorempadd01 36951 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (𝑋 + ∅) = 𝑋)

Theorempadd02 36952 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (∅ + 𝑋) = 𝑋)

Theorempaddcom 36953 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theorempaddssat 36954 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)

Theoremsspadd1 36955 A projective subspace sum is a superset of its first summand. (ssun1 4151 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))

Theoremsspadd2 36956 A projective subspace sum is a superset of its second summand. (ssun2 4152 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑌 + 𝑋))

Theorempaddss1 36957 Subset law for projective subspace sum. (unss1 4158 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑍)))

Theorempaddss2 36958 Subset law for projective subspace sum. (unss2 4160 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑍 + 𝑋) ⊆ (𝑍 + 𝑌)))

Theorempaddss12 36959 Subset law for projective subspace sum. (unss12 4161 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑊𝐴) → ((𝑋𝑌𝑍𝑊) → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑊)))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑥𝐴𝑟𝐴𝑦𝐴) ∧ 𝑥𝑦) ∧ ¬ 𝑟 (𝑥 𝑦)) → ¬ 𝑥 (𝑟 𝑦))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑟𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧))) → 𝑧 (𝑟 𝑦))

Theorempaddasslem3 36962* Lemma for paddass 36978. Restate projective space axiom ps-2 36618. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑥𝐴𝑟𝐴𝑦𝐴) ∧ (𝑝𝐴𝑧𝐴)) → (((¬ 𝑥 (𝑟 𝑦) ∧ 𝑝𝑧) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑧 (𝑟 𝑦))) → ∃𝑠𝐴 (𝑠 (𝑥 𝑦) ∧ 𝑠 (𝑝 𝑧))))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑝𝐴𝑟𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴) ∧ (𝑝𝑧𝑥𝑦 ∧ ¬ 𝑟 (𝑥 𝑦))) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧))) → ∃𝑠𝐴 (𝑠 (𝑥 𝑦) ∧ 𝑠 (𝑝 𝑧)))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑟𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧) ∧ 𝑠 (𝑥 𝑦))) → 𝑠𝑧)

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑝𝐴𝑠𝐴) ∧ 𝑧𝐴) ∧ (𝑠𝑧𝑠 (𝑝 𝑧))) → 𝑝 (𝑠 𝑧))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑝𝐴𝑟𝐴𝑠𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ ((¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧) ∧ 𝑠 (𝑥 𝑦)) ∧ 𝑠 (𝑝 𝑧))) → 𝑝 (𝑠 𝑧))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑠𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ 𝑠 (𝑥 𝑦) ∧ 𝑝 (𝑠 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧)) ∧ (𝑠𝐴𝑠 (𝑥 𝑦) ∧ 𝑠 (𝑝 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑝𝑧𝑥𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Theorempaddasslem11 36970 Lemma for paddass 36978. The case when 𝑝 = 𝑧. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑧𝑍) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Theorempaddasslem12 36971 Lemma for paddass 36978. The case when 𝑥 = 𝑦. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Theorempaddasslem13 36972 Lemma for paddass 36978. The case when 𝑟 (𝑥 𝑦). (Unlike the proof in Maeda and Maeda, we don't need 𝑥𝑦.) (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ∧ (𝑝𝐴 ∧ (𝑥𝑋𝑟 ∈ (𝑌 + 𝑍)) ∧ 𝑝 (𝑥 𝑟))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))

Theorempaddasslem17 36976 Lemma for paddass 36978. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ ¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))

𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))

Theorempaddass 36978 Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (Contributed by NM, 29-Dec-2011.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theorempadd12N 36979 Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Theorempadd4N 36980 Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐴𝑊𝐴)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Theorempaddidm 36981 Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)

TheorempaddclN 36982 The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Theorempaddssw1 36983 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑍𝑌𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍)))

Theorempaddssw2 36984 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → (𝑋𝑍𝑌𝑍)))

Theorempaddss 36985 Subset law for projective subspace sum. (unss 4163 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝑆)) → ((𝑋𝑍𝑌𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍))

Theorempmodlem1 36986* Lemma for pmod1i 36988. (Contributed by NM, 9-Mar-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑍𝑆𝑋𝑍𝑝𝑍) ∧ (𝑞𝑋𝑟𝑌𝑝 (𝑞 𝑟))) → 𝑝 ∈ (𝑋 + (𝑌𝑍)))

Theorempmodlem2 36987 Lemma for pmod1i 36988. (Contributed by NM, 9-Mar-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝑆) ∧ 𝑋𝑍) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌𝑍)))

Theorempmod1i 36988 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝑆)) → (𝑋𝑍 → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑋 + (𝑌𝑍))))

Theorempmod2iN 36989 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑆𝑌𝐴𝑍𝐴)) → (𝑍𝑋 → ((𝑋𝑌) + 𝑍) = (𝑋 ∩ (𝑌 + 𝑍))))

TheorempmodN 36990 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑆𝑌𝐴𝑍𝐴)) → (𝑋 ∩ (𝑌 + (𝑋𝑍))) = ((𝑋𝑌) + (𝑋𝑍)))

Theorempmodl42N 36991 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))

Theorempmapjoin 36992 The projective map of the join of two lattice elements. Part of Equation 15.5.3 of [MaedaMaeda] p. 63. (Contributed by NM, 27-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))

Theorempmapjat1 36993 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (𝑀‘(𝑋 𝑄)) = ((𝑀𝑋) + (𝑀𝑄)))

Theorempmapjat2 36994 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (𝑀‘(𝑄 𝑋)) = ((𝑀𝑄) + (𝑀𝑋)))

Theorempmapjlln1 36995 The projective map of the join of a lattice element and a lattice line (expressed as the join 𝑄 𝑅 of two atoms). (Contributed by NM, 16-Sep-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑅𝐴)) → (𝑀‘(𝑋 (𝑄 𝑅))) = ((𝑀𝑋) + (𝑀‘(𝑄 𝑅))))

Theoremhlmod1i 36996 A version of the modular law pmod1i 36988 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))

Theorematmod1i1 36997 Version of modular law pmod1i 36988 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑌) → (𝑃 (𝑋 𝑌)) = ((𝑃 𝑋) 𝑌))

Theorematmod1i1m 36998 Version of modular law pmod1i 36988 that holds in a Hilbert lattice, when an element meets an atom. (Contributed by NM, 2-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴) ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑃) 𝑍) → ((𝑋 𝑃) (𝑌 𝑍)) = (((𝑋 𝑃) 𝑌) 𝑍))

Theorematmod1i2 36999 Version of modular law pmod1i 36988 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 (𝑃 𝑌)) = ((𝑋 𝑃) 𝑌))

Theoremllnmod1i2 37000 Version of modular law pmod1i 36988 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 𝑄). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑋 𝑌) → (𝑋 ((𝑃 𝑄) 𝑌)) = ((𝑋 (𝑃 𝑄)) 𝑌))

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