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Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheorematpointN 35501 The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (Points‘𝐾)       ((𝐾𝐷𝑋𝐴) → {𝑋} ∈ 𝑃)
 
Theorempsubspset 35502* The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (𝐾𝐵𝑆 = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
 
Theoremispsubsp 35503* The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
 
Theoremispsubsp2 35504* The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
 
Theorempsubspi 35505* Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
 
Theorempsubspi2N 35506 Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ (𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅))) → 𝑃𝑋)
 
Theorem0psubN 35507 The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)       (𝐾𝑉 → ∅ ∈ 𝑆)
 
TheoremsnatpsubN 35508 The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ∈ 𝑆)
 
TheorempointpsubN 35509 A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
𝑃 = (Points‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝑃) → 𝑋𝑆)
 
TheoremlinepsubN 35510 A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
𝑁 = (Lines‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝑁) → 𝑋𝑆)
 
TheorematpsubN 35511 The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (𝐾𝑉𝐴𝑆)
 
Theorempsubssat 35512 A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)
 
TheorempsubatN 35513 A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       ((𝐾𝐵𝑋𝑆𝑌𝑋) → 𝑌𝐴)
 
Theorempmapfval 35514* The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
 
Theorempmapval 35515* Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})
 
Theoremelpmap 35516 Member of a projective map. (Contributed by NM, 27-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ (𝑃𝐴𝑃 𝑋)))
 
Theorempmapssat 35517 The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐴)
 
TheorempmapssbaN 35518 A weakening of pmapssat 35517 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐵)
 
Theorempmaple 35519 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 35520 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 35521 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑀𝑃) = {𝑃})
 
Theoremelpmapat 35522 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 35523 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 35524 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 35525 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 35526 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 35527* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 35528, 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 35528* 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 35529* 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 35530* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 35529, 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 35531 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 35532* 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 35533 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 35534* 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 35535* 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 35536 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 35537* 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 35538* 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 35539 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 35540 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 35541 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 35542 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 35543 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 35544 Generalization of 2llnma1 35545. (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 35545 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 35546 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 35547 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 35548 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)
 
20.23.13  Construction of a vector space from a Hilbert lattice
 
Theoremcdlema1N 35549 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 35550 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 35551 Lemma for cdlemb 35552. (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 35552* 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 35553 Extend class notation with projective subspace sum.
class +𝑃
 
Definitiondf-padd 35554* 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 35555* 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 35556* 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 35557* 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 35558* 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 35559* 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 35560 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 35561 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 35562* 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 35563* 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 35564 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 35565 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 35566 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 35567 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 35568 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 35569 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (𝑋 + ∅) = 𝑋)
 
Theorempadd02 35570 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (∅ + 𝑋) = 𝑋)
 
Theorempaddcom 35571 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theorempaddssat 35572 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
 
Theoremsspadd1 35573 A projective subspace sum is a superset of its first summand. (ssun1 3907 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
 
Theoremsspadd2 35574 A projective subspace sum is a superset of its second summand. (ssun2 3908 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑌 + 𝑋))
 
Theorempaddss1 35575 Subset law for projective subspace sum. (unss1 3913 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑍)))
 
Theorempaddss2 35576 Subset law for projective subspace sum. (unss2 3915 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑍 + 𝑋) ⊆ (𝑍 + 𝑌)))
 
Theorempaddss12 35577 Subset law for projective subspace sum. (unss12 3916 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑊𝐴) → ((𝑋𝑌𝑍𝑊) → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑊)))
 
Theorempaddasslem1 35578 Lemma for paddass 35596. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑥𝐴𝑟𝐴𝑦𝐴) ∧ 𝑥𝑦) ∧ ¬ 𝑟 (𝑥 𝑦)) → ¬ 𝑥 (𝑟 𝑦))
 
Theorempaddasslem2 35579 Lemma for paddass 35596. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑟𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧))) → 𝑧 (𝑟 𝑦))
 
Theorempaddasslem3 35580* Lemma for paddass 35596. Restate projective space axiom ps-2 35236. (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))))
 
Theorempaddasslem4 35581* Lemma for paddass 35596. Combine paddasslem1 35578, paddasslem2 35579, and paddasslem3 35580. (Contributed by NM, 8-Jan-2012.)
= (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)))
 
Theorempaddasslem5 35582 Lemma for paddass 35596. Show 𝑠𝑧 by contradiction. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑟𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧) ∧ 𝑠 (𝑥 𝑦))) → 𝑠𝑧)
 
Theorempaddasslem6 35583 Lemma for paddass 35596. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑝𝐴𝑠𝐴) ∧ 𝑧𝐴) ∧ (𝑠𝑧𝑠 (𝑝 𝑧))) → 𝑝 (𝑠 𝑧))
 
Theorempaddasslem7 35584 Lemma for paddass 35596. Combine paddasslem5 35582 and paddasslem6 35583. (Contributed by NM, 9-Jan-2012.)
= (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))
 
Theorempaddasslem8 35585 Lemma for paddass 35596. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑠𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ 𝑠 (𝑥 𝑦) ∧ 𝑝 (𝑠 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem9 35586 Lemma for paddass 35596. Combine paddasslem7 35584 and paddasslem8 35585. (Contributed by NM, 9-Jan-2012.)
= (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))
 
Theorempaddasslem10 35587 Lemma for paddass 35596. Use paddasslem4 35581 to eliminate 𝑠 from paddasslem9 35586. (Contributed by NM, 9-Jan-2012.)
= (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 35588 Lemma for paddass 35596. 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 35589 Lemma for paddass 35596. 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 35590 Lemma for paddass 35596. 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))
 
Theorempaddasslem14 35591 Lemma for paddass 35596. Remove 𝑝𝑧, 𝑥𝑦, and ¬ 𝑟 (𝑥 𝑦) from antecedent of paddasslem10 35587, using paddasslem11 35588, paddasslem12 35589, and paddasslem13 35590. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem15 35592 Lemma for paddass 35596. Use elpaddn0 35558 to eliminate 𝑦 and 𝑧 from paddasslem14 35591. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ∧ (𝑝𝐴 ∧ (𝑥𝑋𝑟 ∈ (𝑌 + 𝑍)) ∧ 𝑝 (𝑥 𝑟))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem16 35593 Lemma for paddass 35596. Use elpaddn0 35558 to eliminate 𝑥 and 𝑟 from paddasslem15 35592. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem17 35594 Lemma for paddass 35596. 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))
 
Theorempaddasslem18 35595 Lemma for paddass 35596. Combine paddasslem16 35593 and paddasslem17 35594. (Contributed by NM, 12-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddass 35596 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 35597 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 35598 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 35599 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 35600 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)
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