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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlatasymb 17101 A lattice ordering is asymmetric. (eqss 3651 analog.) (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Theoremlatasym 17102 A lattice ordering is asymmetric. (eqss 3651 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))

Theoremlattr 17103 A lattice ordering is transitive. (sstr 3644 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))

Theoremlatasymd 17104 Deduce equality from lattice ordering. (eqssd 3653 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑋)       (𝜑𝑋 = 𝑌)

Theoremlattrd 17105 A lattice ordering is transitive. Deduction version of lattr 17103. (Contributed by NM, 3-Sep-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑍)       (𝜑𝑋 𝑍)

Theoremlatjcom 17106 The join of a lattice commutes. (chjcom 28493 analog.) (Contributed by NM, 16-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremlatlej1 17107 A join's first argument is less than or equal to the join. (chub1 28494 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))

Theoremlatlej2 17108 A join's second argument is less than or equal to the join. (chub2 28495 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (𝑋 𝑌))

Theoremlatjle12 17109 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 28496 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))

Theoremlatleeqj1 17110 Less-than-or-equal-to in terms of join. (chlejb1 28499 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 𝑌) = 𝑌))

Theoremlatleeqj2 17111 Less-than-or-equal-to in terms of join. (chlejb2 28500 analog.) (Contributed by NM, 14-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑌 𝑋) = 𝑌))

Theoremlatjlej1 17112 Add join to both sides of a lattice ordering. (chlej1i 28460 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 𝑍) (𝑌 𝑍)))

Theoremlatjlej2 17113 Add join to both sides of a lattice ordering. (chlej2i 28461 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))

Theoremlatjlej12 17114 Add join to both sides of a lattice ordering. (chlej12i 28462 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌𝑍 𝑊) → (𝑋 𝑍) (𝑌 𝑊)))

Theoremlatnlej 17115 An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → (𝑋𝑌𝑋𝑍))

Theoremlatnlej1l 17116 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → 𝑋𝑌)

Theoremlatnlej1r 17117 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → 𝑋𝑍)

Theoremlatnlej2 17118 An idiom to express that a lattice element differs from two others. (Contributed by NM, 10-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → (¬ 𝑋 𝑌 ∧ ¬ 𝑋 𝑍))

Theoremlatnlej2l 17119 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → ¬ 𝑋 𝑌)

Theoremlatnlej2r 17120 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → ¬ 𝑋 𝑍)

Theoremlatjidm 17121 Lattice join is idempotent. (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)

Theoremlatmcom 17122 The join of a lattice commutes. (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremlatmle1 17123 A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)

Theoremlatmle2 17124 A meet is less than or equal to its second argument. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)

Theoremlatlem12 17125 An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑋 𝑍) ↔ 𝑋 (𝑌 𝑍)))

Theoremlatleeqm1 17126 Less-than-or-equal-to in terms of meet. (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 𝑌) = 𝑋))

Theoremlatleeqm2 17127 Less-than-or-equal-to in terms of meet. (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑌 𝑋) = 𝑋))

Theoremlatmlem1 17128 Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 𝑍) (𝑌 𝑍)))

Theoremlatmlem2 17129 Add meet to both sides of a lattice ordering. (sslin 3872 analog.) (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))

Theoremlatmlem12 17130 Add join to both sides of a lattice ordering. (ss2in 3873 analog.) (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌𝑍 𝑊) → (𝑋 𝑍) (𝑌 𝑊)))

Theoremlatnlemlt 17131 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3889 analog.) (Contributed by NM, 5-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋 𝑌) < 𝑋))

Theoremlatnle 17132 Equivalent expressions for "not less than" in a lattice. (chnle 28501 analog.) (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑌 𝑋𝑋 < (𝑋 𝑌)))

Theoremlatmidm 17133 Lattice join is idempotent. (inidm 3855 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)

Theoremlatabs1 17134 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 28503 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)

Theoremlatabs2 17135 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 28504 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)

Theoremlatledi 17136 An ortholattice is distributive in one ordering direction. (ledi 28527 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍)))

Theoremlatmlej11 17137 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) (𝑋 𝑍))

Theoremlatmlej12 17138 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) (𝑍 𝑋))

Theoremlatmlej21 17139 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑋) (𝑋 𝑍))

Theoremlatmlej22 17140 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑋) (𝑍 𝑋))

Theoremlubsn 17141 The least upper bound of a singleton. (chsupsn 28400 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑈‘{𝑋}) = 𝑋)

Theoremlatjass 17142 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 28520 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Theoremlatj12 17143 Swap 1st and 2nd members of lattice join. (chj12 28521 analog.) (Contributed by NM, 4-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑌 (𝑋 𝑍)))

Theoremlatj32 17144 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Theoremlatj13 17145 Swap 1st and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑍 (𝑌 𝑋)))

Theoremlatj31 17146 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑍 𝑌) 𝑋))

Theoremlatjrot 17147 Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑍 𝑋) 𝑌))

Theoremlatj4 17148 Rearrangement of lattice join of 4 classes. (chj4 28522 analog.) (Contributed by NM, 14-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌) (𝑍 𝑊)) = ((𝑋 𝑍) (𝑌 𝑊)))

Theoremlatj4rot 17149 Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌) (𝑍 𝑊)) = ((𝑊 𝑋) (𝑌 𝑍)))

Theoremlatjjdi 17150 Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Theoremlatjjdir 17151 Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) (𝑌 𝑍)))

Theoremmod1ile 17152 The weak direction of the modular law (e.g., pmod1i 35452, atmod1i1 35461) that holds in any lattice. (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))

Theoremmod2ile 17153 The weak direction of the modular law (e.g., pmod2iN 35453) that holds in any lattice. (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑋 → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍))))

Syntaxccla 17154 Extend class notation with complete lattices.
class CLat

Definitiondf-clat 17155 Define the class of all complete lattices, where every subset of the base set has an LUB and a GLB. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}

Theoremisclat 17156 The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Theoremclatpos 17157 A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.)
(𝐾 ∈ CLat → 𝐾 ∈ Poset)

Theoremclatlem 17158 Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → ((𝑈𝑆) ∈ 𝐵 ∧ (𝐺𝑆) ∈ 𝐵))

Theoremclatlubcl 17159 Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (𝑈𝑆) ∈ 𝐵)

Theoremclatlubcl2 17160 Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝑈)

Theoremclatglbcl 17161 Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (𝐺𝑆) ∈ 𝐵)

Theoremclatglbcl2 17162 Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)

Theoremclatl 17163 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5253 to shorten proof and eliminate joindmss 17054 and meetdmss 17068?
(𝐾 ∈ CLat → 𝐾 ∈ Lat)

Theoremisglbd 17164* Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   ((𝜑𝑦𝑆) → 𝐻 𝑦)    &   ((𝜑𝑥𝐵 ∧ ∀𝑦𝑆 𝑥 𝑦) → 𝑥 𝐻)    &   (𝜑𝐾 ∈ CLat)    &   (𝜑𝑆𝐵)    &   (𝜑𝐻𝐵)       (𝜑 → (𝐺𝑆) = 𝐻)

Theoremlublem 17165* Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))

Theoremlubub 17166 The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝑆) → 𝑋 (𝑈𝑆))

Theoremlubl 17167* The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝐵) → (∀𝑦𝑆 𝑦 𝑋 → (𝑈𝑆) 𝑋))

Theoremlubss 17168 Subset law for least upper bounds. (chsupss 28329 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑇𝐵𝑆𝑇) → (𝑈𝑆) (𝑈𝑇))

Theoremlubel 17169 An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑋𝑆𝑆𝐵) → 𝑋 (𝑈𝑆))

Theoremlubun 17170 The LUB of a union. (Contributed by NM, 5-Mar-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑇𝐵) → (𝑈‘(𝑆𝑇)) = ((𝑈𝑆) (𝑈𝑇)))

Theoremclatglb 17171* Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (∀𝑦𝑆 (𝐺𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝐺𝑆))))

Theoremclatglble 17172 The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝑆) → (𝐺𝑆) 𝑋)

Theoremclatleglb 17173* Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑋𝐵𝑆𝐵) → (𝑋 (𝐺𝑆) ↔ ∀𝑦𝑆 𝑋 𝑦))

Theoremclatglbss 17174 Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑇𝐵𝑆𝑇) → (𝐺𝑇) (𝐺𝑆))

9.2.3  The dual of an ordered set

Syntaxcodu 17175 Class function defining dual orders.
class ODual

Definitiondf-odu 17176 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 17180, oduleval 17178, and oduleg 17179 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 17235. (Contributed by Stefan O'Rear, 29-Jan-2015.)

ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), (le‘𝑤)⟩))

Theoremoduval 17177 Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (le‘𝑂)       𝐷 = (𝑂 sSet ⟨(le‘ndx), ⟩)

Theoremoduleval 17178 Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (le‘𝑂)        = (le‘𝐷)

Theoremoduleg 17179 Truth of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (le‘𝑂)    &   𝐺 = (le‘𝐷)       ((𝐴𝑉𝐵𝑊) → (𝐴𝐺𝐵𝐵 𝐴))

Theoremodubas 17180 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &   𝐵 = (Base‘𝑂)       𝐵 = (Base‘𝐷)

Theorempospropd 17181* Posethood is determined only by structure components and only by the value of the relation within the base set. (Contributed by Stefan O'Rear, 29-Jan-2015.)
(𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(le‘𝐾)𝑦𝑥(le‘𝐿)𝑦))       (𝜑 → (𝐾 ∈ Poset ↔ 𝐿 ∈ Poset))

Theoremodupos 17182 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂 ∈ Poset → 𝐷 ∈ Poset)

Theoremoduposb 17183 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset))

Theoremmeet0 17184 Lemma for odujoin 17189. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 6651 update): This proof increased from 152 bytes to 547 bytes after the df-riota 6651 change. Any way to shorten it? join0 17185 also.
(meet‘∅) = ∅

Theoremjoin0 17185 Lemma for odumeet 17187. (Contributed by Stefan O'Rear, 29-Jan-2015.)
(join‘∅) = ∅

Theoremoduglb 17186 Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &   𝑈 = (lub‘𝑂)       (𝑂𝑉𝑈 = (glb‘𝐷))

Theoremodumeet 17187 Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (join‘𝑂)        = (meet‘𝐷)

Theoremodulub 17188 Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &   𝐿 = (glb‘𝑂)       (𝑂𝑉𝐿 = (lub‘𝐷))

Theoremodujoin 17189 Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (meet‘𝑂)        = (join‘𝐷)

Theoremodulatb 17190 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂𝑉 → (𝑂 ∈ Lat ↔ 𝐷 ∈ Lat))

Theoremoduclatb 17191 Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)

Theoremodulat 17192 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂 ∈ Lat → 𝐷 ∈ Lat)

Theoremposlubmo 17193* Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.)
= (le‘𝐾)    &   𝐵 = (Base‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))

Theoremposglbmo 17194* Greatest lower bounds in a poset are unique if they exist. (Contributed by NM, 20-Sep-2018.)
= (le‘𝐾)    &   𝐵 = (Base‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))

Theoremposlubd 17195* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
= (le‘𝐾)    &   𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝐵)    &   ((𝜑𝑥𝑆) → 𝑥 𝑇)    &   ((𝜑𝑦𝐵 ∧ ∀𝑥𝑆 𝑥 𝑦) → 𝑇 𝑦)       (𝜑 → (𝑈𝑆) = 𝑇)

Theoremposlubdg 17196* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
= (le‘𝐾)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝑈 = (lub‘𝐾))    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝐵)    &   ((𝜑𝑥𝑆) → 𝑥 𝑇)    &   ((𝜑𝑦𝐵 ∧ ∀𝑥𝑆 𝑥 𝑦) → 𝑇 𝑦)       (𝜑 → (𝑈𝑆) = 𝑇)

Theoremposglbd 17197* Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
= (le‘𝐾)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐺 = (glb‘𝐾))    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝐵)    &   ((𝜑𝑥𝑆) → 𝑇 𝑥)    &   ((𝜑𝑦𝐵 ∧ ∀𝑥𝑆 𝑦 𝑥) → 𝑦 𝑇)       (𝜑 → (𝐺𝑆) = 𝑇)

9.2.4  Subset order structures

Syntaxcipo 17198 Class function defining inclusion posets.
class toInc

Definitiondf-ipo 17199* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 17202, ipolerval 17203, and ipole 17205 for its contract.

EDITORIAL: I'm not thrilled with the name. Any suggestions? (Contributed by Stefan O'Rear, 30-Jan-2015.) (New usage is discouraged.)

toInc = (𝑓 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} / 𝑜({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩}))

Theoremipostr 17200 The structure of df-ipo 17199 is a structure defining indexes up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
({⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), ⟩}) Struct ⟨1, 11⟩

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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 42801-42879
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