Theorems · Theorem · order theory
csInf_union
∀ {α : Type u_1} [inst : ConditionallyCompleteLattice α] {s t : Set α},
BddBelow s → s.Nonempty → BddBelow t → t.Nonempty → sInf (s ∪ t) = sInf s ⊓ sInf tThe sInf of a union of two sets is the min of the infima of each subset, under the assumptions
that all sets are bounded below and nonempty.
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- 0 results in Mathlib
- Foundations
- Depth 11 from the axioms · uses no axioms
- Assumes
- ConditionallyCompleteLattice
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Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Set.Nonemptystatement and proof · cited by 2,627
- InfSet.sInfstatement · cited by 935
- BddBelowstatement and proof · cited by 401
- ConditionallyCompleteLatticestatement and proof · cited by 364
- isGLB_csInfproof · cited by 24
- IsGLB.csInf_eqproof · cited by 13
- Set.Nonempty.inlproof · cited by 4
- IsGLB.unionproof · cited by 4
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