[Merged by Bors] - feat(group_theory): subgroups of real numbers #4334
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Co-authored-by: Rob Lewis <Rob.y.lewis@gmail.com>
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| @@ -889,6 +938,9 @@ begin | |||
| refine ⟨-n, neg_gsmul x n⟩ } | |||
| end | |||
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| lemma closure_singleton_zero : closure ({0} : set A) = ⊥ := | |||
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I guess it can't be deduced from closure_singleton_one through to_additive? Even if it can't, it's good to register (after the fact) a to_additive attribute so that later proofs can be converted automatically.
| @@ -226,6 +229,9 @@ lemma mem_closure_singleton {x y : A} : | |||
| y ∈ closure ({x} : set A) ↔ ∃ n:ℕ, n •ℕ x = y := | |||
| by rw [closure_singleton_eq, add_monoid_hom.mem_mrange]; refl | |||
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| lemma closure_singleton_zero : closure ({0} : set A) = ⊥ := | |||
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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bors r+ |
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This fills the last hole in the "Topology of R" section of our undergrad curriculum: additive subgroups of real numbers are either dense or cyclic. Most of this PR is supporting material about ordered abelian groups. Co-Authored-By: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: hrmacbeth <25316162+hrmacbeth@users.noreply.github.com>
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This fills the last hole in the "Topology of R" section of our undergrad curriculum: additive subgroups of real numbers are either dense or cyclic. Most of this PR is supporting material about ordered abelian groups. Co-Authored-By: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: hrmacbeth <25316162+hrmacbeth@users.noreply.github.com>
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This fills the last hole in the "Topology of R" section of our undergrad curriculum: additive subgroups of real numbers are either dense or cyclic. Most of this PR is supporting material about ordered abelian groups.
Co-Authored-By: Heather Macbeth 25316162+hrmacbeth@users.noreply.github.com