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feat: pointwise scalar multiplication is monotone (#11809)
Everywhere we have a `smul_mem_pointwise_smul` lemma, I've added this result.
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/- | ||
Copyright (c) 2024 Eric Wieser. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Eric Wieser | ||
-/ | ||
import Mathlib.Algebra.CovariantAndContravariant | ||
import Mathlib.Order.ConditionallyCompleteLattice.Basic | ||
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/-! | ||
# Results about `CovariantClass G α HSMul.hSMul LE.le` | ||
When working with group actions rather than modules, we drop the `0 < c` condition. | ||
Notably these are relevant for pointwise actions on set-like objects. | ||
-/ | ||
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variable {ι : Sort*} {M α : Type*} | ||
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theorem smul_mono_right [SMul M α] [Preorder α] [CovariantClass M α HSMul.hSMul LE.le] | ||
(m : M) : Monotone (HSMul.hSMul m : α → α) := | ||
fun _ _ => CovariantClass.elim _ | ||
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/-- A copy of `smul_mono_right` that is understood by `gcongr`. -/ | ||
@[gcongr] | ||
theorem smul_le_smul_left [SMul M α] [Preorder α] [CovariantClass M α HSMul.hSMul LE.le] | ||
(m : M) {a b : α} (h : a ≤ b) : | ||
m • a ≤ m • b := | ||
smul_mono_right _ h | ||
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theorem smul_inf_le [SMul M α] [SemilatticeInf α] [CovariantClass M α HSMul.hSMul LE.le] | ||
(m : M) (a₁ a₂ : α) : m • (a₁ ⊓ a₂) ≤ m • a₁ ⊓ m • a₂ := | ||
(smul_mono_right _).map_inf_le _ _ | ||
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theorem smul_iInf_le [SMul M α] [CompleteLattice α] [CovariantClass M α HSMul.hSMul LE.le] | ||
{m : M} {t : ι → α} : | ||
m • iInf t ≤ ⨅ i, m • t i := | ||
le_iInf fun _ => smul_mono_right _ (iInf_le _ _) | ||
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theorem smul_strictMono_right [SMul M α] [Preorder α] [CovariantClass M α HSMul.hSMul LT.lt] | ||
(m : M) : StrictMono (HSMul.hSMul m : α → α) := | ||
fun _ _ => CovariantClass.elim _ |
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