File: Mathlib/Algebra/Group/Embedding.lean

full_name: addLeftEmbedding
code: /-- If left-multiplication by any element is cancellative, left-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If left-addition by any element is cancellative, left-addition by `g` is an
        embedding."]
def mulLeftEmbedding [Mul G] [IsLeftCancelMul G] (g : G) : G ↪ G 
goal: G ↪ G

full_name: mulLeftEmbedding_apply
code: /-- If left-multiplication by any element is cancellative, left-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If left-addition by any element is cancellative, left-addition by `g` is an
        embedding."]
def mulLeftEmbedding [Mul G] [IsLeftCancelMul G] (g : G) : G ↪ G 
goal: (mulLeftEmbedding g) h = g * h

full_name: mulLeftEmbedding
code: /-- If left-multiplication by any element is cancellative, left-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If left-addition by any element is cancellative, left-addition by `g` is an
        embedding."]
def mulLeftEmbedding [Mul G] [IsLeftCancelMul G] (g : G) : G ↪ G 
goal: G ↪ G

full_name: addLeftEmbedding_apply
code: /-- If left-multiplication by any element is cancellative, left-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If left-addition by any element is cancellative, left-addition by `g` is an
        embedding."]
def mulLeftEmbedding [Mul G] [IsLeftCancelMul G] (g : G) : G ↪ G 
goal: (addLeftEmbedding g) h = g + h

full_name: mulRightEmbedding
code: /-- If right-multiplication by any element is cancellative, right-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If right-addition by any element is cancellative, right-addition by `g` is an
        embedding."]
def mulRightEmbedding [Mul G] [IsRightCancelMul G] (g : G) : G ↪ G 
goal: G ↪ G

full_name: addRightEmbedding_apply
code: /-- If right-multiplication by any element is cancellative, right-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If right-addition by any element is cancellative, right-addition by `g` is an
        embedding."]
def mulRightEmbedding [Mul G] [IsRightCancelMul G] (g : G) : G ↪ G 
goal: (addRightEmbedding g) h = h + g

full_name: mulRightEmbedding_apply
code: /-- If right-multiplication by any element is cancellative, right-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If right-addition by any element is cancellative, right-addition by `g` is an
        embedding."]
def mulRightEmbedding [Mul G] [IsRightCancelMul G] (g : G) : G ↪ G 
goal: (mulRightEmbedding g) h = h * g

full_name: addRightEmbedding
code: /-- If right-multiplication by any element is cancellative, right-multiplication by `g` is an
embedding. -/
@[to_additive (attr := simps)
      "If right-addition by any element is cancellative, right-addition by `g` is an
        embedding."]
def mulRightEmbedding [Mul G] [IsRightCancelMul G] (g : G) : G ↪ G 
goal: G ↪ G

full_name: mulLeftEmbedding_eq_mulRightEmbedding
code: @[to_additive]
theorem mulLeftEmbedding_eq_mulRightEmbedding [CommSemigroup G] [IsCancelMul G] (g : G) :
    mulLeftEmbedding g = mulRightEmbedding g 
goal: mulLeftEmbedding g = mulRightEmbedding g

full_name: addLeftEmbedding_eq_addRightEmbedding
code: @[to_additive]
theorem mulLeftEmbedding_eq_mulRightEmbedding [CommSemigroup G] [IsCancelMul G] (g : G) :
    mulLeftEmbedding g = mulRightEmbedding g 
goal: addLeftEmbedding g = addRightEmbedding g