20 lines
724 B
Text
20 lines
724 B
Text
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import algebra.group
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open algebra
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definition is_homomorphism [class] {G₁ G₂ : Type} [group G₁] [group G₂] (φ : G₁ → G₂) : Prop :=
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Π(g h : G₁), φ (g * h) = φ g * φ h
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-- set_option pp.all true
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definition apply_iso {G₁ G₂ : Type} [group G₁] [group G₂] {φ : G₁ → G₂} [H : is_homomorphism φ] (g h : G₁) : φ (g * h) = φ g * φ h :=
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H g h
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variables {G₁ G₂ : Type} (φ : G₁ → G₂) [group G₁] [group G₂] [is_homomorphism φ]
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theorem respect_one : φ 1 = 1 :=
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assert φ (1 * 1) = φ 1 * φ 1, from apply_iso 1 1,
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assert φ 1 * 1 = φ 1 * φ 1, by rewrite one_mul at this; rewrite mul_one; exact this,
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assert 1 = φ 1, from mul_left_cancel this,
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eq.symm this
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