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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | grpsubfval 19001* | Group subtraction (division) operation. For a shorter proof using ax-rep 5279, see grpsubfvalALT 19002. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5279. (Revised by Rohan Ridenour, 17-Aug-2023.) (Proof shortened by AV, 19-Feb-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) | ||
| Theorem | grpsubfvalALT 19002* | Shorter proof of grpsubfval 19001 using ax-rep 5279. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) | ||
| Theorem | grpsubval 19003 | Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + (𝐼‘𝑌))) | ||
| Theorem | grpinvf 19004 | The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) | ||
| Theorem | grpinvcl 19005 | A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) | ||
| Theorem | grpinvcld 19006 | A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) | ||
| Theorem | grplinv 19007 | The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) | ||
| Theorem | grprinv 19008 | The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) | ||
| Theorem | grpinvid1 19009 | The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) | ||
| Theorem | grpinvid2 19010 | The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 )) | ||
| Theorem | isgrpinv 19011* | Properties showing that a function 𝑀 is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀)) | ||
| Theorem | grplinvd 19012 | The left inverse of a group element. Deduction associated with grplinv 19007. (Contributed by SN, 29-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 ) | ||
| Theorem | grprinvd 19013 | The right inverse of a group element. Deduction associated with grprinv 19008. (Contributed by SN, 29-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 ) | ||
| Theorem | grplrinv 19014* | In a group, every member has a left and right inverse. (Contributed by AV, 1-Sep-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 )) | ||
| Theorem | grpidinv2 19015* | A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by AV, 1-Sep-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))) | ||
| Theorem | grpidinv 19016* | A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (Revised by AV, 1-Sep-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢))) | ||
| Theorem | grpinvid 19017 | The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.) |
| ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) | ||
| Theorem | grplcan 19018 | Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌)) | ||
| Theorem | grpasscan1 19019 | An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌) | ||
| Theorem | grpasscan2 19020 | An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) | ||
| Theorem | grpidrcan 19021 | If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 + 𝑍) = 𝑋 ↔ 𝑍 = 0 )) | ||
| Theorem | grpidlcan 19022 | If left adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = 𝑋 ↔ 𝑍 = 0 )) | ||
| Theorem | grpinvinv 19023 | Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) | ||
| Theorem | grpinvcnv 19024 | The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) | ||
| Theorem | grpinv11 19025 | The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.) (Proof shortened by SN, 8-Jul-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) | ||
| Theorem | grpinv11OLD 19026 | Obsolete version of grpinv11 19025 as of 8-Jul-2025. (Contributed by NM, 22-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) | ||
| Theorem | grpinvf1o 19027 | The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵) | ||
| Theorem | grpinvnz 19028 | The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 ) | ||
| Theorem | grpinvnzcl 19029 | The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁‘𝑋) ∈ (𝐵 ∖ { 0 })) | ||
| Theorem | grpsubinv 19030 | Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌)) | ||
| Theorem | grplmulf1o 19031* | Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑋 + 𝑥)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵) | ||
| Theorem | grpraddf1o 19032* | Right addition by a group element is a bijection on any group. (Contributed by SN, 28-Apr-2012.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 + 𝑋)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵) | ||
| Theorem | grpinvpropd 19033* | If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿)) | ||
| Theorem | grpidssd 19034* | If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then both groups have the same identity element. (Contributed by AV, 15-Mar-2019.) |
| ⊢ (𝜑 → 𝑀 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆)) | ||
| Theorem | grpinvssd 19035* | If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.) |
| ⊢ (𝜑 → 𝑀 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋))) | ||
| Theorem | grpinvadd 19036 | The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑁‘𝑋))) | ||
| Theorem | grpsubf 19037 | Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → − :(𝐵 × 𝐵)⟶𝐵) | ||
| Theorem | grpsubcl 19038 | Closure of group subtraction. (Contributed by NM, 31-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) | ||
| Theorem | grpsubrcan 19039 | Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑍) = (𝑌 − 𝑍) ↔ 𝑋 = 𝑌)) | ||
| Theorem | grpinvsub 19040 | Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) | ||
| Theorem | grpinvval2 19041 | A df-neg 11495-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) | ||
| Theorem | grpsubid 19042 | Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = 0 ) | ||
| Theorem | grpsubid1 19043 | Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) | ||
| Theorem | grpsubeq0 19044 | If the difference between two group elements is zero, they are equal. (subeq0 11535 analog.) (Contributed by NM, 31-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) | ||
| Theorem | grpsubadd0sub 19045 | Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ( 0 − 𝑌))) | ||
| Theorem | grpsubadd 19046 | Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋)) | ||
| Theorem | grpsubsub 19047 | Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + (𝑍 − 𝑌))) | ||
| Theorem | grpaddsubass 19048 | Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = (𝑋 + (𝑌 − 𝑍))) | ||
| Theorem | grppncan 19049 | Cancellation law for subtraction (pncan 11514 analog). (Contributed by NM, 16-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑌) = 𝑋) | ||
| Theorem | grpnpcan 19050 | Cancellation law for subtraction (npcan 11517 analog). (Contributed by NM, 19-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) + 𝑌) = 𝑋) | ||
| Theorem | grpsubsub4 19051 | Double group subtraction (subsub4 11542 analog). (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑍 + 𝑌))) | ||
| Theorem | grppnpcan2 19052 | Cancellation law for mixed addition and subtraction. (pnpcan2 11549 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) − (𝑌 + 𝑍)) = (𝑋 − 𝑌)) | ||
| Theorem | grpnpncan 19053 | Cancellation law for group subtraction. (npncan 11530 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) + (𝑌 − 𝑍)) = (𝑋 − 𝑍)) | ||
| Theorem | grpnpncan0 19054 | Cancellation law for group subtraction (npncan2 11536 analog). (Contributed by AV, 24-Nov-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 − 𝑌) + (𝑌 − 𝑋)) = 0 ) | ||
| Theorem | grpnnncan2 19055 | Cancellation law for group subtraction. (nnncan2 11546 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑍) − (𝑌 − 𝑍)) = (𝑋 − 𝑌)) | ||
| Theorem | dfgrp3lem 19056* | Lemma for dfgrp3 19057. (Contributed by AV, 28-Aug-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢)) | ||
| Theorem | dfgrp3 19057* | Alternate definition of a group as semigroup (with at least one element) which is also a quasigroup, i.e. a magma in which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Theorem 3.2 of [Bruck] p. 28. (Contributed by AV, 28-Aug-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))) | ||
| Theorem | dfgrp3e 19058* | Alternate definition of a group as a set with a closed, associative operation, for which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by AV, 28-Aug-2021.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))) | ||
| Theorem | grplactfval 19059* | The left group action of element 𝐴 of group 𝐺. (Contributed by Paul Chapman, 18-Mar-2008.) |
| ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))) | ||
| Theorem | grplactval 19060* | The value of the left group action of element 𝐴 of group 𝐺 at 𝐵. (Contributed by Paul Chapman, 18-Mar-2008.) |
| ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴)‘𝐵) = (𝐴 + 𝐵)) | ||
| Theorem | grplactcnv 19061* | The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴)))) | ||
| Theorem | grplactf1o 19062* | The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴):𝑋–1-1-onto→𝑋) | ||
| Theorem | grpsubpropd 19063 | Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.) |
| ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) ⇒ ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) | ||
| Theorem | grpsubpropd2 19064* | Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐻)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) ⇒ ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) | ||
| Theorem | grp1 19065 | The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.) |
| ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) | ||
| Theorem | grp1inv 19066 | The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Revised by AV, 26-Aug-2021.) |
| ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀) = ( I ↾ {𝐼})) | ||
| Theorem | prdsinvlem 19067* | Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 0 = (0g ∘ 𝑅) & ⊢ 𝑁 = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝐹‘𝑦))) ⇒ ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ (𝑁 + 𝐹) = 0 )) | ||
| Theorem | prdsgrpd 19068 | The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) ⇒ ⊢ (𝜑 → 𝑌 ∈ Grp) | ||
| Theorem | prdsinvgd 19069* | Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑁 = (invg‘𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))) | ||
| Theorem | pwsgrp 19070 | A structure power of a group is a group. (Contributed by Mario Carneiro, 11-Jan-2015.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Grp) | ||
| Theorem | pwsinvg 19071 | Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (invg‘𝑅) & ⊢ 𝑁 = (invg‘𝑌) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑀 ∘ 𝑋)) | ||
| Theorem | pwssub 19072 | Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (-g‘𝑅) & ⊢ − = (-g‘𝑌) ⇒ ⊢ (((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝐹 − 𝐺) = (𝐹 ∘f 𝑀𝐺)) | ||
| Theorem | imasgrp2 19073* | The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧)))) & ⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘ 0 )) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | ||
| Theorem | imasgrp 19074* | The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | ||
| Theorem | imasgrpf1 19075 | The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ 𝑈 = (𝐹 “s 𝑅) & ⊢ 𝑉 = (Base‘𝑅) ⇒ ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑈 ∈ Grp) | ||
| Theorem | qusgrp2 19076* | Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) ∼ (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑁 + 𝑥) ∼ 0 ) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0g‘𝑈))) | ||
| Theorem | xpsgrp 19077 | The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ 𝑇 = (𝑅 ×s 𝑆) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) | ||
| Theorem | xpsinv 19078 | Value of the negation operation in a binary structure product. (Contributed by AV, 18-Mar-2025.) |
| ⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ 𝑀 = (invg‘𝑅) & ⊢ 𝑁 = (invg‘𝑆) & ⊢ 𝐼 = (invg‘𝑇) ⇒ ⊢ (𝜑 → (𝐼‘〈𝐴, 𝐵〉) = 〈(𝑀‘𝐴), (𝑁‘𝐵)〉) | ||
| Theorem | xpsgrpsub 19079 | Value of the subtraction operation in a binary structure product of groups. (Contributed by AV, 24-Mar-2025.) |
| ⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ · = (-g‘𝑅) & ⊢ × = (-g‘𝑆) & ⊢ − = (-g‘𝑇) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 − 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | ||
| Theorem | mhmlem 19080* | Lemma for mhmmnd 19082 and ghmgrp 19084. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))) | ||
| Theorem | mhmid 19081* | A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻)) | ||
| Theorem | mhmmnd 19082* | The image of a monoid 𝐺 under a monoid homomorphism 𝐹 is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐻 ∈ Mnd) | ||
| Theorem | mhmfmhm 19083* | The function fulfilling the conditions of mhmmnd 19082 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) | ||
| Theorem | ghmgrp 19084* | The image of a group 𝐺 under a group homomorphism 𝐹 is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator 𝑂 in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐻 ∈ Grp) | ||
The "group multiple" operation (if the group is multiplicative, also called "group power" or "group exponentiation" operation), can be defined for arbitrary magmas, if the multiplier/exponent is a nonnegative integer. See also the definition in [Lang] p. 6, where an element 𝑥(of a monoid) to the power of a nonnegative integer 𝑛 is defined and denoted by 𝑥↑𝑛. Definition df-mulg 19086, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires Definition df-minusg 18955 of the inverse operation invg. | ||
| Syntax | cmg 19085 | Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group. |
| class .g | ||
| Definition | df-mulg 19086* | Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| ⊢ .g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g‘𝑔), ⦋seq1((+g‘𝑔), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑔)‘(𝑠‘-𝑛)))))) | ||
| Theorem | mulgfval 19087* | Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5279, see mulgfvalALT 19088. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5279. (Revised by Rohan Ridenour, 17-Aug-2023.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) | ||
| Theorem | mulgfvalALT 19088* | Shorter proof of mulgfval 19087 using ax-rep 5279. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) | ||
| Theorem | mulgval 19089 | Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) | ||
| Theorem | mulgfn 19090 | Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · Fn (ℤ × 𝐵) | ||
| Theorem | mulgfvi 19091 | The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.) |
| ⊢ · = (.g‘𝐺) ⇒ ⊢ · = (.g‘( I ‘𝐺)) | ||
| Theorem | mulg0 19092 | Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) | ||
| Theorem | mulgnn 19093 | Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁)) | ||
| Theorem | ressmulgnn 19094 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝐴 ⊆ (Base‘𝐺) & ⊢ ∗ = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) | ||
| Theorem | ressmulgnn0 19095 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝐴 ⊆ (Base‘𝐺) & ⊢ ∗ = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ (0g‘𝐺) = (0g‘𝐻) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) | ||
| Theorem | ressmulgnnd 19096 | Values for the group multiple function in a restricted structure, a deduction version. (Contributed by metakunt, 14-May-2025.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋)) | ||
| Theorem | mulgnngsum 19097* | Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) | ||
| Theorem | mulgnn0gsum 19098* | Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) | ||
| Theorem | mulg1 19099 | Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) | ||
| Theorem | mulgnnp1 19100 | Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) | ||
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