Home | Metamath
Proof Explorer Theorem List (p. 176 of 424) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-27159) |
Hilbert Space Explorer
(27160-28684) |
Users' Mathboxes
(28685-42360) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | resghm2b 17501 | Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈))) | ||
Theorem | ghmghmrn 17502 | A group homomorphism from 𝐺 to 𝐻 is also a group homomorphism from 𝐺 to its image in 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by AV, 26-Aug-2021.) |
⊢ 𝑈 = (𝑇 ↾s ran 𝐹) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑈)) | ||
Theorem | ghmco 17503 | The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈)) | ||
Theorem | ghmima 17504 | The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇)) | ||
Theorem | ghmpreima 17505 | The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆)) | ||
Theorem | ghmeql 17506 | The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆)) | ||
Theorem | ghmnsgima 17507 | The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝑌 = (Base‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇)) | ||
Theorem | ghmnsgpreima 17508 | The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆)) | ||
Theorem | ghmker 17509 | The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 0 = (0g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝑆)) | ||
Theorem | ghmeqker 17510 | Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ − = (-g‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ (𝑈 − 𝑉) ∈ 𝐾)) | ||
Theorem | pwsdiagghm 17511* | Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌)) | ||
Theorem | ghmf1 17512* | Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ 𝑌 = (Base‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝑈 = (0g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1→𝑌 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ))) | ||
Theorem | ghmf1o 17513 | A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ 𝑌 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆))) | ||
Theorem | conjghm 17514* | Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋–1-1-onto→𝑋)) | ||
Theorem | conjsubg 17515* | A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺)) | ||
Theorem | conjsubgen 17516* | A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹) | ||
Theorem | conjnmz 17517* | A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) & ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹) | ||
Theorem | conjnmzb 17518* | Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) & ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹))) | ||
Theorem | conjnsg 17519* | A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 = ran 𝐹) | ||
Theorem | qusghm 17520* | If 𝑌 is a normal subgroup of 𝐺, then the "natural map" from elements to their cosets is a group homomorphism from 𝐺 to 𝐺 / 𝑌. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) ⇒ ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | ||
Theorem | ghmpropd 17521* | Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) ⇒ ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀)) | ||
Syntax | cgim 17522 | The class of group isomorphism sets. |
class GrpIso | ||
Syntax | cgic 17523 | The class of the group isomorphism relation. |
class ≃𝑔 | ||
Definition | df-gim 17524* | An isomorphism of groups is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group operation. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) | ||
Definition | df-gic 17525 | Two groups are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic groups share all global group properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1𝑜)) | ||
Theorem | gimfn 17526 | The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ GrpIso Fn (Grp × Grp) | ||
Theorem | isgim 17527 | An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) | ||
Theorem | gimf1o 17528 | An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) | ||
Theorem | gimghm 17529 | An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | ||
Theorem | isgim2 17530 | A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 21372. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) | ||
Theorem | subggim 17531 | Behavior of subgroups under isomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝐹 “ 𝐴) ∈ (SubGrp‘𝑆))) | ||
Theorem | gimcnv 17532 | The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) → ◡𝐹 ∈ (𝑇 GrpIso 𝑆)) | ||
Theorem | gimco 17533 | The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpIso 𝑈)) | ||
Theorem | brgic 17534 | The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | ||
Theorem | brgici 17535 | Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ≃𝑔 𝑆) | ||
Theorem | gicref 17536 | Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅) | ||
Theorem | giclcl 17537 | Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑔 𝑆 → 𝑅 ∈ Grp) | ||
Theorem | gicrcl 17538 | Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp) | ||
Theorem | gicsym 17539 | Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ≃𝑔 𝑅) | ||
Theorem | gictr 17540 | Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇) | ||
Theorem | gicer 17541 | Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.) |
⊢ ≃𝑔 Er Grp | ||
Theorem | gicerOLD 17542 | Obsolete proof of gicer 17541 as of 1-May-2021. Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ≃𝑔 Er Grp | ||
Theorem | gicen 17543 | Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝑅 ≃𝑔 𝑆 → 𝐵 ≈ 𝐶) | ||
Theorem | gicsubgen 17544 | A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆)) | ||
Syntax | cga 17545 | Extend class definition to include the class of group actions. |
class GrpAct | ||
Definition | df-ga 17546* | Define the class of all group actions. A group 𝐺 acts on a set 𝑆 if a permutation on 𝑆 is associated with every element of 𝐺 in such a way that the identity permutation on 𝑆 is associated with the neutral element of 𝐺, and the composition of the permutations associated with two elements of 𝐺 is identical with the permutation associated with the composition of these two elements (in the same order) in the group 𝐺. (Contributed by Jeff Hankins, 10-Aug-2009.) |
⊢ GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ ⦋(Base‘𝑔) / 𝑏⦌{𝑚 ∈ (𝑠 ↑𝑚 (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))}) | ||
Theorem | isga 17547* | The predicate "is a (left) group action." The group 𝐺 is said to act on the base set 𝑌 of the action, which is not assumed to have any special properties. There is a related notion of right group action, but as the Wikipedia article explains, it is not mathematically interesting. The way actions are usually thought of is that each element 𝑔 of 𝐺 is a permutation of the elements of 𝑌 (see gapm 17562). Since group theory was classically about symmetry groups, it is therefore likely that the notion of group action was useful even in early group theory. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))) | ||
Theorem | gagrp 17548 | The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp) | ||
Theorem | gaset 17549 | The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V) | ||
Theorem | gagrpid 17550 | The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴) | ||
Theorem | gaf 17551 | The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌) | ||
Theorem | gafo 17552 | A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)–onto→𝑌) | ||
Theorem | gaass 17553 | An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))) | ||
Theorem | ga0 17554 | The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅)) | ||
Theorem | gaid 17555 | The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆)) | ||
Theorem | subgga 17556* | A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑌) & ⊢ 𝐹 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) ⇒ ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋)) | ||
Theorem | gass 17557* | A subset of a group action is a group action iff it is closed under the group action operation. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) → (( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍)) | ||
Theorem | gasubg 17558 | The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌)) | ||
Theorem | gaid2 17559* | A group operation is a left group action of the group on itself. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) ⇒ ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpAct 𝑋)) | ||
Theorem | galcan 17560 | The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | gacan 17561 | Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵)) | ||
Theorem | gapm 17562* | The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝑌 ↦ (𝐴 ⊕ 𝑥)) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑌–1-1-onto→𝑌) | ||
Theorem | gaorb 17563* | The orbit equivalence relation puts two points in the group action in the same equivalence class iff there is a group element that takes one element to the other. (Contributed by Mario Carneiro, 14-Jan-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)) | ||
Theorem | gaorber 17564* | The orbit equivalence relation is an equivalence relation on the target set of the group action. (Contributed by NM, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌) | ||
Theorem | gastacl 17565* | The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺)) | ||
Theorem | gastacos 17566* | Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} & ⊢ ∼ = (𝐺 ~QG 𝐻) ⇒ ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∼ 𝐶 ↔ (𝐵 ⊕ 𝐴) = (𝐶 ⊕ 𝐴))) | ||
Theorem | orbstafun 17567* | Existence and uniqueness for the function of orbsta 17569. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} & ⊢ ∼ = (𝐺 ~QG 𝐻) & ⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉) ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → Fun 𝐹) | ||
Theorem | orbstaval 17568* | Value of the function at a given equivalence class element. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} & ⊢ ∼ = (𝐺 ~QG 𝐻) & ⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉) ⇒ ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝐵 ∈ 𝑋) → (𝐹‘[𝐵] ∼ ) = (𝐵 ⊕ 𝐴)) | ||
Theorem | orbsta 17569* | The Orbit-Stabilizer theorem. The mapping 𝐹 is a bijection from the cosets of the stabilizer subgroup of 𝐴 to the orbit of 𝐴. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} & ⊢ ∼ = (𝐺 ~QG 𝐻) & ⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉) & ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂) | ||
Theorem | orbsta2 17570* | Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} & ⊢ ∼ = (𝐺 ~QG 𝐻) & ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝐴]𝑂) · (#‘𝐻))) | ||
Syntax | ccntz 17571 | Syntax for the centralizer of a set in a monoid. |
class Cntz | ||
Syntax | ccntr 17572 | Syntax for the centralizer of a monoid. |
class Cntr | ||
Definition | df-cntz 17573* | Define the centralizer of a subset of a magma, which is the set of elements each of which commutes with each element of the given subset. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)})) | ||
Definition | df-cntr 17574 | Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚))) | ||
Theorem | cntrval 17575 | Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑍‘𝐵) = (Cntr‘𝑀) | ||
Theorem | cntzfval 17576* | First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})) | ||
Theorem | cntzval 17577* | Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) | ||
Theorem | elcntz 17578* | Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))) | ||
Theorem | cntzel 17579* | Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))) | ||
Theorem | cntzsnval 17580* | Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)}) | ||
Theorem | elcntzsn 17581 | Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋)))) | ||
Theorem | sscntz 17582* | A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))) | ||
Theorem | cntzrcl 17583 | Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) | ||
Theorem | cntzssv 17584 | The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑍‘𝑆) ⊆ 𝐵 | ||
Theorem | cntzi 17585 | Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | cntri 17586 | Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑍 = (Cntr‘𝑀) ⇒ ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | resscntz 17587 | Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 𝑌 = (Cntz‘𝐻) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ⊆ 𝐴) → (𝑌‘𝑆) = ((𝑍‘𝑆) ∩ 𝐴)) | ||
Theorem | cntz2ss 17588 | Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇)) | ||
Theorem | cntzrec 17589 | Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆))) | ||
Theorem | cntziinsn 17590* | Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥}))) | ||
Theorem | cntzsubm 17591 | Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀)) | ||
Theorem | cntzsubg 17592 | Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀)) | ||
Theorem | cntzidss 17593 | If the elements of 𝑆 commute, the elements of a subset 𝑇 also commute. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇)) | ||
Theorem | cntzmhm 17594 | Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 𝑌 = (Cntz‘𝐻) ⇒ ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆))) | ||
Theorem | cntzmhm2 17595 | Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 𝑌 = (Cntz‘𝐻) ⇒ ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇))) | ||
Theorem | cntrsubgnsg 17596 | A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝑍 = (Cntr‘𝑀) ⇒ ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀)) | ||
Theorem | cntrnsg 17597 | The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝑍 = (Cntr‘𝑀) ⇒ ⊢ (𝑀 ∈ Grp → 𝑍 ∈ (NrmSGrp‘𝑀)) | ||
Syntax | coppg 17598 | The opposite group operation. |
class oppg | ||
Definition | df-oppg 17599 | Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 18446 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
⊢ oppg = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx), tpos (+g‘𝑤)〉)) | ||
Theorem | oppgval 17600 | Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
⊢ + = (+g‘𝑅) & ⊢ 𝑂 = (oppg‘𝑅) ⇒ ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |