P. 193 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ghmpropd 19201*	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))
10.2.6  Isomorphisms of groups
Syntax	cgim 19202	The class of group isomorphism sets.
class GrpIso
Syntax	cgic 19203	The class of the group isomorphism relation.
class ≃𝑔
Definition	df-gim 19204*	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 19205	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 19206	The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ GrpIso Fn (Grp × Grp)
Theorem	isgim 19207	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 19208	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 19209	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 19210	A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 23650. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 GrpHom 𝑅)))
Theorem	subggim 19211	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 19212	The converse of a group isomorphism is a group isomorphism. (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 19213	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	gim0to0 19214	A group isomorphism maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 23-May-2023.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 ))
Theorem	brgic 19215	The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
Theorem	brgici 19216	Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ≃𝑔 𝑆)
Theorem	gicref 19217	Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅)
Theorem	giclcl 19218	Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑅 ∈ Grp)
Theorem	gicrcl 19219	Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp)
Theorem	gicsym 19220	Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ≃𝑔 𝑅)
Theorem	gictr 19221	Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇)
Theorem	gicer 19222	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	gicen 19223	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 19224	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))
10.2.6.1  The first isomorphism theorem of groups
Theorem	ghmquskerlem1 19225*	Lemma for ghmqusker 19229. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐽‘[𝑋](𝐺 ~QG 𝐾)) = (𝐹‘𝑋))
Theorem	ghmquskerco 19226*	In the case of theorem ghmqusker 19229, the composition of the natural homomorphism 𝐿 with the constructed homomorphism 𝐽 equals the original homomorphism 𝐹. One says that 𝐹 factors through 𝑄. (Proposed by Saveliy Skresanov, 15-Feb-2025.) (Contributed by Thierry Arnoux, 15-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝐾))    ⇒   ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))
Theorem	ghmquskerlem2 19227*	Lemma for ghmqusker 19229. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))
Theorem	ghmquskerlem3 19228*	The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
Theorem	ghmqusker 19229*	A surjective group homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 15-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))
Theorem	gicqusker 19230	The image 𝐻 of a group homomorphism 𝐹 is isomorphic with the quotient group 𝑄 over 𝐹's kernel 𝐾. Together with ghmker 19187 and ghmima 19182, this is sometimes called the first isomorphism theorem for groups. (Contributed by Thierry Arnoux, 10-Mar-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    ⇒   ⊢ (𝜑 → 𝑄 ≃𝑔 𝐻)
10.2.7  Group actions
Syntax	cga 19231	Extend class definition to include the class of group actions.
class GrpAct
Definition	df-ga 19232*	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 19233*	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 19248). 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 19234	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 19235	The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
Theorem	gagrpid 19236	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 19237	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 19238	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 19239	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 19240	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 19241	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 19242*	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 19243*	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 19244	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 19245*	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 19246	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 19247	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 19248*	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 19249*	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 19250*	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 19251*	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 19252*	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 19253*	Existence and uniqueness for the function of orbsta 19255. (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 19254*	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 19255*	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 19256*	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)))
10.2.8  Centralizers and centers
Syntax	ccntz 19257	Syntax for the centralizer of a set in a monoid.
class Cntz
Syntax	ccntr 19258	Syntax for the centralizer of a monoid.
class Cntr
Definition	df-cntz 19259*	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 19260	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 19261	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 19262*	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 19263*	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 19264*	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 19265*	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 19266*	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 19267	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 19268*	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 19269	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 19270	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 19271	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	elcntr 19272*	Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
Theorem	cntrss 19273	The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (Cntr‘𝑀) ⊆ 𝐵
Theorem	cntri 19274	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 19275	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	cntzsgrpcl 19276*	Centralizers are closed under the semigroup operation. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    &   ⊢ 𝐶 = (𝑍‘𝑆)    ⇒   ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
Theorem	cntz2ss 19277	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 19278	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 19279*	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 19280	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 19281	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 19282	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 19283	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 19284	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 19285	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 19286	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))
10.2.9  The opposite group
Syntax	coppg 19287	The opposite group operation.
class oppg
Definition	df-oppg 19288	Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 20262 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 19289	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 + ⟩)
Theorem	oppgplusfval 19290	Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ✚ = (+g‘𝑂)    ⇒   ⊢ ✚ = tpos +
Theorem	oppgplus 19291	Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ✚ = (+g‘𝑂)    ⇒   ⊢ (𝑋 ✚ 𝑌) = (𝑌 + 𝑋)
Theorem	setsplusg 19292	The other components of an extensible structure remain unchanged if the +g component is set/substituted. (Contributed by Stefan O'Rear, 26-Aug-2015.) Generalisation of the former oppglem and mgplem. (Revised by AV, 18-Oct-2024.)
⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), 𝑆⟩)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (+g‘ndx)    ⇒   ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)
Theorem	oppglemOLD 19293	Obsolete version of setsplusg 19292 as of 18-Oct-2024. Lemma for oppgbas 19294. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝑁 ≠ 2    ⇒   ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)
Theorem	oppgbas 19294	Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝐵 = (Base‘𝑂)
Theorem	oppgbasOLD 19295	Obsolete version of oppgbas 19294 as of 18-Oct-2024. Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝐵 = (Base‘𝑂)
Theorem	oppgtset 19296	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopSet‘𝑅)    ⇒   ⊢ 𝐽 = (TopSet‘𝑂)
Theorem	oppgtsetOLD 19297	Obsolete version of oppgtset 19296 as of 18-Oct-2024. Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopSet‘𝑅)    ⇒   ⊢ 𝐽 = (TopSet‘𝑂)
Theorem	oppgtopn 19298	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ 𝐽 = (TopOpen‘𝑂)
Theorem	oppgmnd 19299	The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd)
Theorem	oppgmndb 19300	Bidirectional form of oppgmnd 19299. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)