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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mulgnn0ass 17401 | Product of group multiples, generalized to ℕ0. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
Theorem | mulgass 17402 | Product of group multiples, generalized to ℤ. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
Theorem | mulgassr 17403 | Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑁 · 𝑀) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
Theorem | mulgmodid 17404 | Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ (𝑋 ∈ 𝐵 ∧ (𝑀 · 𝑋) = 0 )) → ((𝑁 mod 𝑀) · 𝑋) = (𝑁 · 𝑋)) | ||
Theorem | mulgsubdir 17405 | Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 − 𝑁) · 𝑋) = ((𝑀 · 𝑋) − (𝑁 · 𝑋))) | ||
Theorem | mhmmulg 17406 | A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ × = (.g‘𝐻) ⇒ ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋))) | ||
Theorem | mulgpropd 17407* | Two structures with the same group-nature have the same group multiple function. 𝐾 is expected to either be V (when strong equality is available) or 𝐵 (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ · = (.g‘𝐺) & ⊢ × = (.g‘𝐻) & ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐻)) & ⊢ (𝜑 → 𝐵 ⊆ 𝐾) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) ⇒ ⊢ (𝜑 → · = × ) | ||
Theorem | submmulgcl 17408 | Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ ∙ = (.g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) ∈ 𝑆) | ||
Theorem | submmulg 17409 | A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ ∙ = (.g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ · = (.g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) = (𝑁 · 𝑋)) | ||
Theorem | pwsmulg 17410 | Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ ∙ = (.g‘𝑌) & ⊢ · = (.g‘𝑅) ⇒ ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴))) | ||
Syntax | csubg 17411 | Extend class notation with all subgroups of a group. |
class SubGrp | ||
Syntax | cnsg 17412 | Extend class notation with all normal subgroups of a group. |
class NrmSGrp | ||
Syntax | cqg 17413 | Quotient group equivalence class. |
class ~QG | ||
Definition | df-subg 17414* | Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17432), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17427), contains the neutral element of the group (see subg0 17423) and contains the inverses for all of its elements (see subginvcl 17426). (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) | ||
Definition | df-nsg 17415* | Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)}) | ||
Definition | df-eqg 17416* | Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)}) | ||
Theorem | issubg 17417 | The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)) | ||
Theorem | subgss 17418 | A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵) | ||
Theorem | subgid 17419 | A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) | ||
Theorem | subggrp 17420 | A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) | ||
Theorem | subgbas 17421 | The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) | ||
Theorem | subgrcl 17422 | Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | ||
Theorem | subg0 17423 | A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘𝐻)) | ||
Theorem | subginv 17424 | The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ 𝐽 = (invg‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋)) | ||
Theorem | subg0cl 17425 | The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆) | ||
Theorem | subginvcl 17426 | The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) ∈ 𝑆) | ||
Theorem | subgcl 17427 | A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
Theorem | subgsubcl 17428 | A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) ∈ 𝑆) | ||
Theorem | subgsub 17429 | The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ − = (-g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ 𝑁 = (-g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋𝑁𝑌)) | ||
Theorem | subgmulgcl 17430 | Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | subgmulg 17431 | A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ · = (.g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ ∙ = (.g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋)) | ||
Theorem | issubg2 17432* | Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆)))) | ||
Theorem | issubgrpd2 17433* | Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) & ⊢ (𝜑 → 0 = (0g‘𝐼)) & ⊢ (𝜑 → + = (+g‘𝐼)) & ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) & ⊢ (𝜑 → 0 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) | ||
Theorem | issubgrpd 17434* | Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) & ⊢ (𝜑 → 0 = (0g‘𝐼)) & ⊢ (𝜑 → + = (+g‘𝐼)) & ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) & ⊢ (𝜑 → 0 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ∈ Grp) ⇒ ⊢ (𝜑 → 𝑆 ∈ Grp) | ||
Theorem | issubg3 17435* | A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆))) | ||
Theorem | issubg4 17436* | A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 − 𝑦) ∈ 𝑆))) | ||
Theorem | grpissubg 17437 | 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 (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (Base‘𝐻) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺))) | ||
Theorem | resgrpisgrp 17438 | 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 other group restricted to the base set of the group is a group. (Contributed by AV, 14-Mar-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (Base‘𝐻) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝐺 ↾s 𝑆) ∈ Grp)) | ||
Theorem | subgsubm 17439 | A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺)) | ||
Theorem | subsubg 17440 | A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆))) | ||
Theorem | subgint 17441 | The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubGrp‘𝐺)) | ||
Theorem | 0subg 17442 | The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) | ||
Theorem | cycsubgcl 17443* | The set of integer powers of an element 𝐴 of a group forms a subgroup containing 𝐴, called the cyclic group generated by the element 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹)) | ||
Theorem | cycsubgss 17444* | The cyclic subgroup generated by an element 𝐴 is a subset of any subgroup containing 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑆) → ran 𝐹 ⊆ 𝑆) | ||
Theorem | cycsubg 17445* | The cyclic group generated by 𝐴 is the smallest subgroup containing 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠}) | ||
Theorem | isnsg 17446* | Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))) | ||
Theorem | isnsg2 17447* | Weaken the condition of isnsg 17446 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))) | ||
Theorem | nsgbi 17448 | Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)) | ||
Theorem | nsgsubg 17449 | A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | ||
Theorem | nsgconj 17450 | The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐵) − 𝐴) ∈ 𝑆) | ||
Theorem | isnsg3 17451* | A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)) | ||
Theorem | subgacs 17452 | Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵)) | ||
Theorem | nsgacs 17453 | Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (NrmSGrp‘𝐺) ∈ (ACS‘𝐵)) | ||
Theorem | cycsubg2 17454* | The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐾‘{𝐴}) = ran 𝐹) | ||
Theorem | cycsubg2cl 17455 | Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴})) | ||
Theorem | elnmz 17456* | Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} ⇒ ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆))) | ||
Theorem | nmzbi 17457* | Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} ⇒ ⊢ ((𝐴 ∈ 𝑁 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)) | ||
Theorem | nmzsubg 17458* | The normalizer NG(S) of a subset 𝑆 of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺)) | ||
Theorem | ssnmz 17459* | A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁) | ||
Theorem | isnsg4 17460* | A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 = 𝑋)) | ||
Theorem | nmznsg 17461* | Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑁) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻)) | ||
Theorem | 0nsg 17462 | The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) | ||
Theorem | nsgid 17463 | The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) | ||
Theorem | releqg 17464 | The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑅 = (𝐺 ~QG 𝑆) ⇒ ⊢ Rel 𝑅 | ||
Theorem | eqgfval 17465* | Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑅 = (𝐺 ~QG 𝑆) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}) | ||
Theorem | eqgval 17466 | Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑅 = (𝐺 ~QG 𝑆) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))) | ||
Theorem | eqger 17467 | The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑌) ⇒ ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋) | ||
Theorem | eqglact 17468* | A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑌) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)) | ||
Theorem | eqgid 17469 | The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑌) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌) | ||
Theorem | eqgen 17470 | Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑌) ⇒ ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝐴) | ||
Theorem | eqgcpbl 17471 | The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑌) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷))) | ||
Theorem | qusgrp 17472 | If 𝑌 is a normal subgroup of 𝐺, then 𝐻 = 𝐺 / 𝑌 is a group, called the quotient of 𝐺 by 𝑌. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) | ||
Theorem | quseccl 17473 | Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) & ⊢ 𝑉 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵) | ||
Theorem | qusadd 17474 | Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) & ⊢ 𝑉 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ✚ = (+g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆) ✚ [𝑌](𝐺 ~QG 𝑆)) = [(𝑋 + 𝑌)](𝐺 ~QG 𝑆)) | ||
Theorem | qus0 17475 | Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g‘𝐻)) | ||
Theorem | qusinv 17476 | Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) & ⊢ 𝑉 = (Base‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ 𝑁 = (invg‘𝐻) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑁‘[𝑋](𝐺 ~QG 𝑆)) = [(𝐼‘𝑋)](𝐺 ~QG 𝑆)) | ||
Theorem | qussub 17477 | Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) & ⊢ 𝑉 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (-g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)𝑁[𝑌](𝐺 ~QG 𝑆)) = [(𝑋 − 𝑌)](𝐺 ~QG 𝑆)) | ||
Theorem | lagsubg2 17478 | Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑌) & ⊢ (𝜑 → 𝑌 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (#‘𝑋) = ((#‘(𝑋 / ∼ )) · (#‘𝑌))) | ||
Theorem | lagsubg 17479 | Lagrange theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. This is Metamath 100 proof #71. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑌) ∥ (#‘𝑋)) | ||
Syntax | cghm 17480 | Extend class notation with the generator of group hom-sets. |
class GrpHom | ||
Definition | df-ghm 17481* | A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∣ [(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑔‘(𝑥(+g‘𝑠)𝑦)) = ((𝑔‘𝑥)(+g‘𝑡)(𝑔‘𝑦)))}) | ||
Theorem | reldmghm 17482 | Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ Rel dom GrpHom | ||
Theorem | isghm 17483* | Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ 𝑌 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))) | ||
Theorem | isghm3 17484* | Property of a group homomorphism, similar to ismhm 17160. (Contributed by Mario Carneiro, 7-Mar-2015.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ 𝑌 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))) | ||
Theorem | ghmgrp1 17485 | A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | ||
Theorem | ghmgrp2 17486 | A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | ||
Theorem | ghmf 17487 | A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ 𝑌 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) | ||
Theorem | ghmlin 17488 | A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) | ||
Theorem | ghmid 17489 | A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝑌 = (0g‘𝑆) & ⊢ 0 = (0g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 ) | ||
Theorem | ghminv 17490 | A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑀 = (invg‘𝑆) & ⊢ 𝑁 = (invg‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋))) | ||
Theorem | ghmsub 17491 | Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ 𝑁 = (-g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘(𝑈 − 𝑉)) = ((𝐹‘𝑈)𝑁(𝐹‘𝑉))) | ||
Theorem | isghmd 17492* | Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ 𝑌 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ (𝜑 → 𝑇 ∈ Grp) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | ||
Theorem | ghmmhm 17493 | A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇)) | ||
Theorem | ghmmhmb 17494 | Group homomorphisms and monoid homomorphisms coincide. (Thus, GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇)) | ||
Theorem | ghmmulg 17495 | A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ × = (.g‘𝐻) ⇒ ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋))) | ||
Theorem | ghmrn 17496 | The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇)) | ||
Theorem | 0ghm 17497 | The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 0 = (0g‘𝑁) & ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) | ||
Theorem | idghm 17498 | The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) | ||
Theorem | resghm 17499 | Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ 𝑈 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇)) | ||
Theorem | resghm2 17500 | One direction of resghm2b 17501. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝑋 ∈ (SubGrp‘𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
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