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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 0idnsgd 19201 | The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → {{ 0 }, 𝐵} ⊆ (NrmSGrp‘𝐺)) | ||
Theorem | trivnsgd 19202 | The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = { 0 }) ⇒ ⊢ (𝜑 → (NrmSGrp‘𝐺) = {𝐵}) | ||
Theorem | triv1nsgd 19203 | A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = { 0 }) ⇒ ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 1o) | ||
Theorem | 1nsgtrivd 19204 | A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 1o) ⇒ ⊢ (𝜑 → 𝐵 = { 0 }) | ||
Theorem | releqg 19205 | The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑅 = (𝐺 ~QG 𝑆) ⇒ ⊢ Rel 𝑅 | ||
Theorem | eqgfval 19206* | 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 19207 | 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 19208 | 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 19209* | 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 19210 | 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 19211 | 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 19212 | 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 | eqg0el 19213 | Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
⊢ ∼ = (𝐺 ~QG 𝐻) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻)) | ||
Theorem | quselbas 19214* | Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.) |
⊢ ∼ = (𝐺 ~QG 𝑆) & ⊢ 𝑈 = (𝐺 /s ∼ ) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ )) | ||
Theorem | quseccl0 19215 | Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 19217 for arbitrary sets 𝐺. (Revised by AV, 24-Feb-2025.) |
⊢ ∼ = (𝐺 ~QG 𝑆) & ⊢ 𝐻 = (𝐺 /s ∼ ) & ⊢ 𝐶 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → [𝑋] ∼ ∈ 𝐵) | ||
Theorem | qusgrp 19216 | 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 19217 | Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 9-Mar-2025.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) & ⊢ 𝑉 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵) | ||
Theorem | qusadd 19218 | 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 19219 | 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 19220 | 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 19221 | 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 | ecqusaddd 19222 | Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.) |
⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∼ = (𝑅 ~QG 𝐼) & ⊢ 𝑄 = (𝑅 /s ∼ ) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ )) | ||
Theorem | ecqusaddcl 19223 | Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.) |
⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∼ = (𝑅 ~QG 𝐼) & ⊢ 𝑄 = (𝑅 /s ∼ ) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄)) | ||
Theorem | lagsubg2 19224 | 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 19225 | Lagrange's 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)) | ||
Theorem | eqg0subg 19226 | The coset equivalence relation for the trivial (zero) subgroup of a group is the identity relation restricted to the base set of the group. (Contributed by AV, 25-Feb-2025.) |
⊢ 0 = (0g‘𝐺) & ⊢ 𝑆 = { 0 } & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑅 = (𝐺 ~QG 𝑆) ⇒ ⊢ (𝐺 ∈ Grp → 𝑅 = ( I ↾ 𝐵)) | ||
Theorem | eqg0subgecsn 19227 | The equivalence classes modulo the coset equivalence relation for the trivial (zero) subgroup of a group are singletons. (Contributed by AV, 26-Feb-2025.) |
⊢ 0 = (0g‘𝐺) & ⊢ 𝑆 = { 0 } & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑅 = (𝐺 ~QG 𝑆) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → [𝑋]𝑅 = {𝑋}) | ||
Theorem | qus0subgbas 19228* | The base set of a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025.) |
⊢ 0 = (0g‘𝐺) & ⊢ 𝑆 = { 0 } & ⊢ ∼ = (𝐺 ~QG 𝑆) & ⊢ 𝑈 = (𝐺 /s ∼ ) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (Base‘𝑈) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) | ||
Theorem | qus0subgadd 19229* | The addition in a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025.) |
⊢ 0 = (0g‘𝐺) & ⊢ 𝑆 = { 0 } & ⊢ ∼ = (𝐺 ~QG 𝑆) & ⊢ 𝑈 = (𝐺 /s ∼ ) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)}) | ||
This section contains some preliminary results about cyclic monoids and groups before the class CycGrp of cyclic groups (see df-cyg 19910) is defined in the context of Abelian groups. | ||
Theorem | cycsubmel 19230* | Characterization of an element of the set of nonnegative integer powers of an element 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) & ⊢ 𝐶 = ran 𝐹 ⇒ ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) | ||
Theorem | cycsubmcl 19231* | The set of nonnegative integer powers of an element 𝐴 contains 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) & ⊢ 𝐶 = ran 𝐹 ⇒ ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶) | ||
Theorem | cycsubm 19232* | The set of nonnegative integer powers of an element 𝐴 of a monoid forms a submonoid containing 𝐴 (see cycsubmcl 19231), called the cyclic monoid generated by the element 𝐴. This corresponds to the statement in [Lang] p. 6. (Contributed by AV, 28-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) & ⊢ 𝐶 = ran 𝐹 ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ (SubMnd‘𝐺)) | ||
Theorem | cyccom 19233* | Condition for an operation to be commutative. Lemma for cycsubmcom 19234 and cygabl 19923. Formerly part of proof for cygabl 19923. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 20-Jan-2024.) |
⊢ (𝜑 → ∀𝑐 ∈ 𝐶 ∃𝑥 ∈ 𝑍 𝑐 = (𝑥 · 𝐴)) & ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑛 ∈ 𝑍 ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴))) & ⊢ (𝜑 → 𝑋 ∈ 𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐶) & ⊢ (𝜑 → 𝑍 ⊆ ℂ) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | cycsubmcom 19234* | The operation of a monoid is commutative over the set of nonnegative integer powers of an element 𝐴 of the monoid. (Contributed by AV, 20-Jan-2024.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) & ⊢ 𝐶 = ran 𝐹 & ⊢ + = (+g‘𝐺) ⇒ ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | cycsubggend 19235* | The cyclic subgroup generated by 𝐴 includes its generator. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ ran 𝐹) | ||
Theorem | cycsubgcl 19236* | 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 19237* | 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 19238* | 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 | cycsubgcld 19239* | The cyclic subgroup generated by 𝐴 is a subgroup. Deduction related to cycsubgcl 19236. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ran 𝐹 ∈ (SubGrp‘𝐺)) | ||
Theorem | cycsubg2 19240* | 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 19241 | 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})) | ||
Syntax | cghm 19242 | Extend class notation with the generator of group hom-sets. |
class GrpHom | ||
Definition | df-ghm 19243* | 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 19244 | Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ Rel dom GrpHom | ||
Theorem | isghm 19245* | Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.) (Proof shortened by SN, 5-Jun-2025.) |
⊢ 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 | isghmOLD 19246* | Obsolete version of isghm 19245 as of 5-Jun-2025. (Contributed by Stefan O'Rear, 31-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 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 19247* | Property of a group homomorphism, similar to ismhm 18810. (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 19248 | 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 19249 | 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 19250 | 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 19251 | 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 19252 | 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 19253 | 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 19254 | 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 19255* | 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 19256 | 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 19257 | 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 19258 | A group homomorphism 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 19259 | 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 19260 | 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 19261 | 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 19262 | 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 19263 | One direction of resghm2b 19264. (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)) | ||
Theorem | resghm2b 19264 | 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 19265 | 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 19266 | 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 19267 | 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 19268 | 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 19269 | 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 19270 | 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 19271 | 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 19272 | 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 19273 | 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 19274* | 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 | f1ghm0to0 19275 | If a group homomorphism 𝐹 is injective, it 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, 13-May-2023.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 𝑁)) | ||
Theorem | ghmf1 19276* | 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.) (Proof shortened by AV, 4-Apr-2025.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))) | ||
Theorem | kerf1ghm 19277 | A group homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁})) | ||
Theorem | ghmf1o 19278 | 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 19279* | 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 19280* | 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 19281* | 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 19282* | 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 19283* | 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 19284* | 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 19285* | 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 19286* | 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 19287 | The class of group isomorphism sets. |
class GrpIso | ||
Syntax | cgic 19288 | The class of the group isomorphism relation. |
class ≃𝑔 | ||
Definition | df-gim 19289* | 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 19290 | 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 19291 | The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ GrpIso Fn (Grp × Grp) | ||
Theorem | isgim 19292 | 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 19293 | 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 19294 | 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 19295 | A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 23782. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) | ||
Theorem | subggim 19296 | 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 19297 | 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 19298 | 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 19299 | 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 19300 | The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) |
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