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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cisomgr 46001 | Extend class notation to include the isomorphy relation for graphs. |
class IsomGr | ||
Definition | df-grisom 46002* | Define the class of all isomorphisms between two graphs. (Contributed by AV, 11-Dec-2022.) |
⊢ GrIsom = (𝑥 ∈ V, 𝑦 ∈ V ↦ {〈𝑓, 𝑔〉 ∣ (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ 𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)))}) | ||
Definition | df-isomgr 46003* | Define the isomorphy relation for graphs. (Contributed by AV, 11-Nov-2022.) |
⊢ IsomGr = {〈𝑥, 𝑦〉 ∣ ∃𝑓(𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖))))} | ||
Theorem | isomgrrel 46004 | The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) |
⊢ Rel IsomGr | ||
Theorem | isomgr 46005* | The isomorphy relation for two graphs. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐼 = (iEdg‘𝐴) & ⊢ 𝐽 = (iEdg‘𝐵) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))) | ||
Theorem | isisomgr 46006* | Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐼 = (iEdg‘𝐴) & ⊢ 𝐽 = (iEdg‘𝐵) ⇒ ⊢ (𝐴 IsomGr 𝐵 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))) | ||
Theorem | isomgreqve 46007 | A set is isomorphic to a hypergraph if it has the same vertices and the same edges. (Contributed by AV, 11-Nov-2022.) |
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → 𝐴 IsomGr 𝐵) | ||
Theorem | isomushgr 46008* | The isomorphy relation for two simple hypergraphs. (Contributed by AV, 28-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ ((𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))))) | ||
Theorem | isomuspgrlem1 46009* | Lemma 1 for isomuspgr 46016. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸)) | ||
Theorem | isomuspgrlem2a 46010* | Lemma 1 for isomuspgrlem2 46015. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) ⇒ ⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) | ||
Theorem | isomuspgrlem2b 46011* | Lemma 2 for isomuspgrlem2 46015. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) ⇒ ⊢ (𝜑 → 𝐺:𝐸⟶𝐾) | ||
Theorem | isomuspgrlem2c 46012* | Lemma 3 for isomuspgrlem2 46015. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐺:𝐸–1-1→𝐾) | ||
Theorem | isomuspgrlem2d 46013* | Lemma 4 for isomuspgrlem2 46015. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ USPGraph) ⇒ ⊢ (𝜑 → 𝐺:𝐸–onto→𝐾) | ||
Theorem | isomuspgrlem2e 46014* | Lemma 5 for isomuspgrlem2 46015. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ USPGraph) ⇒ ⊢ (𝜑 → 𝐺:𝐸–1-1-onto→𝐾) | ||
Theorem | isomuspgrlem2 46015* | Lemma 2 for isomuspgr 46016. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)))) | ||
Theorem | isomuspgr 46016* | The isomorphy relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)))) | ||
Theorem | isomgrref 46017 | The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
⊢ (𝐺 ∈ UHGraph → 𝐺 IsomGr 𝐺) | ||
Theorem | isomgrsym 46018 | The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 𝐵 → 𝐵 IsomGr 𝐴)) | ||
Theorem | isomgrsymb 46019 | The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 IsomGr 𝐵 ↔ 𝐵 IsomGr 𝐴)) | ||
Theorem | isomgrtrlem 46020* | Lemma for isomgrtr 46021. (Contributed by AV, 5-Dec-2022.) |
⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))) | ||
Theorem | isomgrtr 46021 | The isomorphy relation is transitive for hypergraphs. (Contributed by AV, 5-Dec-2022.) |
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) → 𝐴 IsomGr 𝐶)) | ||
Theorem | strisomgrop 46022 | A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝐺 = 〈𝑉, 𝐸〉 & ⊢ 𝐻 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 IsomGr 𝐻) | ||
Theorem | ushrisomgr 46023 | A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐻 = 〈𝑉, ( I ↾ 𝐸)〉 ⇒ ⊢ (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻) | ||
Theorem | 1hegrlfgr 46024* | A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) & ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) & ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) ⇒ ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) | ||
Syntax | cupwlks 46025 | Extend class notation with walks (of a pseudograph). |
class UPWalks | ||
Definition | df-upwlks 46026* |
Define the set of all walks (in a pseudograph), called "simple walks"
in
the following.
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)." According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | upwlksfval 46027* | The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | isupwlk 46028* | Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | isupwlkg 46029* | Generalization of isupwlk 46028: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upwlkbprop 46030 | Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | ||
Theorem | upwlkwlk 46031 | A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.) |
⊢ (𝐹(UPWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlk 46032 | In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.) |
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlkb 46033 | In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.) |
⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ 𝐹(UPWalks‘𝐺)𝑃)) | ||
Theorem | upgrisupwlkALT 46034* | Alternate proof of upgriswlk 28589 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upgredgssspr 46035 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺))) | ||
Theorem | uspgropssxp 46036* | The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of a Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 46046. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ⊆ (𝑊 × 𝑃)) | ||
Theorem | uspgrsprfv 46037* | The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 46043. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) | ||
Theorem | uspgrsprf 46038* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺⟶𝑃 | ||
Theorem | uspgrsprf1 46039* | The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺–1-1→𝑃 | ||
Theorem | uspgrsprfo 46040* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) | ||
Theorem | uspgrsprf1o 46041* | The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 46046. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrex 46042* | The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V) | ||
Theorem | uspgrbispr 46043* | There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrspren 46044* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑃) | ||
Theorem | uspgrymrelen 46045* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 46046. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅) | ||
Theorem | uspgrbisymrel 46046* |
There is a bijection between the "simple pseudographs" for a fixed
set
𝑉 of vertices and the class 𝑅 of the
symmetric relations on the
fixed set 𝑉. The simple pseudographs, which are
graphs without
hyper- or multiedges, but which may contain loops, are expressed as
ordered pairs of the vertices and the edges (as proper or improper
unordered pairs of vertices, not as indexed edges!) in this theorem.
That class 𝐺 of such simple pseudographs is a set
(if 𝑉 is a
set, see uspgrex 46042) of equivalence classes of graphs
abstracting from
the index sets of their edge functions.
Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ 〈𝑣, 𝑒〉 ∈ USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | uspgrbisymrelALT 46047* | Alternate proof of uspgrbisymrel 46046 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | ovn0dmfun 46048 | If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6885. (Contributed by AV, 27-Jan-2020.) |
⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) | ||
Theorem | xpsnopab 46049* | A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} | ||
Theorem | xpiun 46050* | A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.) |
⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | ||
Theorem | ovn0ssdmfun 46051* | If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6885. (Contributed by AV, 27-Jan-2020.) |
⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) | ||
Theorem | fnxpdmdm 46052 | The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.) |
⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴) | ||
Theorem | cnfldsrngbas 46053 | The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ⊆ ℂ → 𝑆 = (Base‘𝑅)) | ||
Theorem | cnfldsrngadd 46054 | The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → + = (+g‘𝑅)) | ||
Theorem | cnfldsrngmul 46055 | The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → · = (.r‘𝑅)) | ||
Theorem | plusfreseq 46056 | If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
Theorem | mgmplusfreseq 46057 | If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
Theorem | 0mgm 46058 | A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.) |
⊢ (Base‘𝑀) = ∅ ⇒ ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) | ||
Theorem | mgmpropd 46059* | If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm)) | ||
Theorem | ismgmd 46060* | Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → + = (+g‘𝐺)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐺 ∈ Mgm) | ||
Syntax | cmgmhm 46061 | Hom-set generator class for magmas. |
class MgmHom | ||
Syntax | csubmgm 46062 | Class function taking a magma to its lattice of submagmas. |
class SubMgm | ||
Definition | df-mgmhm 46063* | A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.) |
⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))}) | ||
Definition | df-submgm 46064* | A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.) |
⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | ||
Theorem | mgmhmrcl 46065 | Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) | ||
Theorem | submgmrcl 46066 | Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) | ||
Theorem | ismgmhm 46067* | Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))) | ||
Theorem | mgmhmf 46068 | A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) | ||
Theorem | mgmhmpropd 46069* | Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐶 ≠ ∅) & ⊢ ((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 MgmHom 𝐾) = (𝐿 MgmHom 𝑀)) | ||
Theorem | mgmhmlin 46070 | A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) | ||
Theorem | mgmhmf1o 46071 | A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MgmHom 𝑅))) | ||
Theorem | idmgmhm 46072 | The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀)) | ||
Theorem | issubmgm 46073* | Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))) | ||
Theorem | issubmgm2 46074 | Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 𝐻 ∈ Mgm))) | ||
Theorem | rabsubmgmd 46075* | Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mgm) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂) & ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃)) & ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏)) & ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂)) ⇒ ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀)) | ||
Theorem | submgmss 46076 | Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 ⊆ 𝐵) | ||
Theorem | submgmid 46077 | Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → 𝐵 ∈ (SubMgm‘𝑀)) | ||
Theorem | submgmcl 46078 | Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.) |
⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
Theorem | submgmmgm 46079 | Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝐻 ∈ Mgm) | ||
Theorem | submgmbas 46080 | The base set of a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 = (Base‘𝐻)) | ||
Theorem | subsubmgm 46081 | A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆))) | ||
Theorem | resmgmhm 46082 | Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇)) | ||
Theorem | resmgmhm2 46083 | One direction of resmgmhm2b 46084. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇)) | ||
Theorem | resmgmhm2b 46084 | Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈))) | ||
Theorem | mgmhmco 46085 | The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.) |
⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom 𝑈)) | ||
Theorem | mgmhmima 46086 | The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.) |
⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMgm‘𝑁)) | ||
Theorem | mgmhmeql 46087 | The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.) |
⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMgm‘𝑆)) | ||
Theorem | submgmacs 46088 | Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵)) | ||
Theorem | ismhm0 46089 | Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝑌 = (0g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌))) | ||
Theorem | mhmismgmhm 46090 | Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.) |
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆)) | ||
Theorem | opmpoismgm 46091* | A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑀 ∈ Mgm) | ||
Theorem | copissgrp 46092* | A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑀 ∈ Smgrp) | ||
Theorem | copisnmnd 46093* | A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 1 < (♯‘𝐵)) ⇒ ⊢ (𝜑 → 𝑀 ∉ Mnd) | ||
Theorem | 0nodd 46094* | 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 0 ∉ 𝑂 | ||
Theorem | 1odd 46095* | 1 is an odd integer. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 1 ∈ 𝑂 | ||
Theorem | 2nodd 46096* | 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 2 ∉ 𝑂 | ||
Theorem | oddibas 46097* | Lemma 1 for oddinmgm 46099: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ 𝑂 = (Base‘𝑀) | ||
Theorem | oddiadd 46098* | Lemma 2 for oddinmgm 46099: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ + = (+g‘𝑀) | ||
Theorem | oddinmgm 46099* | The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 46232, and even a non-unital ring, see 2zrng 46223. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ 𝑀 ∉ Mgm | ||
Theorem | nnsgrpmgm 46100 | The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.) |
⊢ 𝑀 = (ℂfld ↾s ℕ) ⇒ ⊢ 𝑀 ∈ Mgm |
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