P. 488 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48701-48800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	uspgrlimlem4 48701*	Lemma 4 for uspgrlim 48702. (Contributed by AV, 16-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))
Theorem	uspgrlim 48702*	A local isomorphism of simple pseudographs is a bijection between their vertices that preserves neighborhoods, expressed by properties of their edges (not edge functions as in isgrlim2 48693). (Contributed by AV, 15-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))))))
Theorem	usgrlimprop 48703*	Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
Theorem	clnbgrvtxedg 48704*	An edge 𝐸 containing a vertex 𝐴 is an edge in the closed neighborhood of this vertex 𝐴. (Contributed by AV, 25-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐾)
Theorem	grlimedgclnbgr 48705*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there are two bijections 𝑓 and 𝑔 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴) and the edges between the vertices in 𝑁 onto the edges between the vertices in 𝑀, so that the mapped vertices of an edge 𝐸 containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 25-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸)))
Theorem	grlimprclnbgr 48706*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there are two bijections 𝑓 and 𝑔 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴) and the edges between the vertices in 𝑁 onto the edges between the vertices in 𝑀, so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 25-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))
Theorem	grlimprclnbgredg 48707*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 27-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
Theorem	grlimpredg 48708*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge in 𝐻. (Contributed by AV, 27-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽))
Theorem	grlimprclnbgrvtx 48709*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀 containing the vertex (𝐹‘𝐴). (Contributed by AV, 28-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
Theorem	grlimgredgex 48710*	Local isomorphisms between simple pseudographs map an edge onto an edge with an endpoint being the image of one of the endpoints of the first edge under the local isomorphism. (Contributed by AV, 28-Dec-2025.)
⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝐼)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸)
Theorem	grlimgrtrilem1 48711*	Lemma 3 for grlimgrtri 48713. (Contributed by AV, 24-Aug-2025.) (Proof shortened by AV, 27-Dec-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))
Theorem	grlimgrtrilem2 48712*	Lemma 3 for grlimgrtri 48713. (Contributed by AV, 23-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑎))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) ∧ {𝑏, 𝑐} ∈ 𝐾) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)
Theorem	grlimgrtri 48713*	If one of two locally isomorphic graphs has a triangle, so does the other. The triangle in the other graph is not necessarily the image (𝐹 “ 𝑇) of the triangle 𝑇 in the first graph. (Contributed by AV, 24-Aug-2025.)
⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))    ⇒   ⊢ (𝜑 → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐻))
Theorem	brgrlic 48714	The relation "is locally isomorphic to" for graphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
Theorem	brgrilci 48715	Prove that two graphs are locally isomorphic by an explicit local isomorphism. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐹 ∈ (𝑅 GraphLocIso 𝑆) → 𝑅 ≃𝑙𝑔𝑟 𝑆)
Theorem	grlicrel 48716	The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.)
⊢ Rel ≃𝑙𝑔𝑟
Theorem	grlicrcl 48717	Reverse closure of the "is locally isomorphic to" relation for graphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
Theorem	dfgrlic2 48718*	Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))
Theorem	grilcbri 48719*	Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣)))))
Theorem	dfgrlic3 48720*	Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣))    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
Theorem	grilcbri2 48721*	Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑋))    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
Theorem	grlicref 48722	Graph local isomorphism is reflexive for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)
Theorem	grlicsym 48723	Graph local isomorphism is symmetric for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺))
Theorem	grlicsymb 48724	Graph local isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑙𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑙𝑔𝑟 𝐴))
Theorem	grlictr 48725	Graph local isomorphism is transitive. (Contributed by AV, 10-Jun-2025.)
⊢ ((𝑅 ≃𝑙𝑔𝑟 𝑆 ∧ 𝑆 ≃𝑙𝑔𝑟 𝑇) → 𝑅 ≃𝑙𝑔𝑟 𝑇)
Theorem	grlicer 48726	Local isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 11-Jun-2025.)
⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
Theorem	grlicen 48727	Locally isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 11-Jun-2025.)
⊢ 𝐵 = (Vtx‘𝑅)    &   ⊢ 𝐶 = (Vtx‘𝑆)    ⇒   ⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)
Theorem	gricgrlic 48728	Isomorphic hypergraphs are locally isomorphic. (Contributed by AV, 12-Jun-2025.) (Proof shortened by AV, 11-Jul-2025.)
⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐺 ≃𝑔𝑟 𝐻 → 𝐺 ≃𝑙𝑔𝑟 𝐻))
Theorem	clnbgr3stgrgrlim 48729*	If all (closed) neighborhoods of the vertices in two simple graphs with the same order induce a subgraph which is isomorphic to an 𝑁-star, then any bijection between the vertices is a local isomorphism between the two graphs. (Contributed by AV, 28-Dec-2025.)
⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
Theorem	clnbgr3stgrgrlic 48730*	If all (closed) neighborhoods of the vertices in two simple graphs with the same order induce a subgraph which is isomorphic to an 𝑁-star, then the two graphs are locally isomorphic. (Contributed by AV, 29-Sep-2025.)
⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺 ≃𝑙𝑔𝑟 𝐻)
Theorem	usgrexmpl1lem 48731*	Lemma for usgrexmpl1 48732. (Contributed by AV, 2-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    ⇒   ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
Theorem	usgrexmpl1 48732	𝐺 is a simple graph of six vertices 0, 1, 2, 3, 4, 5, with edges {0, 1}, {1, 2}, {0, 2}, {0, 3}, {3, 4}, {3, 5}, {4, 5}. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ 𝐺 ∈ USGraph
Theorem	usgrexmpl1vtx 48733	The vertices 0, 1, 2, 3, 4, 5 of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5})
Theorem	usgrexmpl1edg 48734	The edges {0, 1}, {1, 2}, {0, 2}, {0, 3}, {3, 4}, {3, 5}, {4, 5} of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {0, 2}, {1, 2}} ∪ {{3, 4}, {3, 5}, {4, 5}}))
Theorem	usgrexmpl1tri 48735	𝐺 contains a triangle 0, 1, 2, with corresponding edges {0, 1}, {1, 2}, {0, 2}. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ {0, 1, 2} ∈ (GrTriangles‘𝐺)
Theorem	usgrexmpl2lem 48736*	Lemma for usgrexmpl2 48737. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    ⇒   ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
Theorem	usgrexmpl2 48737	𝐺 is a simple graph of six vertices 0, 1, 2, 3, 4, 5, with edges {0, 1}, {1, 2}, {2, 3}, {0, 3}, {3, 4}, {4, 5}, {0, 5}. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ 𝐺 ∈ USGraph
Theorem	usgrexmpl2vtx 48738	The vertices 0, 1, 2, 3, 4, 5 of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5})
Theorem	usgrexmpl2edg 48739	The edges {0, 1}, {1, 2}, {2, 3}, {0, 3}, {3, 4}, {4, 5}, {0, 5} of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))
Theorem	usgrexmpl2nblem 48740*	Lemma for usgrexmpl2nb0 48741 etc. (Contributed by AV, 9-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (𝐾 ∈ ({0, 1, 2} ∪ {3, 4, 5}) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {𝐾, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))})
Theorem	usgrexmpl2nb0 48741	The neighborhood of the first vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (𝐺 NeighbVtx 0) = {1, 3, 5}
Theorem	usgrexmpl2nb1 48742	The neighborhood of the second vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (𝐺 NeighbVtx 1) = {0, 2}
Theorem	usgrexmpl2nb2 48743	The neighborhood of the third vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (𝐺 NeighbVtx 2) = {1, 3}
Theorem	usgrexmpl2nb3 48744	The neighborhood of the forth vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (𝐺 NeighbVtx 3) = {0, 2, 4}
Theorem	usgrexmpl2nb4 48745	The neighborhood of the fifth vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (𝐺 NeighbVtx 4) = {3, 5}
Theorem	usgrexmpl2nb5 48746	The neighborhood of the sixth vertex of graph 𝐺. (Contributed by AV, 10-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ (𝐺 NeighbVtx 5) = {0, 4}
Theorem	usgrexmpl2trifr 48747*	𝐺 is triangle-free. (Contributed by AV, 10-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ ¬ ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺)
Theorem	usgrexmpl12ngric 48748	The graphs 𝐻 and 𝐺 are not isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 48735, whereas 𝐺 does not, see usgrexmpl2trifr 48747. (Contributed by AV, 10-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    &   ⊢ 𝐾 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   ⊢ 𝐻 = ⟨𝑉, 𝐾⟩    ⇒   ⊢ ¬ 𝐺 ≃𝑔𝑟 𝐻
Theorem	usgrexmpl12ngrlic 48749	The graphs 𝐻 and 𝐺 are not locally isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 48735, whereas 𝐺 does not, see usgrexmpl2trifr 48747. (Contributed by AV, 24-Aug-2025.)
⊢ 𝑉 = (0...5)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    &   ⊢ 𝐾 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   ⊢ 𝐻 = ⟨𝑉, 𝐾⟩    ⇒   ⊢ ¬ 𝐺 ≃𝑙𝑔𝑟 𝐻
21.49.15.8  Generalized Petersen graphs According to Wikipedia "Generalized Petersen graph", 26-Aug-2025, https://en.wikipedia.org/wiki/Generalized_Petersen_graph: "In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. ... Among the generalized Petersen graphs are the n-prism, ...". The vertices of the regular polygon are called "outside vertices", the vertices of the star polygon "inside vertices" (see A. Steimle, W. Stanton, "The isomorphism classes of the generalized Petersen graphs", Discrete Mathematics Volume 309, Issue 1, 6 January 2009, Pages 231-237: https://doi.org/10.1016/j.disc.2007.12.074). Since regular polygons are also considered as star polygons (with density 1), many theorems for "inside vertices" (with labels containing the fragment "vtx1") can be specialized for "outside vertices" (with labels containing the fragment "vtx0").
Syntax	cgpg 48750	Extend class notation with generalized Petersen graphs.
class gPetersenGr
Definition	df-gpg 48751*	Definition of generalized Petersen graphs according to Wikipedia "Generalized Petersen graph", 26-Aug-2025, https://en.wikipedia.org/wiki/Generalized_Petersen_graph: "In Watkins' notation, 𝐺(𝑛, 𝑘) is a graph with vertex set { u0, u1, ... , un-1, v0, v1, ... , vn-1 } and edge set { ui ui+1 , ui vi , vi vi+k | 0 ≤ 𝑖 ≤ (𝑛 − 1) } where subscripts are to be read modulo n and where 𝑘 < (𝑛 / 2). Some authors use the notation GPG(n,k)." Instead of 𝑛 ∈ ℕ, we could restrict the first argument to 𝑛 ∈ (ℤ≥‘3) (i.e., 3 ≤ 𝑛), because for 𝑛 ≤ 2, the definition is not meaningful (since then (⌈‘(𝑛 / 2)) ≤ 1 and therefore (1..^(⌈‘(𝑛 / 2))) = ∅, so that there would be no fitting second argument). (Contributed by AV, 26-Aug-2025.)
⊢ gPetersenGr = (𝑛 ∈ ℕ, 𝑘 ∈ (1..^(⌈‘(𝑛 / 2))) ↦ {⟨(Base‘ndx), ({0, 1} × (0..^𝑛))⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})})⟩})
Theorem	gpgov 48752*	The generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (𝑁 gPetersenGr 𝐾) = {⟨(Base‘ndx), ({0, 1} × 𝐼)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})⟩})
Theorem	gpgvtx 48753	The vertices of the generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × 𝐼))
Theorem	gpgiedg 48754*	The indexed edges of the generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (iEdg‘(𝑁 gPetersenGr 𝐾)) = ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})}))
Theorem	gpgedg 48755*	The edges of the generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (Edg‘(𝑁 gPetersenGr 𝐾)) = {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})})
Theorem	gpgiedgdmellem 48756*	Lemma for gpgiedgdmel 48759 and gpgedgel 48760. (Contributed by AV, 2-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (∃𝑥 ∈ 𝐼 (𝑌 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑌 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑌 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) → 𝑌 ∈ 𝒫 ({0, 1} × 𝐼)))
Theorem	gpgvtxel 48757*	A vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ 𝐼 𝑋 = ⟨𝑥, 𝑦⟩))
Theorem	gpgvtxel2 48758	The second component of a vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ 𝐼)
Theorem	gpgiedgdmel 48759*	An index of edges of the generalized Petersen graph GPG(N,K). (Contributed by AV, 2-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ dom (iEdg‘𝐺) ↔ ∃𝑥 ∈ 𝐼 (𝑋 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑋 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑋 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
Theorem	gpgedgel 48760*	An edge in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.) (Proof shortened by AV, 8-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑌 ∈ 𝐸 ↔ ∃𝑥 ∈ 𝐼 (𝑌 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑌 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑌 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
Theorem	gpgprismgriedgdmel 48761*	An index of edges of the generalized Petersen graph GPG(N,1). (Contributed by AV, 2-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 1)    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑋 ∈ dom (iEdg‘𝐺) ↔ ∃𝑥 ∈ 𝐼 (𝑋 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑋 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑋 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
Theorem	gpgprismgriedgdmss 48762	A subset of the index of edges of the generalized Petersen graph GPG(N,1). (Contributed by AV, 2-Nov-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → ({{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ∪ {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}}) ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
Theorem	gpgvtx0 48763	The outside vertices in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉))
Theorem	gpgvtx1 48764	The inside vertices in a generalized Petersen graph 𝐺. (Contributed by AV, 28-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉))
Theorem	opgpgvtx 48765	A vertex in a generalized Petersen graph 𝐺 as ordered pair. (Contributed by AV, 1-Oct-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (⟨𝑋, 𝑌⟩ ∈ 𝑉 ↔ ((𝑋 = 0 ∨ 𝑋 = 1) ∧ 𝑌 ∈ 𝐼)))
Theorem	gpgusgralem 48766*	Lemma for gpgusgra 48767. (Contributed by AV, 27-Aug-2025.) (Proof shortened by AV, 6-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})} ⊆ {𝑝 ∈ 𝒫 ({0, 1} × 𝐼) ∣ (♯‘𝑝) = 2})
Theorem	gpgusgra 48767	The generalized Petersen graph GPG(N,K) is a simple graph. (Contributed by AV, 27-Aug-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
Theorem	gpgprismgrusgra 48768	The generalized Petersen graphs G(N,1), which are the N-prisms, are simple graphs. (Contributed by AV, 31-Oct-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 gPetersenGr 1) ∈ USGraph)
Theorem	gpgorder 48769	The order of the generalized Petersen graph GPG(N,K). (Contributed by AV, 29-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (♯‘(Vtx‘(𝑁 gPetersenGr 𝐾))) = (2 · 𝑁))
Theorem	gpg5order 48770	The order of a generalized Petersen graph G(5,K), which is either the Petersen graph G(5,2) or the 5-prism G(5,1), is 10. (Contributed by AV, 26-Aug-2025.)
⊢ (𝐾 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 𝐾))) = ;10)
Theorem	gpgedgvtx0 48771	The edges starting at an outside vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ({𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ 𝐸 ∧ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸 ∧ {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ 𝐸))
Theorem	gpgedgvtx1 48772	The edges starting at an inside vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ({𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ 𝐸 ∧ {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ 𝐸 ∧ {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ 𝐸))
Theorem	gpgvtxedg0 48773	The edges starting at an outside vertex 𝑋 in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑌 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
Theorem	gpgvtxedg1 48774	The edges starting at an inside vertex 𝑋 in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑌 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑌 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
Theorem	gpgedgiov 48775	The edges of the generalized Petersen graph GPG(N,K) between an inside and an outside vertex. (Contributed by AV, 11-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ 𝑋 = 𝑌))
Theorem	gpgedg2ov 48776	The edges of the generalized Petersen graph GPG(N,K) between two outside vertices. (Contributed by AV, 15-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))
Theorem	gpgedg2iv 48777	The edges of the generalized Petersen graph GPG(N,K) between two inside vertices. (Contributed by AV, 20-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘5) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ (𝐾 ∈ 𝐽 ∧ ((4 · 𝐾) mod 𝑁) ≠ 0)) → (({⟨1, ((𝑌 − 𝐾) mod 𝑁)⟩, ⟨1, 𝑋⟩} ∈ 𝐸 ∧ {⟨1, 𝑋⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))
Theorem	gpg5nbgrvtx03starlem1 48778	Lemma 1 for gpg5nbgrvtx03star 48790. (Contributed by AV, 5-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨1, 𝑋⟩} ∉ 𝐸)
Theorem	gpg5nbgrvtx03starlem2 48779	Lemma 2 for gpg5nbgrvtx03star 48790. (Contributed by AV, 6-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → {⟨0, ((𝑋 + 1) mod 𝑁)⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∉ 𝐸)
Theorem	gpg5nbgrvtx03starlem3 48780	Lemma 3 for gpg5nbgrvtx03star 48790. (Contributed by AV, 5-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊) → {⟨1, 𝑋⟩, ⟨0, ((𝑋 − 1) mod 𝑁)⟩} ∉ 𝐸)
Theorem	gpg5nbgrvtx13starlem1 48781	Lemma 1 for gpg5nbgr3star 48791. (Contributed by AV, 7-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑊) → {⟨1, ((𝑋 + 𝐾) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∉ 𝐸)
Theorem	gpg5nbgrvtx13starlem2 48782	Lemma 2 for gpg5nbgr3star 48791. (Contributed by AV, 8-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ ℤ) → {⟨1, ((𝑋 + 𝐾) mod 𝑁)⟩, ⟨1, ((𝑋 − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
Theorem	gpg5nbgrvtx13starlem3 48783	Lemma 3 for gpg5nbgr3star 48791. (Contributed by AV, 8-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑊) → {⟨0, 𝑋⟩, ⟨1, ((𝑋 − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
Theorem	gpgnbgrvtx0 48784	The (open) neighborhood of an outside vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 28-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑈 = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
Theorem	gpgnbgrvtx1 48785	The (open) neighborhood of an inside vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑈 = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
Theorem	gpg3nbgrvtx0 48786	In a generalized Petersen graph 𝐺, every outside vertex has exactly three (different) neighbors. (Contributed by AV, 30-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
Theorem	gpg3nbgrvtx0ALT 48787	In a generalized Petersen graph 𝐺, every outside vertex has exactly three (different) neighbors. (Contributed by AV, 30-Aug-2025.) The proof of gpg3nbgrvtx0 48786 can be shortened using modmknepk 48050, but then theorem 2ltceilhalf 48014 is required which is based on an "example" ex-ceil 30765. If these theorems were moved to main, the "example" should also be moved up to become a full-fledged theorem. (Proof shortened by AV, 4-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
Theorem	gpg3nbgrvtx1 48788	In a generalized Petersen graph 𝐺, every inside vertex has exactly three (different) neighbors. (Contributed by AV, 3-Sep-2025.) (Proof shortened by AV, 22-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)
Theorem	gpgcubic 48789	Every generalized Petersen graph is a cubic graph, i.e., it is a 3-regular graph, i.e., every vertex has degree 3 (see gpgvtxdg3 48792), i.e., every vertex has exactly three (different) neighbors. (Contributed by AV, 3-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (♯‘𝑈) = 3)
Theorem	gpg5nbgrvtx03star 48790*	In a generalized Petersen graph G(N,K) of order greater than 8 (3 < 𝑁), every outside vertex has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every outside vertex induces a subgraph which is isomorphic to a 3-star). (Contributed by AV, 31-Aug-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))
Theorem	gpg5nbgr3star 48791*	In a generalized Petersen graph G(N,K) of order 10 (𝑁 = 5), these are the Petersen graph G(5,2) and the 5-prism G(5,1), every vertex has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every vertex induces a subgraph which is isomorphic to a 3-star). This does not hold for every generalized Petersen graph: for example, in the 3-prism G(3,1) (see gpg31grim3prism TODO) and the Dürer graph G(6,2) there are vertices which have neighborhoods containing triangles. In general, all generalized Petersen graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles, see gpg3kgrtriex 48799. (Contributed by AV, 8-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))
Theorem	gpgvtxdg3 48792	Every vertex in a generalized Petersen graph has degree 3. (Contributed by AV, 4-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑋) = 3)
Theorem	gpg3kgrtriexlem1 48793	Lemma 1 for gpg3kgrtriex 48799. (Contributed by AV, 1-Oct-2025.)
⊢ (𝐾 ∈ ℕ → 𝐾 < (⌈‘((3 · 𝐾) / 2)))
Theorem	gpg3kgrtriexlem2 48794	Lemma 2 for gpg3kgrtriex 48799. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → (-𝐾 mod 𝑁) = (((𝐾 mod 𝑁) + 𝐾) mod 𝑁))
Theorem	gpg3kgrtriexlem3 48795	Lemma 3 for gpg3kgrtriex 48799. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ (ℤ≥‘3))
Theorem	gpg3kgrtriexlem4 48796	Lemma 4 for gpg3kgrtriex 48799. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
Theorem	gpg3kgrtriexlem5 48797	Lemma 5 for gpg3kgrtriex 48799. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → (𝐾 mod 𝑁) ≠ (-𝐾 mod 𝑁))
Theorem	gpg3kgrtriexlem6 48798	Lemma 6 for gpg3kgrtriex 48799: 𝐸 is an edge in the generalized Petersen graph G(N,K) with 𝑁 = 3 · 𝐾. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩}    ⇒   ⊢ (𝐾 ∈ ℕ → 𝐸 ∈ (Edg‘𝐺))
Theorem	gpg3kgrtriex 48799*	All generalized Petersen graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺))
Theorem	gpg5gricstgr3 48800	Each closed neighborhood in a generalized Petersen graph G(N,K) of order 10 (𝑁 = 5), which is either the Petersen graph G(5,2) or the 5-prism G(5,1), is isomorphic to a 3-star. (Contributed by AV, 13-Sep-2025.)
⊢ 𝐺 = (5 gPetersenGr 𝐾)    ⇒   ⊢ ((𝐾 ∈ (1...2) ∧ 𝑉 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑉)) ≃𝑔𝑟 (StarGr‘3))