P. 490 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48901-49000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isgrlim2 48901*	A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. Definitions expanded. (Contributed by AV, 29-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 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 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
Theorem	grlimprop 48902*	Properties of a local isomorphism of graphs. (Contributed by AV, 21-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣)))))
Theorem	grlimf1o 48903	A local isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 21-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)
Theorem	grlimprop2 48904*	Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ (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 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
Theorem	uhgrimgrlim 48905	An isomorphism of hypergraphs is a local isomorphism between the two graphs. (Contributed by AV, 2-Jun-2025.)
⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
Theorem	uspgrlimlem1 48906*	Lemma 1 for uspgrlim 48910. (Contributed by AV, 16-Aug-2025.)
⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
Theorem	uspgrlimlem2 48907*	Lemma 2 for uspgrlim 48910. (Contributed by AV, 16-Aug-2025.)
⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (𝐻 ∈ USPGraph → (◡(iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀})
Theorem	uspgrlimlem3 48908*	Lemma 3 for uspgrlim 48910. (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:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) → (𝑒 ∈ 𝐾 → (𝑓 “ 𝑒) = ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒)))
Theorem	uspgrlimlem4 48909*	Lemma 4 for uspgrlim 48910. (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 48910*	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 48901). (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 48911*	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 48912*	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 48913*	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 48914*	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 48915*	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 48916*	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 48917*	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 48918*	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 48919*	Lemma 3 for grlimgrtri 48921. (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 48920*	Lemma 3 for grlimgrtri 48921. (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 48921*	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 48922	The relation "is locally isomorphic to" for graphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
Theorem	brgrilci 48923	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 48924	The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.)
⊢ Rel ≃𝑙𝑔𝑟
Theorem	grlicrcl 48925	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 48926*	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 48927*	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 48928*	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 48929*	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 48930	Graph local isomorphism is reflexive for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)
Theorem	grlicsym 48931	Graph local isomorphism is symmetric for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺))
Theorem	grlicsymb 48932	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 48933	Graph local isomorphism is transitive. (Contributed by AV, 10-Jun-2025.)
⊢ ((𝑅 ≃𝑙𝑔𝑟 𝑆 ∧ 𝑆 ≃𝑙𝑔𝑟 𝑇) → 𝑅 ≃𝑙𝑔𝑟 𝑇)
Theorem	grlicer 48934	Local isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 11-Jun-2025.)
⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
Theorem	grlicen 48935	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 48936	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 48937*	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 48938*	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 48939*	Lemma for usgrexmpl1 48940. (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 48940	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 48941	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 48942	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 48943	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 48944*	Lemma for usgrexmpl2 48945. (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 48945	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 48946	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 48947	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 48948*	Lemma for usgrexmpl2nb0 48949 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 48949	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 48950	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 48951	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 48952	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 48953	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 48954	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 48955*	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 48956	The graphs 𝐻 and 𝐺 are not isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 48943, whereas 𝐺 does not, see usgrexmpl2trifr 48955. (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 48957	The graphs 𝐻 and 𝐺 are not locally isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 48943, whereas 𝐺 does not, see usgrexmpl2trifr 48955. (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
21.50.15.8  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 48958	Extend class notation with generalized Petersen graphs.
class gPetersenGr
Definition	df-gpg 48959*	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 48960*	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 48961	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 48962*	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 48963*	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 48964*	Lemma for gpgiedgdmel 48967 and gpgedgel 48968. (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 48965*	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 48966	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 48967*	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 48968*	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 48969*	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 48970	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 48971	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 48972	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 48973	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 48974*	Lemma for gpgusgra 48975. (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 48975	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 48976	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 48977	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 48978	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 48979	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 48980	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 48981	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 48982	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 48983	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 48984	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 48985	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 48986	Lemma 1 for gpg5nbgrvtx03star 48998. (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 48987	Lemma 2 for gpg5nbgrvtx03star 48998. (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 48988	Lemma 3 for gpg5nbgrvtx03star 48998. (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 48989	Lemma 1 for gpg5nbgr3star 48999. (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 48990	Lemma 2 for gpg5nbgr3star 48999. (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 48991	Lemma 3 for gpg5nbgr3star 48999. (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 48992	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 48993	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 48994	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 48995	In a generalized Petersen graph 𝐺, every outside vertex has exactly three (different) neighbors. (Contributed by AV, 30-Aug-2025.) The proof of gpg3nbgrvtx0 48994 can be shortened using modmknepk 48258, but then theorem 2ltceilhalf 48222 is required which is based on an "example" ex-ceil 30929. 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 48996	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 48997	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 49000), 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 48998*	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 48999*	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 49007. (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 49000	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)