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Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtrlsegvdeglem4 27201 Lemma for trlsegvdeg 27205. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))

Theoremtrlsegvdeglem5 27202 Lemma for trlsegvdeg 27205. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})

Theoremtrlsegvdeglem6 27203 Lemma for trlsegvdeg 27205. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑋) ∈ Fin)

Theoremtrlsegvdeglem7 27204 Lemma for trlsegvdeg 27205. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑌) ∈ Fin)

Theoremtrlsegvdeg 27205 Formerly part of proof of eupth2lem3 27214: If a trail in a graph 𝐺 induces a subgraph 𝑍 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the walk, then the vertex degree of any vertex 𝑈 of 𝐺 within 𝑍 is the sum of the vertex degree of 𝑈 within 𝑋 and the vertex degree of 𝑈 within 𝑌. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Theoremeupth2lem3lem1 27206 Lemma for eupth2lem3 27214. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0)

Theoremeupth2lem3lem2 27207 Lemma for eupth2lem3 27214. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ0)

Theoremeupth2lem3lem3 27208* Lemma for eupth2lem3 27214, formerly part of proof of eupth2lem3 27214: If a loop {(𝑃𝑁), (𝑃‘(𝑁 + 1))} is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))       ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lem3lem4 27209* Lemma for eupth2lem3 27214, formerly part of proof of eupth2lem3 27214: If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))    &   (𝜑 → (𝐼‘(𝐹𝑁)) ∈ 𝒫 𝑉)       ((𝜑 ∧ (𝑃𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lem3lem5 27210* Lemma for eupth2 27217. (Contributed by AV, 25-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁), (𝑃‘(𝑁 + 1))})       (𝜑 → (𝐼‘(𝐹𝑁)) ∈ 𝒫 𝑉)

Theoremeupth2lem3lem6 27211* Formerly part of proof of eupth2lem3 27214: If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than (𝑃‘0) and (𝑃𝑁): if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁), (𝑃‘(𝑁 + 1))})       ((𝜑 ∧ (𝑃𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lem3lem7 27212* Lemma for eupth2lem3 27214: Combining trlsegvdeg 27205, eupth2lem3lem3 27208, eupth2lem3lem4 27209 and eupth2lem3lem6 27211. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁), (𝑃‘(𝑁 + 1))})       (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupthvdres 27213 Formerly part of proof of eupth2 27217: The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺𝑊)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩       (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Theoremeupth2lem3 27214* Lemma for eupth2 27217. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩    &   𝑋 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑁 + 1) ≤ (#‘𝐹))    &   (𝜑𝑈𝑉)    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐻)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))       (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lemb 27215* Lemma for eupth2 27217 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). Formerly part of proof for eupth2 27217. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)

Theoremeupth2lems 27216* Lemma for eupth2 27217 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. Formerly part of proof for eupth2 27217. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))

Theoremeupth2 27217* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))

Theoremeulerpathpr 27218* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Theoremeulerpath 27219* A pseudograph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Theoremeulercrct 27220* A pseudograph with an Eulerian circuit 𝐹, 𝑃 (an "Eulerian pseudograph") has only vertices of even degree. (Contributed by AV, 12-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃𝐹(Circuits‘𝐺)𝑃) → ∀𝑥𝑉 2 ∥ ((VtxDeg‘𝐺)‘𝑥))

Theoremeucrctshift 27221* Cyclically shifting the indices of an Eulerian circuit 𝐹, 𝑃 results in an Eulerian circuit 𝐻, 𝑄. (Contributed by AV, 15-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       (𝜑 → (𝐻(EulerPaths‘𝐺)𝑄𝐻(Circuits‘𝐺)𝑄))

Theoremeucrct2eupth1 27222 Removing one edge (𝐼‘(𝐹𝑁)) from a nonempty graph 𝐺 with an Eulerian circuit 𝐹, 𝑃 results in a graph 𝑆 with an Eulerian path 𝐻, 𝑄. This is the special case of eucrct2eupth 27223 (with 𝐽 = (𝑁 − 1)) where the last segment/edge of the circuit is removed. (Contributed by AV, 11-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   (Vtx‘𝑆) = 𝑉    &   (𝜑 → 0 < (#‘𝐹))    &   (𝜑𝑁 = ((#‘𝐹) − 1))    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   𝐻 = (𝐹 ↾ (0..^𝑁))    &   𝑄 = (𝑃 ↾ (0...𝑁))       (𝜑𝐻(EulerPaths‘𝑆)𝑄)

Theoremeucrct2eupth 27223* Removing one edge (𝐼‘(𝐹𝐽)) from a graph 𝐺 with an Eulerian circuit 𝐹, 𝑃 results in a graph 𝑆 with an Eulerian path 𝐻, 𝑄. (Contributed by AV, 17-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   (Vtx‘𝑆) = 𝑉    &   (𝜑𝑁 = (#‘𝐹))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))))    &   𝐾 = (𝐽 + 1)    &   𝐻 = ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1)))    &   𝑄 = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁))))       (𝜑𝐻(EulerPaths‘𝑆)𝑄)

16.4.2  The Königsberg Bridge problem

According to Wikipedia ("Seven Bridges of Königsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in [East] Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands - Kneiphof and Lomse - which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once.". Euler proved that the problem has no solution by applying Euler's theorem to the Königsberg graph, which is obtained by replacing each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which connects two land masses/vertices. The Königsberg graph 𝐺 is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: 𝐺 = ⟨(0...3), ⟨“{0, 1}{0, 2} {0, 3}{1, 2}{1, 2}{2, 3}{2, 3}”⟩⟩, see konigsbergumgr 27229. konigsberg 27235 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

Theoremkonigsbergvtx 27224 The set of vertices of the Königsberg graph 𝐺. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (Vtx‘𝐺) = (0...3)

Theoremkonigsbergiedg 27225 The indexed edges of the Königsberg graph 𝐺. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩

Theoremkonigsbergiedgw 27226* The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}

Theoremkonigsbergssiedgwpr 27227* Each subset of the indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremkonigsbergssiedgw 27228* Each subset of the indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremkonigsbergumgr 27229 The Königsberg graph 𝐺 is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐺 ∈ UMGraph

Theoremkonigsberglem1 27230 Lemma 1 for konigsberg 27235: Vertex 0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((VtxDeg‘𝐺)‘0) = 3

Theoremkonigsberglem2 27231 Lemma 2 for konigsberg 27235: Vertex 1 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((VtxDeg‘𝐺)‘1) = 3

Theoremkonigsberglem3 27232 Lemma 3 for konigsberg 27235: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((VtxDeg‘𝐺)‘3) = 3

Theoremkonigsberglem4 27233* Lemma 4 for konigsberg 27235: Vertices 0, 1, 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       {0, 1, 3} ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}

Theoremkonigsberglem5 27234* Lemma 5 for konigsberg 27235: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})

Theoremkonigsberg 27235 The Königsberg Bridge problem. If 𝐺 is the Königsberg graph, i.e. a graph on four vertices 0, 1, 2, 3, with edges {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 2}, {2, 3}, {2, 3}, then vertices 0, 1, 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eulerpath 27219 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (EulerPaths‘𝐺) = ∅

16.5  The Friendship Theorem

16.5.1  Friendship graphs - basics

Syntaxcfrgr 27236 Extend class notation with friendship graphs.
class FriendGraph

Definitiondf-frgr 27237* Define the class of all friendship graphs: a simple graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. This condition is called the friendship condition , see definition in [MertziosUnger] p. 152. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.) (Revised by AV, 29-Mar-2021.)
FriendGraph = {𝑔 ∣ (𝑔 ∈ USGraph ∧ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒)}

Theoremisfrgr 27238* The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))

Theoremfrgrusgrfrcond 27239* A friendship graph is a simple graph which fulfils the friendship condition. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

Theoremfrgrusgr 27240 A friendship graph is a simple graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
(𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

Theoremfrgr0v 27241 Any null graph (set with no vertices) is a friendship graph iff its edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ FriendGraph ↔ (iEdg‘𝐺) = ∅))

Theoremfrgr0vb 27242 Any null graph (without vertices and edges) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 29-Mar-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ FriendGraph )

Theoremfrgruhgr0v 27243 Any null graph (without vertices) represented as hypergraph is a friendship graph. (Contributed by AV, 29-Mar-2021.)
((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ FriendGraph )

Theoremfrgr0 27244 The null graph (graph without vertices) is a friendship graph. (Contributed by AV, 29-Mar-2021.)
∅ ∈ FriendGraph

Theoremrspc2vd 27245* Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class 𝐷 for the second set variable 𝑦 may depend on the first set variable 𝑥. (Contributed by AV, 29-Mar-2021.)
(𝑥 = 𝐴 → (𝜃𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐸)    &   (𝜑𝐵𝐸)       (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃𝜓))

Theoremfrcond1 27246* The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))

Theoremfrcond2 27247* The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))

Theoremfrgreu 27248* Variant of frcond2 27247: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))

Theoremfrcond3 27249* The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhood of a vertex and the neighborhood of a second, different vertex have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))

Theoremfrcond4 27250* The friendship condition, alternatively expressed by neighborhoods: in a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})

16.5.2  The friendship theorem for small graphs

Theoremfrgr1v 27251 Any graph with (at most) one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph )

Theoremnfrgr2v 27252 Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Proof shortened by Alexander van der Vekens, 13-Sep-2018.) (Revised by AV, 29-Mar-2021.)
(((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )

Theoremfrgr3vlem1 27253* Lemma 1 for frgr3v 27255. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))

Theoremfrgr3vlem2 27254* Lemma 2 for frgr3v 27255. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))

Theoremfrgr3v 27255 Any graph with three vertices which are completely connected with each other is a friendship graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))))

Theorem1vwmgr 27256* Every graph with one vertex (which may be connect with itself by (multiple) loops!) is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Revised by AV, 31-Mar-2021.)
((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))

Theorem3vfriswmgrlem 27257* Lemma for 3vfriswmgr 27258. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))

Theorem3vfriswmgr 27258* Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))

Theorem1to2vfriswmgr 27259* Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))

Theorem1to3vfriswmgr 27260* Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))

Theorem1to3vfriendship 27261* The friendship theorem for small graphs: In every friendship graph with one, two or three vertices, there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))

16.5.3  Theorems according to Mertzios and Unger

Theorem2pthfrgrrn 27262* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.) (Revised by AV, 1-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))

Theorem2pthfrgrrn2 27263* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 1-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐)))

Theorem2pthfrgr 27264* Between any two (different) vertices in a friendship graph, tere is a 2-path (simple path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 1-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓𝑝(𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑝 ∧ (#‘𝑓) = 2))

Theorem3cyclfrgrrn1 27265* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))

Theorem3cyclfrgrrn 27266* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))

Theorem3cyclfrgrrn2 27267* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))

Theorem3cyclfrgr 27268* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑣𝑉𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣))

Theorem4cycl2v2nb 27269 In a (maybe degenerated) 4-cycle, two vertice have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))

Theorem4cycl2vnunb 27270* In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)

Theoremn4cyclfrgr 27271 There is no 4-cycle in a friendship graph, see Proposition 1(a) of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
((𝐺 ∈ FriendGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (#‘𝐹) ≠ 4)

Theorem4cyclusnfrgr 27272 A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → 𝐺 ∉ FriendGraph ))

Theoremfrgrnbnb 27273 If two neighbors 𝑈 and 𝑊 of a vertex 𝑋 have a common neighbor 𝐴 in a friendship graph, then this common neighbor 𝐴 must be the vertex 𝑋. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.) (Proof shortened by AV, 13-Feb-2022.)
𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)       ((𝐺 ∈ FriendGraph ∧ (𝑈𝐷𝑊𝐷) ∧ 𝑈𝑊) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))

Theoremfrgrconngr 27274 A friendship graph is connected, see remark 1 in [MertziosUnger] p. 153 (after Proposition 1): "An arbitrary friendship graph has to be connected, ... ". (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 1-Apr-2021.)
(𝐺 ∈ FriendGraph → 𝐺 ∈ ConnGraph)

Theoremvdgn0frgrv2 27275 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0))

Theoremvdgn1frgrv2 27276 Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Theoremvdgn1frgrv3 27277* Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)

Theoremvdgfrgrgt2 27278 Any vertex in a friendship graph (with more than one vertex - then, actually, the graph must have at least three vertices, because otherwise, it would not be a friendship graph) has at least degree 2, see remark 3 in [MertziosUnger] p. 153 (after Proposition 1): "It follows that deg(v) >= 2 for every node v of a friendship graph". (Contributed by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 5-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → 2 ≤ ((VtxDeg‘𝐺)‘𝑁)))

16.5.4  Huneke's Proof of the Friendship Theorem

In this section, the friendship theorem friendship 27386 is proven by formalizing Huneke's proof, see [Huneke] pp. 1-2. The three claims (see frgrncvvdeq 27289, frgrregorufr 27305 and frrusgrord0 27320) and additional statements (numbered in the order of their occurence in the paper) in Huneke's proof are cited in the corresponding theorems.

Theoremfrgrncvvdeqlem1 27279 Lemma 1 for frgrncvvdeq 27289. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝑋𝑁)

Theoremfrgrncvvdeqlem2 27280* Lemma 2 for frgrncvvdeq 27289. In a friendship graph, for each neighbor of a vertex there is exactly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)

Theoremfrgrncvvdeqlem3 27281* Lemma 3 for frgrncvvdeq 27289. The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Theoremfrgrncvvdeqlem4 27282* Lemma 4 for frgrncvvdeq 27289. The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷𝑁)

Theoremfrgrncvvdeqlem5 27283* Lemma 5 for frgrncvvdeq 27289. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Theoremfrgrncvvdeqlem6 27284* Lemma 6 for frgrncvvdeq 27289. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸)

Theoremfrgrncvvdeqlem7 27285* Lemma 7 for frgrncvvdeq 27289. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)

Theoremfrgrncvvdeqlem8 27286* Lemma 8 for frgrncvvdeq 27289. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Revised by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷1-1𝑁)

Theoremfrgrncvvdeqlem9 27287* Lemma 9 for frgrncvvdeq 27289. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷onto𝑁)

Theoremfrgrncvvdeqlem10 27288* Lemma 10 for frgrncvvdeq 27289. (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷1-1-onto𝑁)

Theoremfrgrncvvdeq 27289* In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))

Theoremfrgrwopreglem4a 27290 In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 27295 without a fixed degree and without using the sets 𝐴 and 𝐵. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 4-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → {𝑋, 𝑌} ∈ 𝐸)

Theoremfrgrwopreglem5a 27291 If a friendship graph has two vertices with the same degree and two other vertices with different degrees, then there is a 4-cycle in the graph. Alternate version of frgrwopreglem5 27301 without a fixed degree and without using the sets 𝐴 and 𝐵. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 4-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ ((𝐴𝑉𝑋𝑉) ∧ (𝐵𝑉𝑌𝑉)) ∧ ((𝐷𝐴) = (𝐷𝑋) ∧ (𝐷𝐴) ≠ (𝐷𝐵) ∧ (𝐷𝑋) ≠ (𝐷𝑌))) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝑋} ∈ 𝐸) ∧ ({𝑋, 𝑌} ∈ 𝐸 ∧ {𝑌, 𝐴} ∈ 𝐸)))

Theoremfrgrwopreglem1 27292* Lemma 1 for frgrwopreg 27303: the classes 𝐴 and 𝐵 are sets. The definition of 𝐴 and 𝐵 corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       (𝐴 ∈ V ∧ 𝐵 ∈ V)

Theoremfrgrwopreglem2 27293* Lemma 2 for frgrwopreg 27303. If the set 𝐴 of vertices of degree 𝐾 is not empty in a friendship graph with at least two vertices, then 𝐾 must be greater than 1 . This is only an observation, which is not required for the proof the friendship theorem. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 2-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉) ∧ 𝐴 ≠ ∅) → 2 ≤ 𝐾)

Theoremfrgrwopreglem3 27294* Lemma 3 for frgrwopreg 27303. The vertices in the sets 𝐴 and 𝐵 have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 2-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))

Theoremfrgrwopreglem4 27295* Lemma 4 for frgrwopreg 27303. In a friendship graph each vertex with degree 𝐾 is connected with any vertex with degree other than 𝐾. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 4-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ 𝐸)

Theoremfrgrwopregasn 27296* According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". This version of frgrwopreg1 27298 is stricter (claiming that the singleton itself is a universal friend instead of claiming the existence of a universal friend only) and therefore closer to Huneke's statement. This strict variant, however, is not required for the proof of the friendship theorem. (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 4-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝐴 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)

Theoremfrgrwopregbsn 27297* According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". This version of frgrwopreg2 27299 is stricter (claiming that the singleton itself is a universal friend instead of claiming the existence of a universal friend only) and therefore closer to Huneke's statement. This strict variant, however, is not required for the proof of the friendship theorem. (Contributed by AV, 4-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝐵 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)

Theoremfrgrwopreg1 27298* According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Proof shortened by AV, 4-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (#‘𝐴) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theoremfrgrwopreg2 27299* According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Proof shortened by AV, 4-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (#‘𝐵) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theoremfrgrwopreglem5lem 27300* Lemma for frgrwopreglem5 27301. (Contributed by AV, 5-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))

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