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Type | Label | Description |
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Statement | ||
Theorem | 0Cycl 41301 | A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) |
⊢ ((𝐺 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍) → (∅(CycleS‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶(Vtx‘𝐺))) | ||
Theorem | 1pthdlem1 41302 | Lemma 1 for 1pthd 41310. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 ⇒ ⊢ Fun ◡(𝑃 ↾ (1..^(#‘𝐹))) | ||
Theorem | 1pthdlem2 41303 | Lemma 2 for 1pthd 41310. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 ⇒ ⊢ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ | ||
Theorem | 11wlkdlem1 41304 | Lemma 1 for 11wlkd 41308. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑃:(0...(#‘𝐹))⟶𝑉) | ||
Theorem | 11wlkdlem2 41305 | Lemma 2 for 11wlkd 41308. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐼‘𝐽)) | ||
Theorem | 11wlkdlem3 41306 | Lemma 3 for 11wlkd 41308. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) ⇒ ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) | ||
Theorem | 11wlkdlem4 41307* | Lemma 4 for 11wlkd 41308. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) | ||
Theorem | 11wlkd 41308 | In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃) | ||
Theorem | 1trld 41309 | In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) | ||
Theorem | 1pthd 41310 | In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(PathS‘𝐺)𝑃) | ||
Theorem | 1pthond 41311 | In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) | ||
Theorem | upgr11wlkdlem1 41312 | Lemma 1 for upgr11wlkd 41314. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) ⇒ ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋}) | ||
Theorem | upgr11wlkdlem2 41313 | Lemma 2 for upgr11wlkd 41314. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) ⇒ ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽)) | ||
Theorem | upgr11wlkd 41314 | In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph ) ⇒ ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃) | ||
Theorem | upgr1trld 41315 | In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph ) ⇒ ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) | ||
Theorem | upgr1pthd 41316 | In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph ) ⇒ ⊢ (𝜑 → 𝐹(PathS‘𝐺)𝑃) | ||
Theorem | upgr1pthond 41317 | In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph ) ⇒ ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) | ||
Theorem | lppthon 41318 | A loop (which is an edge at index 𝐽) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(𝐴(PathsOn‘𝐺)𝐴)〈“𝐴𝐴”〉) | ||
Theorem | lp1cycl 41319 | A loop (which is an edge at index 𝐽) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(CycleS‘𝐺)〈“𝐴𝐴”〉) | ||
Theorem | 1pthon2v-av 41320* | For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝) | ||
Theorem | 1pthon2ve 41321* | For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Proof shortened by AV, 15-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝) | ||
Theorem | 1wlk2v2elem1 41322 | Lemma 1 for 1wlk2v2e 41324: 𝐹 is a length 2 word of over {0}, the domain of the singleton word 𝐼. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 & ⊢ 𝐹 = 〈“00”〉 ⇒ ⊢ 𝐹 ∈ Word dom 𝐼 | ||
Theorem | 1wlk2v2elem2 41323* | Lemma 2 for 1wlk2v2e 41324: The values of 𝐼 after 𝐹 are edges between two vertices enumerated by 𝑃. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 & ⊢ 𝐹 = 〈“00”〉 & ⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 ⇒ ⊢ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} | ||
Theorem | 1wlk2v2e 41324 | In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that 𝐺 is a simple graph (without loops) only if 𝑋 ≠ 𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) |
⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 & ⊢ 𝐹 = 〈“00”〉 & ⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 & ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 ⇒ ⊢ 𝐹(1Walks‘𝐺)𝑃 | ||
Theorem | ntrl2v2e 41325 | A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see 1wlk2v2e 41324, but not a trail. Notice that 𝐺 is a simple graph (without loops) only if 𝑋 ≠ 𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) |
⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 & ⊢ 𝐹 = 〈“00”〉 & ⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 & ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 ⇒ ⊢ ¬ 𝐹(TrailS‘𝐺)𝑃 | ||
Theorem | 31wlkdlem1 41326 | Lemma 1 for 31wlkd 41337. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 ⇒ ⊢ (#‘𝑃) = ((#‘𝐹) + 1) | ||
Theorem | 31wlkdlem2 41327 | Lemma 2 for 31wlkd 41337. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 ⇒ ⊢ (0..^(#‘𝐹)) = {0, 1, 2} | ||
Theorem | 31wlkdlem3 41328 | Lemma 3 for 31wlkd 41337. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) ⇒ ⊢ (𝜑 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷))) | ||
Theorem | 31wlkdlem4 41329* | Lemma 4 for 31wlkd 41337. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0...(#‘𝐹))(𝑃‘𝑘) ∈ 𝑉) | ||
Theorem | 31wlkdlem5 41330* | Lemma 5 for 31wlkd 41337. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | ||
Theorem | 3pthdlem1 41331* | Lemma 1 for 3pthd 41341. (Contributed by AV, 9-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^(#‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗))) | ||
Theorem | 31wlkdlem6 41332 | Lemma 6 for 31wlkd 41337. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → (𝐴 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐿))) | ||
Theorem | 31wlkdlem7 41333 | Lemma 7 for 31wlkd 41337. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V)) | ||
Theorem | 31wlkdlem8 41334 | Lemma 8 for 31wlkd 41337. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿)) | ||
Theorem | 31wlkdlem9 41335 | Lemma 9 for 31wlkd 41337. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2)))) | ||
Theorem | 31wlkdlem10 41336* | Lemma 10 for 31wlkd 41337. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) | ||
Theorem | 31wlkd 41337 | Construction of a walk from two given edges in a graph. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃) | ||
Theorem | 31wlkond 41338 | A 1-walk of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3trld 41339 | Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) | ||
Theorem | 3trlond 41340 | A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(𝐴(TrailsOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3pthd 41341 | A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(PathS‘𝐺)𝑃) | ||
Theorem | 3pthond 41342 | A path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(𝐴(PathsOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3spthd 41343 | A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 ≠ 𝐷) ⇒ ⊢ (𝜑 → 𝐹(SPathS‘𝐺)𝑃) | ||
Theorem | 3spthond 41344 | A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 ≠ 𝐷) ⇒ ⊢ (𝜑 → 𝐹(𝐴(SPathsOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3cycld 41345 | Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 = 𝐷) ⇒ ⊢ (𝜑 → 𝐹(CycleS‘𝐺)𝑃) | ||
Theorem | 3cyclpd 41346 | Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴)) | ||
Theorem | upgr3v3e3cycl 41347* | If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))) | ||
Theorem | uhgr3cyclexlem 41348 | Lemma for uhgr3cyclex 41349. (Contributed by AV, 12-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)))) → 𝐽 ≠ 𝐾) | ||
Theorem | uhgr3cyclex 41349* | If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)) | ||
Theorem | umgr3cyclex 41350* | If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)) | ||
Theorem | umgr3v3e3cycl 41351* | If and only if there is a 3-cycle in a multigraph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.) (Revised by AV, 12-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ UMGraph → (∃𝑓∃𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))) | ||
Theorem | upgr4cycl4dv4e 41352* | If there is a cycle of length 4 in a pseudograph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 13-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ∧ ({𝑐, 𝑑} ∈ 𝐸 ∧ {𝑑, 𝑎} ∈ 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))) | ||
Syntax | cconngr 41353 | Extend class notation with connected graphs. |
class ConnGraph | ||
Definition | df-conngr 41354* | Define the class of all connected graphs. A graph is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv-av 41297 and dfconngr1 41355. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} | ||
Theorem | dfconngr1 41355* | Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ (𝑣 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} | ||
Theorem | isconngr 41356* | The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) | ||
Theorem | isconngr1 41357* | The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) | ||
Theorem | cusconngr 41358 | A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ((𝐺 ∈ UHGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ ConnGraph) | ||
Theorem | 0conngr 41359 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ∅ ∈ ConnGraph | ||
Theorem | 0vconngr 41360 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph) | ||
Theorem | 1conngr 41361 | A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ ConnGraph) | ||
Theorem | conngrv2edg 41362* | A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrav2 26551. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ ConnGraph ∧ 𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉)) → ∃𝑒 ∈ ran 𝐼 𝑁 ∈ 𝑒) | ||
Theorem | vdn0conngrumgrv2 41363 | A vertex in a connected multigraph with more than one vertex cannot have degree 0. Formerly vdgn0frgrav2 26551. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0) | ||
According to Wikipedia ("Eulerian path", 9-Mar-2021, https://en.wikipedia.org/wiki/Eulerian_path): "In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. ... The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. ... A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian." Correspondingly, an Eulerian path is defined as "a trail containing all edges" (see definition in [Bollobas] p. 16) in df-eupth 41365 resp. iseupth 41368. (EulerPaths‘𝐺) is the set of all Eulerian paths in graph 𝐺, see eupths 41367. An Eulerian circuit (called Euler tour in the definition in [Diestel] p. 22) is "a circuit in a graph containing all the edges" (see definition in [Bollobas] p. 16), or, with other words, a circuit which is an Eulerian path. The function mapping a graph to the set of its Eulerian paths is defined as EulerPaths in df-eupth 41365, whereas there is no explicit definition for Eulerian circuits (yet): The statement "〈𝐹, 𝑃〉 is an Eulerian circuit" is formally expressed by (𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝐹(CircuitS‘𝐺)𝑃). Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 41385). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 41413. An Eulerian path does not have to be a path in the meaning of definition df-pths 40923, because it may traverse some vertices more than once. Therefore, "Eulerian trail" would be a more appropriate name. The main result of this section is (one direction of) Euler's Theorem: "A non-trivial connected graph has an Euler[ian] circuit iff each vertex has even degree." (see part 1 of theorem 12 in [Bollobas] p. 16 and theorem 1.8.1 in [Diestel] p. 22) or, expressed with Eulerian paths: "A connected graph has an Euler[ian] trail from a vertex x to a vertex y (not equal with x) iff x and y are the only vertices of odd degree." (see part 2 of theorem 12 in [Bollobas] p. 17). In eulerpath 41409, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 41410 shows that a pseudograph with an Eulerian circuit has only vertices of even degree. | ||
Syntax | ceupth 41364 | Extend class notation with Eulerian paths. |
class EulerPaths | ||
Definition | df-eupth 41365* | Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ 𝑓:(0..^(#‘𝑓))–onto→dom (iEdg‘𝑔))}) | ||
Theorem | releupth 41366 | The set (EulerPaths‘𝐺) of all Eulerian paths on 𝐺 is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ Rel (EulerPaths‘𝐺) | ||
Theorem | eupths 41367* | The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑋 → (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝐺)𝑝 ∧ 𝑓:(0..^(#‘𝑓))–onto→dom 𝐼)}) | ||
Theorem | iseupth 41368 | The property "〈𝐹, 𝑃〉 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌 ∧ 𝑃 ∈ 𝑍) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃 ∧ 𝐹:(0..^(#‘𝐹))–onto→dom 𝐼))) | ||
Theorem | iseupthf1o 41369 | The property "〈𝐹, 𝑃〉 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌 ∧ 𝑃 ∈ 𝑍) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼))) | ||
Theorem | eupthtrli 41370 | Properties of an Eulerian path as a trail. (Contributed by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 ∧ 𝐹:(0..^(#‘𝐹))–onto→dom 𝐼)) | ||
Theorem | eupthi 41371 | Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(1Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼)) | ||
Theorem | eupthf1o 41372 | The 𝐹 function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼) | ||
Theorem | eupthfi 41373 | Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → dom 𝐼 ∈ Fin) | ||
Theorem | eupthseg 41374 | The 𝑁-th edge in an eulerian path is the edge having 𝑃(𝑁) and 𝑃(𝑁 + 1) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑁 ∈ (0..^(#‘𝐹))) → {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁))) | ||
Theorem | upgriseupth 41375* | The property "〈𝐹, 𝑃〉 is an Eulerian path on the pseudograph 𝐺". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upgreupthi 41376* | Properties of an Eulerian path in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) | ||
Theorem | upgreupthseg 41377 | The 𝑁-th edge in an eulerian path is the edge from 𝑃(𝑁) to 𝑃(𝑁 + 1). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑁 ∈ (0..^(#‘𝐹))) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) | ||
Theorem | eupthcl 41378 | An Eulerian path has length #(𝐹), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0) | ||
Theorem | eupthistrl 41379 | An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(TrailS‘𝐺)𝑃) | ||
Theorem | eupthis1wlk 41380 | An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃) | ||
Theorem | eupthpf 41381 | The 𝑃 function in an Eulerian path is a function from a finite sequence of nonnegative integers to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) | ||
Theorem | eupth0 41382 | There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → ∅(EulerPaths‘𝐺){〈0, 𝐴〉}) | ||
Theorem | eupthres 41383 | The restriction 〈𝐻, 𝑄〉 of an Eulerian path 〈𝐹, 𝑃〉 to an initial segment of the path (of length 𝑁) forms an Eulerian path on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) & ⊢ (Vtx‘𝑆) = 𝑉 ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
Theorem | eupthp1 41384 | Append one path segment to an Eulerian path 〈𝐹, 𝑃〉 to become an Eulerian path 〈𝐻, 𝑄〉 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝑁 = (#‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉}) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
Theorem | eupth2eucrct 41385 | Append one path segment to an Eulerian path 〈𝐹, 𝑃〉 which may not be an (Eulerian) circuit to become an Eulerian circuit 〈𝐻, 𝑄〉 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by AV, 11-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝑁 = (#‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉}) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) & ⊢ (𝜑 → 𝐶 = (𝑃‘0)) ⇒ ⊢ (𝜑 → (𝐻(EulerPaths‘𝑆)𝑄 ∧ 𝐻(CircuitS‘𝑆)𝑄)) | ||
Theorem | eupth2lem1 41386 | TODO-AV: Duplicate of eupath2lem1 26504! Lemma for eupath2 26507. (Contributed by Mario Carneiro, 8-Apr-2015.) |
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))) | ||
Theorem | eupth2lem2 41387 | TODO-AV: Duplicate of eupath2lem2 26505! Lemma for eupath2 26507. (Contributed by Mario Carneiro, 8-Apr-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}))) | ||
Theorem | trlsegvdeglem1 41388 | Lemma for trlsegvdeg 41395. (Contributed by AV, 20-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) ⇒ ⊢ (𝜑 → ((𝑃‘𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉)) | ||
Theorem | trlsegvdeglem2 41389 | Lemma for trlsegvdeg 41395. (Contributed by AV, 20-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → Fun (iEdg‘𝑋)) | ||
Theorem | trlsegvdeglem3 41390 | Lemma for trlsegvdeg 41395. (Contributed by AV, 20-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → Fun (iEdg‘𝑌)) | ||
Theorem | trlsegvdeglem4 41391 | Lemma for trlsegvdeg 41395. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) | ||
Theorem | trlsegvdeglem5 41392 | Lemma for trlsegvdeg 41395. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) | ||
Theorem | trlsegvdeglem6 41393 | Lemma for trlsegvdeg 41395. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) | ||
Theorem | trlsegvdeglem7 41394 | Lemma for trlsegvdeg 41395. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) | ||
Theorem | trlsegvdeg 41395 | Formerly part of proof of eupath2lem3 26506: 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 1-walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the 1-walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the 1-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 1-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) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) | ||
Theorem | eupth2lem3lem1 41396 | Lemma for eupth2lem3 41404. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0) | ||
Theorem | eupth2lem3lem2 41397 | Lemma for eupth2lem3 41404. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ0) | ||
Theorem | eupth2lem3lem3 41398* | Lemma for eupth2lem3 41404, formerly part of proof of eupath2lem3 26506: If a loop {(𝑃‘𝑁), (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) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) ⇒ ⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | ||
Theorem | eupth2lem3lem4 41399* | Lemma for eupth2lem3 41404, formerly part of proof of eupath2lem3 26506: 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) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) & ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ 𝒫 𝑉) ⇒ ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | ||
Theorem | eupth2lem3lem5 41400* | Lemma for eupath2 26507. (Contributed by AV, 25-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(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) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) ⇒ ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ 𝒫 𝑉) |
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