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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | gpgov 48001* | 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 48002 | 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 48003* | 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 48004* | 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 | gpgvtxel 48005* | 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 48006 | 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 | gpgedgel 48007* | An edge in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-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 | gpgvtx0 48008 | 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 48009 | 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 48010 | 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 48011* | Lemma for gpgusgra 48012. (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 48012 | 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 | gpgorder 48013 | 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 48014 | 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 | 2ltceilhalf 48015 | The ceiling of half of an integer greater than 2 is greater than or equal to 2. (Contributed by AV, 4-Sep-2025.) |
| ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ≤ (⌈‘(𝑁 / 2))) | ||
| Theorem | ceilhalfelfzo1 48016 | A positive integer less than (the ceiling of) half of another integer is in the half-open range of positive integers up to the other integer. (Contributed by AV, 7-Sep-2025.) |
| ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^𝑁))) | ||
| Theorem | gpgedgvtx1lem 48017 | Lemma for gpgedgvtx1 48020. (Contributed by AV, 1-Sep-2025.) (Proof shortened by AV, 8-Sep-2025.) |
| ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑋 ∈ 𝐽) → 𝑋 ∈ 𝐼) | ||
| Theorem | 2tceilhalfelfzo1 48018 | Two times a positive integer less than (the ceiling of) half of another integer is less than the other integer. This theorem would hold even for integers less than 3, but then a corresponding 𝐾 would not exist. (Contributed by AV, 9-Sep-2025.) |
| ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (2 · 𝐾) < 𝑁) | ||
| Theorem | gpgedgvtx0 48019 | 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 48020 | 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 48021 | 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 48022 | 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 | gpg3nbgrvtxlem 48023 | Lemma for gpg3nbgrvtx0ALT 48033 and gpg3nbgrvtx1 48034. For this theorem, it is essential that 2 < 𝑁 and 𝐾 < (𝑁 / 2)! (Contributed by AV, 3-Sep-2025.) (Proof shortened by AV, 9-Sep-2025.) |
| ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))) ∧ 𝐴 ∈ (0..^𝑁)) → ((𝐴 + 𝐾) mod 𝑁) ≠ ((𝐴 − 𝐾) mod 𝑁)) | ||
| Theorem | gpg5nbgrvtx03starlem1 48024 | Lemma 1 for gpg5nbgrvtx03star 48036. (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 48025 | Lemma 2 for gpg5nbgrvtx03star 48036. (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 48026 | Lemma 3 for gpg5nbgrvtx03star 48036. (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 48027 | Lemma 1 for gpg5nbgr3star 48037. (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 48028 | Lemma 2 for gpg5nbgr3star 48037. (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 48029 | Lemma 3 for gpg5nbgr3star 48037. (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 48030 | 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 48031 | 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 48032 | 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 48033 |
In a generalized Petersen graph 𝐺, every outside vertex has exactly
three (different) neighbors. (Contributed by AV, 30-Aug-2025.)
The proof of gpg3nbgrvtx0 48032 can be shortened using lemma gpg3nbgrvtxlem 48023, but then theorem 2ltceilhalf 48015 is required which is based on an "example" ex-ceil 30467. 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 48034 | In a generalized Petersen graph 𝐺, every inside 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 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3) | ||
| Theorem | gpgcubic 48035 | 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 48038), 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 48036* | 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 48037* | 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 Peterson graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles, see gpg3kgrtriex 48045. (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 48038 | Every vertex in a generalized Petersen graph has degree 3. (Contributed by AV, 4-Sep-2025.) |
| ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑋) = 3) | ||
| Theorem | gpg3kgrtriexlem1 48039 | Lemma 1 for gpg3kgrtriex 48045. (Contributed by AV, 1-Oct-2025.) |
| ⊢ (𝐾 ∈ ℕ → 𝐾 < (⌈‘((3 · 𝐾) / 2))) | ||
| Theorem | gpg3kgrtriexlem2 48040 | Lemma 2 for gpg3kgrtriex 48045. (Contributed by AV, 1-Oct-2025.) |
| ⊢ 𝑁 = (3 · 𝐾) ⇒ ⊢ (𝐾 ∈ ℕ → (-𝐾 mod 𝑁) = (((𝐾 mod 𝑁) + 𝐾) mod 𝑁)) | ||
| Theorem | gpg3kgrtriexlem3 48041 | Lemma 3 for gpg3kgrtriex 48045. (Contributed by AV, 1-Oct-2025.) |
| ⊢ 𝑁 = (3 · 𝐾) ⇒ ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ (ℤ≥‘3)) | ||
| Theorem | gpg3kgrtriexlem4 48042 | Lemma 4 for gpg3kgrtriex 48045. (Contributed by AV, 1-Oct-2025.) |
| ⊢ 𝑁 = (3 · 𝐾) ⇒ ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) | ||
| Theorem | gpg3kgrtriexlem5 48043 | Lemma 5 for gpg3kgrtriex 48045. (Contributed by AV, 1-Oct-2025.) |
| ⊢ 𝑁 = (3 · 𝐾) ⇒ ⊢ (𝐾 ∈ ℕ → (𝐾 mod 𝑁) ≠ (-𝐾 mod 𝑁)) | ||
| Theorem | gpg3kgrtriexlem6 48044 | Lemma 6 for gpg3kgrtriex 48045: 𝐸 is an edge in the generalized Petersen graph G(N,K) with 𝑁 = 3 · 𝐾. (Contributed by AV, 1-Oct-2025.) |
| ⊢ 𝑁 = (3 · 𝐾) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝐸 = {〈1, (𝐾 mod 𝑁)〉, 〈1, (-𝐾 mod 𝑁)〉} ⇒ ⊢ (𝐾 ∈ ℕ → 𝐸 ∈ (Edg‘𝐺)) | ||
| Theorem | gpg3kgrtriex 48045* | All generalized Petersen graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles. (Contributed by AV, 1-Oct-2025.) |
| ⊢ 𝑁 = (3 · 𝐾) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) ⇒ ⊢ (𝐾 ∈ ℕ → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺)) | ||
| Theorem | gpg5gricstgr3 48046 | Each closed neighborhood in a generalized Petersen graph G(N,K) of order 10 (𝑁 = 5), which is either the Petersen graph G(5,2) or the 5-prism G(5,1), is isomorphic to a 3-star. (Contributed by AV, 13-Sep-2025.) |
| ⊢ 𝐺 = (5 gPetersenGr 𝐾) ⇒ ⊢ ((𝐾 ∈ (1...2) ∧ 𝑉 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑉)) ≃𝑔𝑟 (StarGr‘3)) | ||
| Theorem | gpg5grlic 48047 | The two generalized Petersen graphs G(N,K) of order 10 (𝑁 = 5), which are the Petersen graph G(5,2) and the 5-prism G(5,1), are locally isomorphic. (Contributed by AV, 29-Sep-2025.) |
| ⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2) | ||
| Theorem | 1hegrlfgr 48048* | A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) & ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) & ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) ⇒ ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) | ||
| Syntax | cupwlks 48049 | Extend class notation with walks (of a pseudograph). |
| class UPWalks | ||
| Definition | df-upwlks 48050* |
Define the set of all walks (in a pseudograph), called "simple walks"
in
the following.
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)." According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
| ⊢ UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
| Theorem | upwlksfval 48051* | The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
| Theorem | isupwlk 48052* | Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| Theorem | isupwlkg 48053* | Generalization of isupwlk 48052: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| Theorem | upwlkbprop 48054 | Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | ||
| Theorem | upwlkwlk 48055 | A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.) |
| ⊢ (𝐹(UPWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
| Theorem | upgrwlkupwlk 48056 | In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.) |
| ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃) | ||
| Theorem | upgrwlkupwlkb 48057 | In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.) |
| ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ 𝐹(UPWalks‘𝐺)𝑃)) | ||
| Theorem | upgrisupwlkALT 48058* | Alternate proof of upgriswlk 29659 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| Theorem | upgredgssspr 48059 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
| ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺))) | ||
| Theorem | uspgropssxp 48060* | The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of a Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 48070. (Contributed by AV, 24-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ⊆ (𝑊 × 𝑃)) | ||
| Theorem | uspgrsprfv 48061* | The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 48067. (Contributed by AV, 24-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) | ||
| Theorem | uspgrsprf 48062* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺⟶𝑃 | ||
| Theorem | uspgrsprf1 48063* | The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺–1-1→𝑃 | ||
| Theorem | uspgrsprfo 48064* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) | ||
| Theorem | uspgrsprf1o 48065* | The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 48070. (Contributed by AV, 25-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–1-1-onto→𝑃) | ||
| Theorem | uspgrex 48066* | The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V) | ||
| Theorem | uspgrbispr 48067* | There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑃) | ||
| Theorem | uspgrspren 48068* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑃) | ||
| Theorem | uspgrymrelen 48069* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 48070. (Contributed by AV, 27-Nov-2021.) |
| ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅) | ||
| Theorem | uspgrbisymrel 48070* |
There is a bijection between the "simple pseudographs" for a fixed
set
𝑉 of vertices and the class 𝑅 of the
symmetric relations on the
fixed set 𝑉. The simple pseudographs, which are
graphs without
hyper- or multiedges, but which may contain loops, are expressed as
ordered pairs of the vertices and the edges (as proper or improper
unordered pairs of vertices, not as indexed edges!) in this theorem.
That class 𝐺 of such simple pseudographs is a set
(if 𝑉 is a
set, see uspgrex 48066) of equivalence classes of graphs
abstracting from
the index sets of their edge functions.
Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ 〈𝑣, 𝑒〉 ∈ USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
| ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
| Theorem | uspgrbisymrelALT 48071* | Alternate proof of uspgrbisymrel 48070 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
| Theorem | ovn0dmfun 48072 | If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6949. (Contributed by AV, 27-Jan-2020.) |
| ⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) | ||
| Theorem | xpsnopab 48073* | A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
| ⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} | ||
| Theorem | xpiun 48074* | A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.) |
| ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | ||
| Theorem | ovn0ssdmfun 48075* | If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6949. (Contributed by AV, 27-Jan-2020.) |
| ⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) | ||
| Theorem | fnxpdmdm 48076 | The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.) |
| ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴) | ||
| Theorem | cnfldsrngbas 48077 | The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
| ⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ⊆ ℂ → 𝑆 = (Base‘𝑅)) | ||
| Theorem | cnfldsrngadd 48078 | The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
| ⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → + = (+g‘𝑅)) | ||
| Theorem | cnfldsrngmul 48079 | The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
| ⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → · = (.r‘𝑅)) | ||
| Theorem | plusfreseq 48080 | If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
| Theorem | mgmplusfreseq 48081 | If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
| Theorem | 0mgm 48082 | A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.) |
| ⊢ (Base‘𝑀) = ∅ ⇒ ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) | ||
| Theorem | opmpoismgm 48083* | A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑀 ∈ Mgm) | ||
| Theorem | copissgrp 48084* | A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑀 ∈ Smgrp) | ||
| Theorem | copisnmnd 48085* | A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 1 < (♯‘𝐵)) ⇒ ⊢ (𝜑 → 𝑀 ∉ Mnd) | ||
| Theorem | 0nodd 48086* | 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 0 ∉ 𝑂 | ||
| Theorem | 1odd 48087* | 1 is an odd integer. (Contributed by AV, 3-Feb-2020.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 1 ∈ 𝑂 | ||
| Theorem | 2nodd 48088* | 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 2 ∉ 𝑂 | ||
| Theorem | oddibas 48089* | Lemma 1 for oddinmgm 48091: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ 𝑂 = (Base‘𝑀) | ||
| Theorem | oddiadd 48090* | Lemma 2 for oddinmgm 48091: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ + = (+g‘𝑀) | ||
| Theorem | oddinmgm 48091* | The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 48166, and even a non-unital ring, see 2zrng 48157. (Contributed by AV, 3-Feb-2020.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ 𝑀 ∉ Mgm | ||
| Theorem | nnsgrpmgm 48092 | The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.) |
| ⊢ 𝑀 = (ℂfld ↾s ℕ) ⇒ ⊢ 𝑀 ∈ Mgm | ||
| Theorem | nnsgrp 48093 | The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.) |
| ⊢ 𝑀 = (ℂfld ↾s ℕ) ⇒ ⊢ 𝑀 ∈ Smgrp | ||
| Theorem | nnsgrpnmnd 48094 | The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.) |
| ⊢ 𝑀 = (ℂfld ↾s ℕ) ⇒ ⊢ 𝑀 ∉ Mnd | ||
| Theorem | nn0mnd 48095 | The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: 𝑀 could have also been written as (ℂfld ↾s ℕ0). (Contributed by AV, 27-Dec-2023.) |
| ⊢ 𝑀 = {〈(Base‘ndx), ℕ0〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ 𝑀 ∈ Mnd | ||
| Theorem | gsumsplit2f 48096* | Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) | ||
| Theorem | gsumdifsndf 48097* | Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.) |
| ⊢ Ⅎ𝑘𝑌 & ⊢ Ⅎ𝑘𝜑 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) | ||
| Theorem | gsumfsupp 48098 | A group sum of a family can be restricted to the support of that family without changing its value, provided that that support is finite. This corresponds to the definition of an (infinite) product in [Lang] p. 5, last two formulas. (Contributed by AV, 27-Dec-2023.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (𝐹 supp 0 ) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐼)) = (𝐺 Σg 𝐹)) | ||
With df-mpo 7436, binary operations are defined by a rule, and with df-ov 7434, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation 7434 (19-Jan-2020), "... a binary operation on a set 𝑆 is a mapping of the elements of the Cartesian product 𝑆 × 𝑆 to S: 𝑓:𝑆 × 𝑆⟶𝑆. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, we call binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 internal binary operations, see df-intop 48115. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 48116. Therefore, a "binary operation on a set 𝑆 " according to Wikipedia is a "closed internal binary operation" in our terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations 48116 ). Taking a step back, we define "laws" applicable for "binary operations" (which even need not to be functions), according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7. These laws are used, on the one hand, to specialize internal binary operations (see df-clintop 48116 and df-assintop 48117), and on the other hand to define the common algebraic structures like magmas, groups, rings, etc. Internal binary operations, which obey these laws, are defined afterwards. Notice that in [BourbakiAlg1] p. 1, p. 4 and p. 7, these operations are called "laws" by themselves. In the following, an alternate definition df-cllaw 48102 for an internal binary operation is provided, which does not require function-ness, but only closure. Therefore, this definition could be used as binary operation (Slot 2) defined for a magma as extensible structure, see mgmplusgiopALT 48110, or for an alternate definition df-mgm2 48135 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups). | ||
In this subsection, the "laws" applicable for "binary operations" according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7 are defined. These laws are called "internal laws" in [BourbakiAlg1] p. xxi. | ||
| Syntax | ccllaw 48099 | Extend class notation for the closure law. |
| class clLaw | ||
| Syntax | casslaw 48100 | Extend class notation for the associative law. |
| class assLaw | ||
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