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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | acycgrsubgr 34901 | The subgraph of an acyclic graph is also acyclic. (Contributed by BTernaryTau, 23-Oct-2023.) |
⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ AcyclicGraph) | ||
Axiom | ax-7d 34902* | Distinct variable version of ax-11 2146. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Axiom | ax-8d 34903* | Distinct variable version of ax-7 2003. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Axiom | ax-9d1 34904 | Distinct variable version of ax-6 1963, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥 | ||
Axiom | ax-9d2 34905* | Distinct variable version of ax-6 1963, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Axiom | ax-10d 34906* | Distinct variable version of axc11n 2419. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Axiom | ax-11d 34907* | Distinct variable version of ax-12 2166. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | quartfull 34908 | The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)) ≠ 0) & ⊢ (𝜑 → -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3) ≠ 0) ⇒ ⊢ (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = 0 ↔ ((𝑋 = ((-(𝐴 / 4) − ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) − ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))) ∨ (𝑋 = ((-(𝐴 / 4) + ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) + ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))))))) | ||
Theorem | deranglem 34909* | Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ (𝐴 ∈ Fin → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ 𝜑)} ∈ Fin) | ||
Theorem | derangval 34910* | Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)})) | ||
Theorem | derangf 34911* | The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ 𝐷:Fin⟶ℕ0 | ||
Theorem | derang0 34912* | The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ (𝐷‘∅) = 1 | ||
Theorem | derangsn 34913* | The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐷‘{𝐴}) = 0) | ||
Theorem | derangenlem 34914* | One half of derangen 34915. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) ≤ (𝐷‘𝐵)) | ||
Theorem | derangen 34915* | The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) = (𝐷‘𝐵)) | ||
Theorem | subfacval 34916* | The subfactorial is defined as the number of derangements (see derangval 34910) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁))) | ||
Theorem | derangen2 34917* | Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (𝑆‘(♯‘𝐴))) | ||
Theorem | subfacf 34918* | The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ 𝑆:ℕ0⟶ℕ0 | ||
Theorem | subfaclefac 34919* | The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) ≤ (!‘𝑁)) | ||
Theorem | subfac0 34920* | The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑆‘0) = 1 | ||
Theorem | subfac1 34921* | The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑆‘1) = 0 | ||
Theorem | subfacp1lem1 34922* | Lemma for subfacp1 34929. The set 𝐾 together with {1, 𝑀} partitions the set 1...(𝑁 + 1). (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) ⇒ ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1))) | ||
Theorem | subfacp1lem2a 34923* | Lemma for subfacp1 34929. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐹 = (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) & ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾) ⇒ ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1)) | ||
Theorem | subfacp1lem2b 34924* | Lemma for subfacp1 34929. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐹 = (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) & ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋)) | ||
Theorem | subfacp1lem3 34925* | Lemma for subfacp1 34929. In subfacp1lem6 34928 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements that satisfy this for fixed 𝑀 = (𝑓‘1) is in bijection with 𝑁 − 1 derangements, by simply dropping the 𝑥 = 1 and 𝑥 = 𝑀 points from the function to get a derangement on 𝐾 = (1...(𝑁 − 1)) ∖ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)} & ⊢ 𝐶 = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)} ⇒ ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1))) | ||
Theorem | subfacp1lem4 34926* | Lemma for subfacp1 34929. The function 𝐹, which swaps 1 with 𝑀 and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)} & ⊢ 𝐹 = (( I ↾ 𝐾) ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ⇒ ⊢ (𝜑 → ◡𝐹 = 𝐹) | ||
Theorem | subfacp1lem5 34927* | Lemma for subfacp1 34929. In subfacp1lem6 34928 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements with (𝑓‘(𝑓‘1)) ≠ 1 for fixed 𝑀 = (𝑓‘1) is in bijection with derangements of 2...(𝑁 + 1), because pre-composing with the function 𝐹 swaps 1 and 𝑀 and turns the function into a bijection with (𝑓‘1) = 1 and (𝑓‘𝑥) ≠ 𝑥 for all other 𝑥, so dropping the point at 1 yields a derangement on the 𝑁 remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)} & ⊢ 𝐹 = (( I ↾ 𝐾) ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) & ⊢ 𝐶 = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⇒ ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘𝑁)) | ||
Theorem | subfacp1lem6 34928* | Lemma for subfacp1 34929. By induction, we cut up the set of all derangements on 𝑁 + 1 according to the 𝑁 possible values of (𝑓‘1) (since (𝑓‘1) ≠ 1), and for each set for fixed 𝑀 = (𝑓‘1), the subset of derangements with (𝑓‘𝑀) = 1 has size 𝑆(𝑁 − 1) (by subfacp1lem3 34925), while the subset with (𝑓‘𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 34927). Adding it all up yields the desired equation 𝑁(𝑆(𝑁) + 𝑆(𝑁 − 1)) for the number of derangements on 𝑁 + 1. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⇒ ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) | ||
Theorem | subfacp1 34929* | A two-term recurrence for the subfactorial. This theorem allows to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 34920, subfac1 34921. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) | ||
Theorem | subfacval2 34930* | A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘)))) | ||
Theorem | subfaclim 34931* | The subfactorial converges rapidly to 𝑁! / e. This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ → (abs‘(((!‘𝑁) / e) − (𝑆‘𝑁))) < (1 / 𝑁)) | ||
Theorem | subfacval3 34932* | Another closed form expression for the subfactorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝑆‘𝑁) = (⌊‘(((!‘𝑁) / e) + (1 / 2)))) | ||
Theorem | derangfmla 34933* | The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (𝐷‘𝐴) = (⌊‘(((!‘(♯‘𝐴)) / e) + (1 / 2)))) | ||
Theorem | erdszelem1 34934* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⇒ ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)) | ||
Theorem | erdszelem2 34935* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⇒ ⊢ ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ) | ||
Theorem | erdszelem3 34936* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) ⇒ ⊢ (𝐴 ∈ (1...𝑁) → (𝐾‘𝐴) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), ℝ, < )) | ||
Theorem | erdszelem4 34937* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → {𝐴} ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) | ||
Theorem | erdszelem5 34938* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)})) | ||
Theorem | erdszelem6 34939* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ ⇒ ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ) | ||
Theorem | erdszelem7 34940* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ & ⊢ (𝜑 → 𝐴 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → ¬ (𝐾‘𝐴) ∈ (1...(𝑅 − 1))) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)))) | ||
Theorem | erdszelem8 34941* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ & ⊢ (𝜑 → 𝐴 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ((𝐾‘𝐴) = (𝐾‘𝐵) → ¬ (𝐹‘𝐴)𝑂(𝐹‘𝐵))) | ||
Theorem | erdszelem9 34942* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ 〈(𝐼‘𝑛), (𝐽‘𝑛)〉) ⇒ ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ)) | ||
Theorem | erdszelem10 34943* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ 〈(𝐼‘𝑛), (𝐽‘𝑛)〉) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) | ||
Theorem | erdszelem11 34944* | Lemma for erdsze 34945. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ 〈(𝐼‘𝑛), (𝐽‘𝑛)〉) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | erdsze 34945* | The Erdős-Szekeres theorem. For any injective sequence 𝐹 on the reals of length at least (𝑅 − 1) · (𝑆 − 1) + 1, there is either a subsequence of length at least 𝑅 on which 𝐹 is increasing (i.e. a < , < order isomorphism) or a subsequence of length at least 𝑆 on which 𝐹 is decreasing (i.e. a < , ◡ < order isomorphism, recalling that ◡ < is the "greater than" relation). This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | erdsze2lem1 34946* | Lemma for erdsze2 34948. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1)) & ⊢ (𝜑 → 𝑁 < (♯‘𝐴)) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))) | ||
Theorem | erdsze2lem2 34947* | Lemma for erdsze2 34948. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1)) & ⊢ (𝜑 → 𝑁 < (♯‘𝐴)) & ⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))–1-1→𝐴) & ⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺)) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | erdsze2 34948* | Generalize the statement of the Erdős-Szekeres theorem erdsze 34945 to "sequences" indexed by an arbitrary subset of ℝ, which can be infinite. This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (♯‘𝐴)) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | kur14lem1 34949 | Lemma for kur14 34959. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ 𝐴 ⊆ 𝑋 & ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇 & ⊢ (𝐾‘𝐴) ∈ 𝑇 ⇒ ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) | ||
Theorem | kur14lem2 34950 | Lemma for kur14 34959. Write interior in terms of closure and complement: 𝑖𝐴 = 𝑐𝑘𝑐𝐴 where 𝑐 is complement and 𝑘 is closure. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) | ||
Theorem | kur14lem3 34951 | Lemma for kur14 34959. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝐾‘𝐴) ⊆ 𝑋 | ||
Theorem | kur14lem4 34952 | Lemma for kur14 34959. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 | ||
Theorem | kur14lem5 34953 | Lemma for kur14 34959. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴) | ||
Theorem | kur14lem6 34954 | Lemma for kur14 34959. If 𝑘 is the complementation operator and 𝑘 is the closure operator, this expresses the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴 for any subset 𝐴 of the topological space. This is the key result that lets us cut down long enough sequences of 𝑐𝑘𝑐𝑘... that arise when applying closure and complement repeatedly to 𝐴, and explains why we end up with a number as large as 14, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) ⇒ ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵) | ||
Theorem | kur14lem7 34955 | Lemma for kur14 34959: main proof. The set 𝑇 here contains all the distinct combinations of 𝑘 and 𝑐 that can arise, and we prove here that applying 𝑘 or 𝑐 to any element of 𝑇 yields another element of 𝑇. In operator shorthand, we have 𝑇 = {𝐴, 𝑐𝐴, 𝑘𝐴 , 𝑐𝑘𝐴, 𝑘𝑐𝐴, 𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝑐𝐴}. From the identities 𝑐𝑐𝐴 = 𝐴 and 𝑘𝑘𝐴 = 𝑘𝐴, we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴, proved in kur14lem6 34954. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) & ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴)) & ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴)) & ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) ⇒ ⊢ (𝑁 ∈ 𝑇 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) | ||
Theorem | kur14lem8 34956 | Lemma for kur14 34959. Show that the set 𝑇 contains at most 14 elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of 14 is tight in the sense that there exist topological spaces and subsets of these spaces for which all 14 generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) & ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴)) & ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴)) & ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) ⇒ ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14) | ||
Theorem | kur14lem9 34957* | Lemma for kur14 34959. Since the set 𝑇 is closed under closure and complement, it contains the minimal set 𝑆 as a subset, so 𝑆 also has at most 14 elements. (Indeed 𝑆 = 𝑇, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) & ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴)) & ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴)) & ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) & ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⇒ ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) | ||
Theorem | kur14lem10 34958* | Lemma for kur14 34959. Discharge the set 𝑇. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) | ||
Theorem | kur14 34959* | Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)) | ||
Syntax | cretr 34960 | Extend class notation with the retract relation. |
class Retr | ||
Definition | df-retr 34961* | Define the set of retractions on two topological spaces. We say that 𝑅 is a retraction from 𝐽 to 𝐾. or 𝑅 ∈ (𝐽 Retr 𝐾) iff there is an 𝑆 such that 𝑅:𝐽⟶𝐾, 𝑆:𝐾⟶𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅 ∘ 𝑆 is homotopic to the identity map. If a retraction exists, we say 𝐽 is a retract of 𝐾. (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}) | ||
Syntax | cpconn 34962 | Extend class notation with the class of path-connected topologies. |
class PConn | ||
Syntax | csconn 34963 | Extend class notation with the class of simply connected topologies. |
class SConn | ||
Definition | df-pconn 34964* | Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the closed unit interval) that goes from 𝑥 to 𝑦 for any points 𝑥, 𝑦 in the space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} | ||
Definition | df-sconn 34965* | Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))} | ||
Theorem | ispconn 34966* | The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) | ||
Theorem | pconncn 34967* | The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)) | ||
Theorem | pconntop 34968 | A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) | ||
Theorem | issconn 34969* | The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | ||
Theorem | sconnpconn 34970 | A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) | ||
Theorem | sconntop 34971 | A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top) | ||
Theorem | sconnpht 34972 | A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) | ||
Theorem | cnpconn 34973 | An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn) | ||
Theorem | pconnconn 34974 | A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn) | ||
Theorem | txpconn 34975 | The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn) | ||
Theorem | ptpconn 34976 | The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn) | ||
Theorem | indispconn 34977 | The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ {∅, 𝐴} ∈ PConn | ||
Theorem | connpconn 34978 | A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) |
⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ PConn) | ||
Theorem | qtoppconn 34979 | A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ PConn) | ||
Theorem | pconnpi1 34980 | All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑃 = (𝐽 π1 𝐴) & ⊢ 𝑄 = (𝐽 π1 𝐵) & ⊢ 𝑆 = (Base‘𝑃) & ⊢ 𝑇 = (Base‘𝑄) ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑃 ≃𝑔 𝑄) | ||
Theorem | sconnpht2 34981 | Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ (𝜑 → 𝐽 ∈ SConn) & ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) & ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1)) ⇒ ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺) | ||
Theorem | sconnpi1 34982 | A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) → (𝐽 ∈ SConn ↔ (Base‘(𝐽 π1 𝑌)) ≈ 1o)) | ||
Theorem | txsconnlem 34983 | Lemma for txsconn 34984. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ (𝜑 → 𝑅 ∈ Top) & ⊢ (𝜑 → 𝑆 ∈ Top) & ⊢ (𝜑 → 𝐹 ∈ (II Cn (𝑅 ×t 𝑆))) & ⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) & ⊢ 𝐵 = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) & ⊢ (𝜑 → 𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)}))) & ⊢ (𝜑 → 𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)}))) ⇒ ⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) | ||
Theorem | txsconn 34984 | The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn) | ||
Theorem | cvxpconn 34985* | A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11225. (Revised by GG, 19-Apr-2025.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ (𝜑 → 𝐾 ∈ PConn) | ||
Theorem | cvxsconn 34986* | A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11225. (Revised by GG, 19-Apr-2025.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ (𝜑 → 𝐾 ∈ SConn) | ||
Theorem | blsconn 34987 | An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝐾 ∈ SConn) | ||
Theorem | cnllysconn 34988 | The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Locally SConn | ||
Theorem | resconn 34989 | A subset of ℝ is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴) ⇒ ⊢ (𝐴 ⊆ ℝ → (𝐽 ∈ SConn ↔ 𝐽 ∈ Conn)) | ||
Theorem | ioosconn 34990 | An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ ((topGen‘ran (,)) ↾t (𝐴(,)𝐵)) ∈ SConn | ||
Theorem | iccsconn 34991 | A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ SConn) | ||
Theorem | retopsconn 34992 | The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ (topGen‘ran (,)) ∈ SConn | ||
Theorem | iccllysconn 34993 | A closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Locally SConn) | ||
Theorem | rellysconn 34994 | The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ (topGen‘ran (,)) ∈ Locally SConn | ||
Theorem | iisconn 34995 | The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ II ∈ SConn | ||
Theorem | iillysconn 34996 | The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ II ∈ Locally SConn | ||
Theorem | iinllyconn 34997 | The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.) |
⊢ II ∈ 𝑛-Locally Conn | ||
Syntax | ccvm 34998 | Extend class notation with the class of covering maps. |
class CovMap | ||
Definition | df-cvm 34999* | Define the class of covering maps on two topological spaces. A function 𝑓:𝑐⟶𝑗 is a covering map if it is continuous and for every point 𝑥 in the target space there is a neighborhood 𝑘 of 𝑥 and a decomposition 𝑠 of the preimage of 𝑘 as a disjoint union such that 𝑓 is a homeomorphism of each set 𝑢 ∈ 𝑠 onto 𝑘. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)))))}) | ||
Theorem | fncvm 35000 | Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ CovMap Fn (Top × Top) |
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