MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eupth2 Structured version   Visualization version   GIF version

Theorem eupth2 29183
Description: The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
Hypotheses
Ref Expression
eupth2.v 𝑉 = (Vtx‘𝐺)
eupth2.i 𝐼 = (iEdg‘𝐺)
eupth2.g (𝜑𝐺 ∈ UPGraph)
eupth2.f (𝜑 → Fun 𝐼)
eupth2.p (𝜑𝐹(EulerPaths‘𝐺)𝑃)
Assertion
Ref Expression
eupth2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐹   𝑥,𝐼   𝑥,𝑉
Allowed substitution hints:   𝑃(𝑥)   𝐺(𝑥)

Proof of Theorem eupth2
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eupth2.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
2 eupth2.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
3 eupth2.g . . . . . . 7 (𝜑𝐺 ∈ UPGraph)
4 eupth2.f . . . . . . 7 (𝜑 → Fun 𝐼)
5 eupth2.p . . . . . . 7 (𝜑𝐹(EulerPaths‘𝐺)𝑃)
6 eqid 2736 . . . . . . 7 𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩
71, 2, 3, 4, 5, 6eupthvdres 29179 . . . . . 6 (𝜑 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = (VtxDeg‘𝐺))
87fveq1d 6844 . . . . 5 (𝜑 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) = ((VtxDeg‘𝐺)‘𝑥))
98breq2d 5117 . . . 4 (𝜑 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
109notbid 317 . . 3 (𝜑 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
1110rabbidv 3415 . 2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
12 eupthiswlk 29156 . . . 4 (𝐹(EulerPaths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
13 wlkcl 28563 . . . 4 (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
145, 12, 133syl 18 . . 3 (𝜑 → (♯‘𝐹) ∈ ℕ0)
15 nn0re 12422 . . . . 5 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℝ)
1615leidd 11721 . . . 4 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ≤ (♯‘𝐹))
17 breq1 5108 . . . . . . 7 (𝑚 = 0 → (𝑚 ≤ (♯‘𝐹) ↔ 0 ≤ (♯‘𝐹)))
18 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (0..^𝑚) = (0..^0))
1918imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^0)))
2019reseq2d 5937 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^0))))
2120opeq2d 4837 . . . . . . . . . . . . 13 (𝑚 = 0 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)
2221fveq2d 6846 . . . . . . . . . . . 12 (𝑚 = 0 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩))
2322fveq1d 6844 . . . . . . . . . . 11 (𝑚 = 0 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥))
2423breq2d 5117 . . . . . . . . . 10 (𝑚 = 0 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
2524notbid 317 . . . . . . . . 9 (𝑚 = 0 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
2625rabbidv 3415 . . . . . . . 8 (𝑚 = 0 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)})
27 fveq2 6842 . . . . . . . . . 10 (𝑚 = 0 → (𝑃𝑚) = (𝑃‘0))
2827eqeq2d 2747 . . . . . . . . 9 (𝑚 = 0 → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘0)))
2927preq2d 4701 . . . . . . . . 9 (𝑚 = 0 → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘0)})
3028, 29ifbieq2d 4512 . . . . . . . 8 (𝑚 = 0 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
3126, 30eqeq12d 2752 . . . . . . 7 (𝑚 = 0 → ({𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)})))
3217, 31imbi12d 344 . . . . . 6 (𝑚 = 0 → ((𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ (0 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))))
3332imbi2d 340 . . . . 5 (𝑚 = 0 → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → (0 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)})))))
34 breq1 5108 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ≤ (♯‘𝐹) ↔ 𝑛 ≤ (♯‘𝐹)))
35 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → (0..^𝑚) = (0..^𝑛))
3635imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^𝑛)))
3736reseq2d 5937 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^𝑛))))
3837opeq2d 4837 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)
3938fveq2d 6846 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩))
4039fveq1d 6844 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥))
4140breq2d 5117 . . . . . . . . . 10 (𝑚 = 𝑛 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
4241notbid 317 . . . . . . . . 9 (𝑚 = 𝑛 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
4342rabbidv 3415 . . . . . . . 8 (𝑚 = 𝑛 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)})
44 fveq2 6842 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑃𝑚) = (𝑃𝑛))
4544eqeq2d 2747 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃𝑛)))
4644preq2d 4701 . . . . . . . . 9 (𝑚 = 𝑛 → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃𝑛)})
4745, 46ifbieq2d 4512 . . . . . . . 8 (𝑚 = 𝑛 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))
4843, 47eqeq12d 2752 . . . . . . 7 (𝑚 = 𝑛 → ({𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})))
4934, 48imbi12d 344 . . . . . 6 (𝑚 = 𝑛 → ((𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ (𝑛 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))))
5049imbi2d 340 . . . . 5 (𝑚 = 𝑛 → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → (𝑛 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})))))
51 breq1 5108 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ≤ (♯‘𝐹) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
52 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 + 1) → (0..^𝑚) = (0..^(𝑛 + 1)))
5352imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 + 1) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(𝑛 + 1))))
5453reseq2d 5937 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 + 1) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1)))))
5554opeq2d 4837 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)
5655fveq2d 6846 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩))
5756fveq1d 6844 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥))
5857breq2d 5117 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
5958notbid 317 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
6059rabbidv 3415 . . . . . . . 8 (𝑚 = (𝑛 + 1) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)})
61 fveq2 6842 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (𝑃𝑚) = (𝑃‘(𝑛 + 1)))
6261eqeq2d 2747 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(𝑛 + 1))))
6361preq2d 4701 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(𝑛 + 1))})
6462, 63ifbieq2d 4512 . . . . . . . 8 (𝑚 = (𝑛 + 1) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
6560, 64eqeq12d 2752 . . . . . . 7 (𝑚 = (𝑛 + 1) → ({𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
6651, 65imbi12d 344 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
6766imbi2d 340 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))))
68 breq1 5108 . . . . . . 7 (𝑚 = (♯‘𝐹) → (𝑚 ≤ (♯‘𝐹) ↔ (♯‘𝐹) ≤ (♯‘𝐹)))
69 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑚 = (♯‘𝐹) → (0..^𝑚) = (0..^(♯‘𝐹)))
7069imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝑚 = (♯‘𝐹) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(♯‘𝐹))))
7170reseq2d 5937 . . . . . . . . . . . . . 14 (𝑚 = (♯‘𝐹) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
7271opeq2d 4837 . . . . . . . . . . . . 13 (𝑚 = (♯‘𝐹) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
7372fveq2d 6846 . . . . . . . . . . . 12 (𝑚 = (♯‘𝐹) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩))
7473fveq1d 6844 . . . . . . . . . . 11 (𝑚 = (♯‘𝐹) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥))
7574breq2d 5117 . . . . . . . . . 10 (𝑚 = (♯‘𝐹) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
7675notbid 317 . . . . . . . . 9 (𝑚 = (♯‘𝐹) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
7776rabbidv 3415 . . . . . . . 8 (𝑚 = (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)})
78 fveq2 6842 . . . . . . . . . 10 (𝑚 = (♯‘𝐹) → (𝑃𝑚) = (𝑃‘(♯‘𝐹)))
7978eqeq2d 2747 . . . . . . . . 9 (𝑚 = (♯‘𝐹) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
8078preq2d 4701 . . . . . . . . 9 (𝑚 = (♯‘𝐹) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
8179, 80ifbieq2d 4512 . . . . . . . 8 (𝑚 = (♯‘𝐹) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
8277, 81eqeq12d 2752 . . . . . . 7 (𝑚 = (♯‘𝐹) → ({𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) ↔ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})))
8368, 82imbi12d 344 . . . . . 6 (𝑚 = (♯‘𝐹) → ((𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)})) ↔ ((♯‘𝐹) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))))
8483imbi2d 340 . . . . 5 (𝑚 = (♯‘𝐹) → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}))) ↔ (𝜑 → ((♯‘𝐹) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})))))
851, 2, 3, 4, 5eupth2lemb 29181 . . . . . . 7 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)
86 eqid 2736 . . . . . . . 8 (𝑃‘0) = (𝑃‘0)
8786iftruei 4493 . . . . . . 7 if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}) = ∅
8885, 87eqtr4di 2794 . . . . . 6 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
8988a1d 25 . . . . 5 (𝜑 → (0 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)})))
901, 2, 3, 4, 5eupth2lems 29182 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
9190expcom 414 . . . . . 6 (𝑛 ∈ ℕ0 → (𝜑 → ((𝑛 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))))
9291a2d 29 . . . . 5 (𝑛 ∈ ℕ0 → ((𝜑 → (𝑛 ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝜑 → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))))
9333, 50, 67, 84, 89, 92nn0ind 12598 . . . 4 ((♯‘𝐹) ∈ ℕ0 → (𝜑 → ((♯‘𝐹) ≤ (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))))
9416, 93mpid 44 . . 3 ((♯‘𝐹) ∈ ℕ0 → (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})))
9514, 94mpcom 38 . 2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
9611, 95eqtr3d 2778 1 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  wcel 2106  {crab 3407  c0 4282  ifcif 4486  {cpr 4588  cop 4592   class class class wbr 5105  cres 5635  cima 5636  Fun wfun 6490  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  cle 11190  2c2 12208  0cn0 12413  ..^cfzo 13567  chash 14230  cdvds 16136  Vtxcvtx 27947  iEdgciedg 27948  UPGraphcupgr 28031  VtxDegcvtxdg 28413  Walkscwlks 28544  EulerPathsceupth 29141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-xadd 13034  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-vtx 27949  df-iedg 27950  df-edg 27999  df-uhgr 28009  df-ushgr 28010  df-upgr 28033  df-uspgr 28101  df-vtxdg 28414  df-wlks 28547  df-trls 28640  df-eupth 29142
This theorem is referenced by:  eulerpathpr  29184  eulercrct  29186
  Copyright terms: Public domain W3C validator