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Theorem eupth2 27217
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 2651 . . . . . . 7 𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩
71, 2, 3, 4, 5, 6eupthvdres 27213 . . . . . 6 (𝜑 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩) = (VtxDeg‘𝐺))
87fveq1d 6231 . . . . 5 (𝜑 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥) = ((VtxDeg‘𝐺)‘𝑥))
98breq2d 4697 . . . 4 (𝜑 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
109notbid 307 . . 3 (𝜑 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
1110rabbidv 3220 . 2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
12 eupthiswlk 27190 . . . 4 (𝐹(EulerPaths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
13 wlkcl 26567 . . . 4 (𝐹(Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)
145, 12, 133syl 18 . . 3 (𝜑 → (#‘𝐹) ∈ ℕ0)
15 nn0re 11339 . . . . 5 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℝ)
1615leidd 10632 . . . 4 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ≤ (#‘𝐹))
17 breq1 4688 . . . . . . 7 (𝑚 = 0 → (𝑚 ≤ (#‘𝐹) ↔ 0 ≤ (#‘𝐹)))
18 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (0..^𝑚) = (0..^0))
1918imaeq2d 5501 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^0)))
2019reseq2d 5428 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^0))))
2120opeq2d 4440 . . . . . . . . . . . . 13 (𝑚 = 0 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)
2221fveq2d 6233 . . . . . . . . . . . 12 (𝑚 = 0 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩))
2322fveq1d 6231 . . . . . . . . . . 11 (𝑚 = 0 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥))
2423breq2d 4697 . . . . . . . . . 10 (𝑚 = 0 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
2524notbid 307 . . . . . . . . 9 (𝑚 = 0 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
2625rabbidv 3220 . . . . . . . 8 (𝑚 = 0 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)})
27 fveq2 6229 . . . . . . . . . 10 (𝑚 = 0 → (𝑃𝑚) = (𝑃‘0))
2827eqeq2d 2661 . . . . . . . . 9 (𝑚 = 0 → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘0)))
2927preq2d 4307 . . . . . . . . 9 (𝑚 = 0 → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘0)})
3028, 29ifbieq2d 4144 . . . . . . . 8 (𝑚 = 0 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
3126, 30eqeq12d 2666 . . . . . . 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 333 . . . . . 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 329 . . . . 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 4688 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ≤ (#‘𝐹) ↔ 𝑛 ≤ (#‘𝐹)))
35 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → (0..^𝑚) = (0..^𝑛))
3635imaeq2d 5501 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^𝑛)))
3736reseq2d 5428 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^𝑛))))
3837opeq2d 4440 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)
3938fveq2d 6233 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩))
4039fveq1d 6231 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥))
4140breq2d 4697 . . . . . . . . . 10 (𝑚 = 𝑛 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
4241notbid 307 . . . . . . . . 9 (𝑚 = 𝑛 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
4342rabbidv 3220 . . . . . . . 8 (𝑚 = 𝑛 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)})
44 fveq2 6229 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑃𝑚) = (𝑃𝑛))
4544eqeq2d 2661 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃𝑛)))
4644preq2d 4307 . . . . . . . . 9 (𝑚 = 𝑛 → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃𝑛)})
4745, 46ifbieq2d 4144 . . . . . . . 8 (𝑚 = 𝑛 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))
4843, 47eqeq12d 2666 . . . . . . 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 333 . . . . . 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 329 . . . . 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 4688 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ≤ (#‘𝐹) ↔ (𝑛 + 1) ≤ (#‘𝐹)))
52 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 + 1) → (0..^𝑚) = (0..^(𝑛 + 1)))
5352imaeq2d 5501 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 + 1) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(𝑛 + 1))))
5453reseq2d 5428 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 + 1) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1)))))
5554opeq2d 4440 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)
5655fveq2d 6233 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩))
5756fveq1d 6231 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥))
5857breq2d 4697 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
5958notbid 307 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
6059rabbidv 3220 . . . . . . . 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 6229 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (𝑃𝑚) = (𝑃‘(𝑛 + 1)))
6261eqeq2d 2661 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(𝑛 + 1))))
6361preq2d 4307 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(𝑛 + 1))})
6462, 63ifbieq2d 4144 . . . . . . . 8 (𝑚 = (𝑛 + 1) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
6560, 64eqeq12d 2666 . . . . . . 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 333 . . . . . 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 329 . . . . 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 4688 . . . . . . 7 (𝑚 = (#‘𝐹) → (𝑚 ≤ (#‘𝐹) ↔ (#‘𝐹) ≤ (#‘𝐹)))
69 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑚 = (#‘𝐹) → (0..^𝑚) = (0..^(#‘𝐹)))
7069imaeq2d 5501 . . . . . . . . . . . . . . 15 (𝑚 = (#‘𝐹) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(#‘𝐹))))
7170reseq2d 5428 . . . . . . . . . . . . . 14 (𝑚 = (#‘𝐹) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹)))))
7271opeq2d 4440 . . . . . . . . . . . . 13 (𝑚 = (#‘𝐹) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)
7372fveq2d 6233 . . . . . . . . . . . 12 (𝑚 = (#‘𝐹) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩))
7473fveq1d 6231 . . . . . . . . . . 11 (𝑚 = (#‘𝐹) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥))
7574breq2d 4697 . . . . . . . . . 10 (𝑚 = (#‘𝐹) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥)))
7675notbid 307 . . . . . . . . 9 (𝑚 = (#‘𝐹) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥)))
7776rabbidv 3220 . . . . . . . 8 (𝑚 = (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩)‘𝑥)})
78 fveq2 6229 . . . . . . . . . 10 (𝑚 = (#‘𝐹) → (𝑃𝑚) = (𝑃‘(#‘𝐹)))
7978eqeq2d 2661 . . . . . . . . 9 (𝑚 = (#‘𝐹) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(#‘𝐹))))
8078preq2d 4307 . . . . . . . . 9 (𝑚 = (#‘𝐹) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(#‘𝐹))})
8179, 80ifbieq2d 4144 . . . . . . . 8 (𝑚 = (#‘𝐹) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))
8277, 81eqeq12d 2666 . . . . . . 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 333 . . . . . 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 329 . . . . 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 27215 . . . . . . 7 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)
86 eqid 2651 . . . . . . . 8 (𝑃‘0) = (𝑃‘0)
8786iftruei 4126 . . . . . . 7 if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}) = ∅
8885, 87syl6eqr 2703 . . . . . 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 27216 . . . . . . 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 450 . . . . . 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 11510 . . . 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 2687 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 1523  wcel 2030  {crab 2945  c0 3948  ifcif 4119  {cpr 4212  cop 4216   class class class wbr 4685  cres 5145  cima 5146  Fun wfun 5920  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cle 10113  2c2 11108  0cn0 11330  ..^cfzo 12504  #chash 13157  cdvds 15027  Vtxcvtx 25919  iEdgciedg 25920  UPGraphcupgr 26020  VtxDegcvtxdg 26417  Walkscwlks 26548  EulerPathsceupth 27175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-xadd 11985  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-ushgr 25999  df-upgr 26022  df-uspgr 26090  df-vtxdg 26418  df-wlks 26551  df-trls 26645  df-eupth 27176
This theorem is referenced by:  eulerpathpr  27218  eulercrct  27220
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