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Theorem eupth2 27945
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 2818 . . . . . . 7 𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩
71, 2, 3, 4, 5, 6eupthvdres 27941 . . . . . 6 (𝜑 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = (VtxDeg‘𝐺))
87fveq1d 6665 . . . . 5 (𝜑 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) = ((VtxDeg‘𝐺)‘𝑥))
98breq2d 5069 . . . 4 (𝜑 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
109notbid 319 . . 3 (𝜑 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
1110rabbidv 3478 . 2 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
12 eupthiswlk 27918 . . . 4 (𝐹(EulerPaths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
13 wlkcl 27324 . . . 4 (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
145, 12, 133syl 18 . . 3 (𝜑 → (♯‘𝐹) ∈ ℕ0)
15 nn0re 11894 . . . . 5 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℝ)
1615leidd 11194 . . . 4 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ≤ (♯‘𝐹))
17 breq1 5060 . . . . . . 7 (𝑚 = 0 → (𝑚 ≤ (♯‘𝐹) ↔ 0 ≤ (♯‘𝐹)))
18 oveq2 7153 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (0..^𝑚) = (0..^0))
1918imaeq2d 5922 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^0)))
2019reseq2d 5846 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^0))))
2120opeq2d 4802 . . . . . . . . . . . . 13 (𝑚 = 0 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)
2221fveq2d 6667 . . . . . . . . . . . 12 (𝑚 = 0 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩))
2322fveq1d 6665 . . . . . . . . . . 11 (𝑚 = 0 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥))
2423breq2d 5069 . . . . . . . . . 10 (𝑚 = 0 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
2524notbid 319 . . . . . . . . 9 (𝑚 = 0 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
2625rabbidv 3478 . . . . . . . 8 (𝑚 = 0 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)})
27 fveq2 6663 . . . . . . . . . 10 (𝑚 = 0 → (𝑃𝑚) = (𝑃‘0))
2827eqeq2d 2829 . . . . . . . . 9 (𝑚 = 0 → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘0)))
2927preq2d 4668 . . . . . . . . 9 (𝑚 = 0 → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘0)})
3028, 29ifbieq2d 4488 . . . . . . . 8 (𝑚 = 0 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
3126, 30eqeq12d 2834 . . . . . . 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 346 . . . . . 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 342 . . . . 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 5060 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ≤ (♯‘𝐹) ↔ 𝑛 ≤ (♯‘𝐹)))
35 oveq2 7153 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → (0..^𝑚) = (0..^𝑛))
3635imaeq2d 5922 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^𝑛)))
3736reseq2d 5846 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^𝑛))))
3837opeq2d 4802 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)
3938fveq2d 6667 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩))
4039fveq1d 6665 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥))
4140breq2d 5069 . . . . . . . . . 10 (𝑚 = 𝑛 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
4241notbid 319 . . . . . . . . 9 (𝑚 = 𝑛 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
4342rabbidv 3478 . . . . . . . 8 (𝑚 = 𝑛 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)})
44 fveq2 6663 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑃𝑚) = (𝑃𝑛))
4544eqeq2d 2829 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃𝑛)))
4644preq2d 4668 . . . . . . . . 9 (𝑚 = 𝑛 → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃𝑛)})
4745, 46ifbieq2d 4488 . . . . . . . 8 (𝑚 = 𝑛 → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))
4843, 47eqeq12d 2834 . . . . . . 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 346 . . . . . 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 342 . . . . 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 5060 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ≤ (♯‘𝐹) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
52 oveq2 7153 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 + 1) → (0..^𝑚) = (0..^(𝑛 + 1)))
5352imaeq2d 5922 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 + 1) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(𝑛 + 1))))
5453reseq2d 5846 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 + 1) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1)))))
5554opeq2d 4802 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)
5655fveq2d 6667 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩))
5756fveq1d 6665 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥))
5857breq2d 5069 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
5958notbid 319 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
6059rabbidv 3478 . . . . . . . 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 6663 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (𝑃𝑚) = (𝑃‘(𝑛 + 1)))
6261eqeq2d 2829 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(𝑛 + 1))))
6361preq2d 4668 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(𝑛 + 1))})
6462, 63ifbieq2d 4488 . . . . . . . 8 (𝑚 = (𝑛 + 1) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
6560, 64eqeq12d 2834 . . . . . . 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 346 . . . . . 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 342 . . . . 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 5060 . . . . . . 7 (𝑚 = (♯‘𝐹) → (𝑚 ≤ (♯‘𝐹) ↔ (♯‘𝐹) ≤ (♯‘𝐹)))
69 oveq2 7153 . . . . . . . . . . . . . . . 16 (𝑚 = (♯‘𝐹) → (0..^𝑚) = (0..^(♯‘𝐹)))
7069imaeq2d 5922 . . . . . . . . . . . . . . 15 (𝑚 = (♯‘𝐹) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(♯‘𝐹))))
7170reseq2d 5846 . . . . . . . . . . . . . 14 (𝑚 = (♯‘𝐹) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
7271opeq2d 4802 . . . . . . . . . . . . 13 (𝑚 = (♯‘𝐹) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
7372fveq2d 6667 . . . . . . . . . . . 12 (𝑚 = (♯‘𝐹) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩))
7473fveq1d 6665 . . . . . . . . . . 11 (𝑚 = (♯‘𝐹) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥))
7574breq2d 5069 . . . . . . . . . 10 (𝑚 = (♯‘𝐹) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
7675notbid 319 . . . . . . . . 9 (𝑚 = (♯‘𝐹) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
7776rabbidv 3478 . . . . . . . 8 (𝑚 = (♯‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)})
78 fveq2 6663 . . . . . . . . . 10 (𝑚 = (♯‘𝐹) → (𝑃𝑚) = (𝑃‘(♯‘𝐹)))
7978eqeq2d 2829 . . . . . . . . 9 (𝑚 = (♯‘𝐹) → ((𝑃‘0) = (𝑃𝑚) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
8078preq2d 4668 . . . . . . . . 9 (𝑚 = (♯‘𝐹) → {(𝑃‘0), (𝑃𝑚)} = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
8179, 80ifbieq2d 4488 . . . . . . . 8 (𝑚 = (♯‘𝐹) → if((𝑃‘0) = (𝑃𝑚), ∅, {(𝑃‘0), (𝑃𝑚)}) = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
8277, 81eqeq12d 2834 . . . . . . 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 346 . . . . . 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 342 . . . . 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 27943 . . . . . . 7 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)
86 eqid 2818 . . . . . . . 8 (𝑃‘0) = (𝑃‘0)
8786iftruei 4470 . . . . . . 7 if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}) = ∅
8885, 87syl6eqr 2871 . . . . . 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 27944 . . . . . . 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 12065 . . . 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 2855 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 1528  wcel 2105  {crab 3139  c0 4288  ifcif 4463  {cpr 4559  cop 4563   class class class wbr 5057  cres 5550  cima 5551  Fun wfun 6342  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526   + caddc 10528  cle 10664  2c2 11680  0cn0 11885  ..^cfzo 13021  chash 13678  cdvds 15595  Vtxcvtx 26708  iEdgciedg 26709  UPGraphcupgr 26792  VtxDegcvtxdg 27174  Walkscwlks 27305  EulerPathsceupth 27903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-xadd 12496  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-word 13850  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-vtx 26710  df-iedg 26711  df-edg 26760  df-uhgr 26770  df-ushgr 26771  df-upgr 26794  df-uspgr 26862  df-vtxdg 27175  df-wlks 27308  df-trls 27401  df-eupth 27904
This theorem is referenced by:  eulerpathpr  27946  eulercrct  27948
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