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Theorem eupth2 29759
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 = (Vtxβ€˜πΊ)
eupth2.i 𝐼 = (iEdgβ€˜πΊ)
eupth2.g (πœ‘ β†’ 𝐺 ∈ UPGraph)
eupth2.f (πœ‘ β†’ Fun 𝐼)
eupth2.p (πœ‘ β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
Assertion
Ref Expression
eupth2 (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜πΊ)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))}))
Distinct variable groups:   πœ‘,π‘₯   π‘₯,𝐹   π‘₯,𝐼   π‘₯,𝑉
Allowed substitution hints:   𝑃(π‘₯)   𝐺(π‘₯)
Proof of Theorem eupth2
Dummy variables 𝑛 π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eupth2.v . . . . . . 7 𝑉 = (Vtxβ€˜πΊ)
2 eupth2.i . . . . . . 7 𝐼 = (iEdgβ€˜πΊ)
3 eupth2.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ UPGraph)
4 eupth2.f . . . . . . 7 (πœ‘ β†’ Fun 𝐼)
5 eupth2.p . . . . . . 7 (πœ‘ β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
6 eqid 2730 . . . . . . 7 βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩ = βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩
71, 2, 3, 4, 5, 6eupthvdres 29755 . . . . . 6 (πœ‘ β†’ (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩) = (VtxDegβ€˜πΊ))
87fveq1d 6892 . . . . 5 (πœ‘ β†’ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯) = ((VtxDegβ€˜πΊ)β€˜π‘₯))
98breq2d 5159 . . . 4 (πœ‘ β†’ (2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜πΊ)β€˜π‘₯)))
109notbid 317 . . 3 (πœ‘ β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜πΊ)β€˜π‘₯)))
1110rabbidv 3438 . 2 (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)} = {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜πΊ)β€˜π‘₯)})
12 eupthiswlk 29732 . . . 4 (𝐹(EulerPathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
13 wlkcl 29139 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
145, 12, 133syl 18 . . 3 (πœ‘ β†’ (β™―β€˜πΉ) ∈ β„•0)
15 nn0re 12485 . . . . 5 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ ℝ)
1615leidd 11784 . . . 4 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ≀ (β™―β€˜πΉ))
17 breq1 5150 . . . . . . 7 (π‘š = 0 β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ 0 ≀ (β™―β€˜πΉ)))
18 oveq2 7419 . . . . . . . . . . . . . . . 16 (π‘š = 0 β†’ (0..^π‘š) = (0..^0))
1918imaeq2d 6058 . . . . . . . . . . . . . . 15 (π‘š = 0 β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^0)))
2019reseq2d 5980 . . . . . . . . . . . . . 14 (π‘š = 0 β†’ (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š))) = (𝐼 β†Ύ (𝐹 β€œ (0..^0))))
2120opeq2d 4879 . . . . . . . . . . . . 13 (π‘š = 0 β†’ βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩ = βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)
2221fveq2d 6894 . . . . . . . . . . . 12 (π‘š = 0 β†’ (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩) = (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩))
2322fveq1d 6892 . . . . . . . . . . 11 (π‘š = 0 β†’ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) = ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯))
2423breq2d 5159 . . . . . . . . . 10 (π‘š = 0 β†’ (2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)))
2524notbid 317 . . . . . . . . 9 (π‘š = 0 β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)))
2625rabbidv 3438 . . . . . . . 8 (π‘š = 0 β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)})
27 fveq2 6890 . . . . . . . . . 10 (π‘š = 0 β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜0))
2827eqeq2d 2741 . . . . . . . . 9 (π‘š = 0 β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜0)))
2927preq2d 4743 . . . . . . . . 9 (π‘š = 0 β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜0)})
3028, 29ifbieq2d 4553 . . . . . . . 8 (π‘š = 0 β†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) = if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)}))
3126, 30eqeq12d 2746 . . . . . . 7 (π‘š = 0 β†’ ({π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) ↔ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)})))
3217, 31imbi12d 343 . . . . . 6 (π‘š = 0 β†’ ((π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)})) ↔ (0 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)}))))
3332imbi2d 339 . . . . 5 (π‘š = 0 β†’ ((πœ‘ β†’ (π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}))) ↔ (πœ‘ β†’ (0 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)})))))
34 breq1 5150 . . . . . . 7 (π‘š = 𝑛 β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ 𝑛 ≀ (β™―β€˜πΉ)))
35 oveq2 7419 . . . . . . . . . . . . . . . 16 (π‘š = 𝑛 β†’ (0..^π‘š) = (0..^𝑛))
3635imaeq2d 6058 . . . . . . . . . . . . . . 15 (π‘š = 𝑛 β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^𝑛)))
3736reseq2d 5980 . . . . . . . . . . . . . 14 (π‘š = 𝑛 β†’ (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š))) = (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛))))
3837opeq2d 4879 . . . . . . . . . . . . 13 (π‘š = 𝑛 β†’ βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩ = βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)
3938fveq2d 6894 . . . . . . . . . . . 12 (π‘š = 𝑛 β†’ (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩) = (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩))
4039fveq1d 6892 . . . . . . . . . . 11 (π‘š = 𝑛 β†’ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) = ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯))
4140breq2d 5159 . . . . . . . . . 10 (π‘š = 𝑛 β†’ (2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)))
4241notbid 317 . . . . . . . . 9 (π‘š = 𝑛 β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)))
4342rabbidv 3438 . . . . . . . 8 (π‘š = 𝑛 β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)})
44 fveq2 6890 . . . . . . . . . 10 (π‘š = 𝑛 β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜π‘›))
4544eqeq2d 2741 . . . . . . . . 9 (π‘š = 𝑛 β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›)))
4644preq2d 4743 . . . . . . . . 9 (π‘š = 𝑛 β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)})
4745, 46ifbieq2d 4553 . . . . . . . 8 (π‘š = 𝑛 β†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)}))
4843, 47eqeq12d 2746 . . . . . . 7 (π‘š = 𝑛 β†’ ({π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) ↔ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)})))
4934, 48imbi12d 343 . . . . . 6 (π‘š = 𝑛 β†’ ((π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)})) ↔ (𝑛 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)}))))
5049imbi2d 339 . . . . 5 (π‘š = 𝑛 β†’ ((πœ‘ β†’ (π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}))) ↔ (πœ‘ β†’ (𝑛 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)})))))
51 breq1 5150 . . . . . . 7 (π‘š = (𝑛 + 1) β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ (𝑛 + 1) ≀ (β™―β€˜πΉ)))
52 oveq2 7419 . . . . . . . . . . . . . . . 16 (π‘š = (𝑛 + 1) β†’ (0..^π‘š) = (0..^(𝑛 + 1)))
5352imaeq2d 6058 . . . . . . . . . . . . . . 15 (π‘š = (𝑛 + 1) β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^(𝑛 + 1))))
5453reseq2d 5980 . . . . . . . . . . . . . 14 (π‘š = (𝑛 + 1) β†’ (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š))) = (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1)))))
5554opeq2d 4879 . . . . . . . . . . . . 13 (π‘š = (𝑛 + 1) β†’ βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩ = βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)
5655fveq2d 6894 . . . . . . . . . . . 12 (π‘š = (𝑛 + 1) β†’ (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩) = (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩))
5756fveq1d 6892 . . . . . . . . . . 11 (π‘š = (𝑛 + 1) β†’ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) = ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯))
5857breq2d 5159 . . . . . . . . . 10 (π‘š = (𝑛 + 1) β†’ (2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)))
5958notbid 317 . . . . . . . . 9 (π‘š = (𝑛 + 1) β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)))
6059rabbidv 3438 . . . . . . . 8 (π‘š = (𝑛 + 1) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)})
61 fveq2 6890 . . . . . . . . . 10 (π‘š = (𝑛 + 1) β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜(𝑛 + 1)))
6261eqeq2d 2741 . . . . . . . . 9 (π‘š = (𝑛 + 1) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1))))
6361preq2d 4743 . . . . . . . . 9 (π‘š = (𝑛 + 1) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})
6462, 63ifbieq2d 4553 . . . . . . . 8 (π‘š = (𝑛 + 1) β†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}))
6560, 64eqeq12d 2746 . . . . . . 7 (π‘š = (𝑛 + 1) β†’ ({π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) ↔ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})))
6651, 65imbi12d 343 . . . . . 6 (π‘š = (𝑛 + 1) β†’ ((π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)})) ↔ ((𝑛 + 1) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}))))
6766imbi2d 339 . . . . 5 (π‘š = (𝑛 + 1) β†’ ((πœ‘ β†’ (π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}))) ↔ (πœ‘ β†’ ((𝑛 + 1) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})))))
68 breq1 5150 . . . . . . 7 (π‘š = (β™―β€˜πΉ) β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ (β™―β€˜πΉ) ≀ (β™―β€˜πΉ)))
69 oveq2 7419 . . . . . . . . . . . . . . . 16 (π‘š = (β™―β€˜πΉ) β†’ (0..^π‘š) = (0..^(β™―β€˜πΉ)))
7069imaeq2d 6058 . . . . . . . . . . . . . . 15 (π‘š = (β™―β€˜πΉ) β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^(β™―β€˜πΉ))))
7170reseq2d 5980 . . . . . . . . . . . . . 14 (π‘š = (β™―β€˜πΉ) β†’ (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š))) = (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ)))))
7271opeq2d 4879 . . . . . . . . . . . . 13 (π‘š = (β™―β€˜πΉ) β†’ βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩ = βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)
7372fveq2d 6894 . . . . . . . . . . . 12 (π‘š = (β™―β€˜πΉ) β†’ (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩) = (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩))
7473fveq1d 6892 . . . . . . . . . . 11 (π‘š = (β™―β€˜πΉ) β†’ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) = ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯))
7574breq2d 5159 . . . . . . . . . 10 (π‘š = (β™―β€˜πΉ) β†’ (2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)))
7675notbid 317 . . . . . . . . 9 (π‘š = (β™―β€˜πΉ) β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)))
7776rabbidv 3438 . . . . . . . 8 (π‘š = (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)})
78 fveq2 6890 . . . . . . . . . 10 (π‘š = (β™―β€˜πΉ) β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜(β™―β€˜πΉ)))
7978eqeq2d 2741 . . . . . . . . 9 (π‘š = (β™―β€˜πΉ) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
8078preq2d 4743 . . . . . . . . 9 (π‘š = (β™―β€˜πΉ) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))})
8179, 80ifbieq2d 4553 . . . . . . . 8 (π‘š = (β™―β€˜πΉ) β†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))}))
8277, 81eqeq12d 2746 . . . . . . 7 (π‘š = (β™―β€˜πΉ) β†’ ({π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}) ↔ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))})))
8368, 82imbi12d 343 . . . . . 6 (π‘š = (β™―β€˜πΉ) β†’ ((π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)})) ↔ ((β™―β€˜πΉ) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))}))))
8483imbi2d 339 . . . . 5 (π‘š = (β™―β€˜πΉ) β†’ ((πœ‘ β†’ (π‘š ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)}))) ↔ (πœ‘ β†’ ((β™―β€˜πΉ) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))})))))
851, 2, 3, 4, 5eupth2lemb 29757 . . . . . . 7 (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)} = βˆ…)
86 eqid 2730 . . . . . . . 8 (π‘ƒβ€˜0) = (π‘ƒβ€˜0)
8786iftruei 4534 . . . . . . 7 if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)}) = βˆ…
8885, 87eqtr4di 2788 . . . . . 6 (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)}))
8988a1d 25 . . . . 5 (πœ‘ β†’ (0 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^0)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)})))
901, 2, 3, 4, 5eupth2lems 29758 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)})) β†’ ((𝑛 + 1) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}))))
9190expcom 412 . . . . . 6 (𝑛 ∈ β„•0 β†’ (πœ‘ β†’ ((𝑛 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)})) β†’ ((𝑛 + 1) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})))))
9291a2d 29 . . . . 5 (𝑛 ∈ β„•0 β†’ ((πœ‘ β†’ (𝑛 ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)}))) β†’ (πœ‘ β†’ ((𝑛 + 1) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})))))
9333, 50, 67, 84, 89, 92nn0ind 12661 . . . 4 ((β™―β€˜πΉ) ∈ β„•0 β†’ (πœ‘ β†’ ((β™―β€˜πΉ) ≀ (β™―β€˜πΉ) β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))}))))
9416, 93mpid 44 . . 3 ((β™―β€˜πΉ) ∈ β„•0 β†’ (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))})))
9514, 94mpcom 38 . 2 (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))}))
9611, 95eqtr3d 2772 1 (πœ‘ β†’ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜πΊ)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(β™―β€˜πΉ))}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1539   ∈ wcel 2104  {crab 3430  βˆ…c0 4321  ifcif 4527  {cpr 4629  βŸ¨cop 4633   class class class wbr 5147   β†Ύ cres 5677   β€œ cima 5678  Fun wfun 6536  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115   ≀ cle 11253  2c2 12271  β„•0cn0 12476  ..^cfzo 13631  β™―chash 14294   βˆ₯ cdvds 16201  Vtxcvtx 28523  iEdgciedg 28524  UPGraphcupgr 28607  VtxDegcvtxdg 28989  Walkscwlks 29120  EulerPathsceupth 29717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12979  df-xadd 13097  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-word 14469  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-dvds 16202  df-vtx 28525  df-iedg 28526  df-edg 28575  df-uhgr 28585  df-ushgr 28586  df-upgr 28609  df-uspgr 28677  df-vtxdg 28990  df-wlks 29123  df-trls 29216  df-eupth 29718
This theorem is referenced by:  eulerpathpr  29760  eulercrct  29762
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