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Theorem eupth2 29186
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 2737 . . . . . . 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 29182 . . . . . 6 (πœ‘ β†’ (VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(β™―β€˜πΉ))))⟩) = (VtxDegβ€˜πΊ))
87fveq1d 6845 . . . . 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 5118 . . . 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 318 . . 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 3416 . 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 29159 . . . 4 (𝐹(EulerPathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
13 wlkcl 28566 . . . 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 12423 . . . . 5 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ ℝ)
1615leidd 11722 . . . 4 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ≀ (β™―β€˜πΉ))
17 breq1 5109 . . . . . . 7 (π‘š = 0 β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ 0 ≀ (β™―β€˜πΉ)))
18 oveq2 7366 . . . . . . . . . . . . . . . 16 (π‘š = 0 β†’ (0..^π‘š) = (0..^0))
1918imaeq2d 6014 . . . . . . . . . . . . . . 15 (π‘š = 0 β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^0)))
2019reseq2d 5938 . . . . . . . . . . . . . 14 (π‘š = 0 β†’ (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š))) = (𝐼 β†Ύ (𝐹 β€œ (0..^0))))
2120opeq2d 4838 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . 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 6845 . . . . . . . . . . 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 5118 . . . . . . . . . 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 318 . . . . . . . . 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 3416 . . . . . . . 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 6843 . . . . . . . . . 10 (π‘š = 0 β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜0))
2827eqeq2d 2748 . . . . . . . . 9 (π‘š = 0 β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜0)))
2927preq2d 4702 . . . . . . . . 9 (π‘š = 0 β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜0)})
3028, 29ifbieq2d 4513 . . . . . . . 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 2753 . . . . . . 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 345 . . . . . 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 341 . . . . 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 5109 . . . . . . 7 (π‘š = 𝑛 β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ 𝑛 ≀ (β™―β€˜πΉ)))
35 oveq2 7366 . . . . . . . . . . . . . . . 16 (π‘š = 𝑛 β†’ (0..^π‘š) = (0..^𝑛))
3635imaeq2d 6014 . . . . . . . . . . . . . . 15 (π‘š = 𝑛 β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^𝑛)))
3736reseq2d 5938 . . . . . . . . . . . . . 14 (π‘š = 𝑛 β†’ (𝐼 β†Ύ (𝐹 β€œ (0..^π‘š))) = (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛))))
3837opeq2d 4838 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . 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 6845 . . . . . . . . . . 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 5118 . . . . . . . . . 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 318 . . . . . . . . 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 3416 . . . . . . . 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 6843 . . . . . . . . . 10 (π‘š = 𝑛 β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜π‘›))
4544eqeq2d 2748 . . . . . . . . 9 (π‘š = 𝑛 β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›)))
4644preq2d 4702 . . . . . . . . 9 (π‘š = 𝑛 β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)})
4745, 46ifbieq2d 4513 . . . . . . . 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 2753 . . . . . . 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 345 . . . . . 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 341 . . . . 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 5109 . . . . . . 7 (π‘š = (𝑛 + 1) β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ (𝑛 + 1) ≀ (β™―β€˜πΉ)))
52 oveq2 7366 . . . . . . . . . . . . . . . 16 (π‘š = (𝑛 + 1) β†’ (0..^π‘š) = (0..^(𝑛 + 1)))
5352imaeq2d 6014 . . . . . . . . . . . . . . 15 (π‘š = (𝑛 + 1) β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^(𝑛 + 1))))
5453reseq2d 5938 . . . . . . . . . . . . . 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 4838 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . 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 6845 . . . . . . . . . . 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 5118 . . . . . . . . . 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 318 . . . . . . . . 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 3416 . . . . . . . 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 6843 . . . . . . . . . 10 (π‘š = (𝑛 + 1) β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜(𝑛 + 1)))
6261eqeq2d 2748 . . . . . . . . 9 (π‘š = (𝑛 + 1) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘š) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1))))
6361preq2d 4702 . . . . . . . . 9 (π‘š = (𝑛 + 1) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘š)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})
6462, 63ifbieq2d 4513 . . . . . . . 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 2753 . . . . . . 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 345 . . . . . 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 341 . . . . 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 5109 . . . . . . 7 (π‘š = (β™―β€˜πΉ) β†’ (π‘š ≀ (β™―β€˜πΉ) ↔ (β™―β€˜πΉ) ≀ (β™―β€˜πΉ)))
69 oveq2 7366 . . . . . . . . . . . . . . . 16 (π‘š = (β™―β€˜πΉ) β†’ (0..^π‘š) = (0..^(β™―β€˜πΉ)))
7069imaeq2d 6014 . . . . . . . . . . . . . . 15 (π‘š = (β™―β€˜πΉ) β†’ (𝐹 β€œ (0..^π‘š)) = (𝐹 β€œ (0..^(β™―β€˜πΉ))))
7170reseq2d 5938 . . . . . . . . . . . . . 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 4838 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . 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 6845 . . . . . . . . . . 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 5118 . . . . . . . . . 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 318 . . . . . . . . 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 3416 . . . . . . . 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 6843 . . . . . . . . . 10 (π‘š = (β™―β€˜πΉ) β†’ (π‘ƒβ€˜π‘š) = (π‘ƒβ€˜(β™―β€˜πΉ)))
7978eqeq2d 2748 . . . . . . . . 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 4702 . . . . . . . . 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 4513 . . . . . . . 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 2753 . . . . . . 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 345 . . . . . 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 341 . . . . 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 29184 . . . . . . 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 2737 . . . . . . . 8 (π‘ƒβ€˜0) = (π‘ƒβ€˜0)
8786iftruei 4494 . . . . . . 7 if((π‘ƒβ€˜0) = (π‘ƒβ€˜0), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜0)}) = βˆ…
8885, 87eqtr4di 2795 . . . . . 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 29185 . . . . . . 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 415 . . . . . 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 12599 . . . 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 2779 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 1542   ∈ wcel 2107  {crab 3408  βˆ…c0 4283  ifcif 4487  {cpr 4589  βŸ¨cop 4593   class class class wbr 5106   β†Ύ cres 5636   β€œ cima 5637  Fun wfun 6491  β€˜cfv 6497  (class class class)co 7358  0cc0 11052  1c1 11053   + caddc 11055   ≀ cle 11191  2c2 12209  β„•0cn0 12414  ..^cfzo 13568  β™―chash 14231   βˆ₯ cdvds 16137  Vtxcvtx 27950  iEdgciedg 27951  UPGraphcupgr 28034  VtxDegcvtxdg 28416  Walkscwlks 28547  EulerPathsceupth 29144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-er 8649  df-map 8768  df-pm 8769  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9379  df-inf 9380  df-dju 9838  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-3 12218  df-n0 12415  df-xnn0 12487  df-z 12501  df-uz 12765  df-rp 12917  df-xadd 13035  df-fz 13426  df-fzo 13569  df-seq 13908  df-exp 13969  df-hash 14232  df-word 14404  df-cj 14985  df-re 14986  df-im 14987  df-sqrt 15121  df-abs 15122  df-dvds 16138  df-vtx 27952  df-iedg 27953  df-edg 28002  df-uhgr 28012  df-ushgr 28013  df-upgr 28036  df-uspgr 28104  df-vtxdg 28417  df-wlks 28550  df-trls 28643  df-eupth 29145
This theorem is referenced by:  eulerpathpr  29187  eulercrct  29189
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