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Theorem eupth2lems 30090
Description: Lemma for eupth2 30091 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. Formerly part of proof for eupth2 30091. (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
eupth2lems ((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))}))))
Distinct variable groups:   πœ‘,π‘₯   π‘₯,𝐹   π‘₯,𝐼   π‘₯,𝑉   π‘₯,𝑛
Allowed substitution hints:   πœ‘(𝑛)   𝑃(π‘₯,𝑛)   𝐹(𝑛)   𝐺(π‘₯,𝑛)   𝐼(𝑛)   𝑉(𝑛)

Proof of Theorem eupth2lems
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nn0re 12509 . . . . . 6 (𝑛 ∈ β„•0 β†’ 𝑛 ∈ ℝ)
21adantl 480 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ ℝ)
32lep1d 12173 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ≀ (𝑛 + 1))
4 peano2re 11415 . . . . . 6 (𝑛 ∈ ℝ β†’ (𝑛 + 1) ∈ ℝ)
52, 4syl 17 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + 1) ∈ ℝ)
6 eupth2.p . . . . . . . 8 (πœ‘ β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
7 eupthiswlk 30064 . . . . . . . 8 (𝐹(EulerPathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
8 wlkcl 29471 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
96, 7, 83syl 18 . . . . . . 7 (πœ‘ β†’ (β™―β€˜πΉ) ∈ β„•0)
109nn0red 12561 . . . . . 6 (πœ‘ β†’ (β™―β€˜πΉ) ∈ ℝ)
1110adantr 479 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (β™―β€˜πΉ) ∈ ℝ)
12 letr 11336 . . . . 5 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (β™―β€˜πΉ) ∈ ℝ) β†’ ((𝑛 ≀ (𝑛 + 1) ∧ (𝑛 + 1) ≀ (β™―β€˜πΉ)) β†’ 𝑛 ≀ (β™―β€˜πΉ)))
132, 5, 11, 12syl3anc 1368 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 ≀ (𝑛 + 1) ∧ (𝑛 + 1) ≀ (β™―β€˜πΉ)) β†’ 𝑛 ≀ (β™―β€˜πΉ)))
143, 13mpand 693 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + 1) ≀ (β™―β€˜πΉ) β†’ 𝑛 ≀ (β™―β€˜πΉ)))
1514imim1d 82 . 2 ((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)))⟩)β€˜π‘₯)} = if((π‘ƒβ€˜0) = (π‘ƒβ€˜π‘›), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜π‘›)}))))
16 fveq2 6891 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯) = ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘¦))
1716breq2d 5155 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯) ↔ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘¦)))
1817notbid 317 . . . . . . 7 (π‘₯ = 𝑦 β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯) ↔ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘¦)))
1918elrab 3675 . . . . . 6 (𝑦 ∈ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} ↔ (𝑦 ∈ 𝑉 ∧ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘¦)))
20 eupth2.v . . . . . . . . 9 𝑉 = (Vtxβ€˜πΊ)
21 eupth2.i . . . . . . . . 9 𝐼 = (iEdgβ€˜πΊ)
22 eupth2.g . . . . . . . . . 10 (πœ‘ β†’ 𝐺 ∈ UPGraph)
2322ad3antrrr 728 . . . . . . . . 9 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) β†’ 𝐺 ∈ UPGraph)
24 eupth2.f . . . . . . . . . 10 (πœ‘ β†’ Fun 𝐼)
2524ad3antrrr 728 . . . . . . . . 9 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) β†’ Fun 𝐼)
266ad3antrrr 728 . . . . . . . . 9 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
27 eqid 2725 . . . . . . . . 9 βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩ = βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^𝑛)))⟩
28 eqid 2725 . . . . . . . . 9 βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩ = βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩
29 simpr 483 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
3029ad2antrr 724 . . . . . . . . 9 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) β†’ 𝑛 ∈ β„•0)
31 simprl 769 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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))
3231adantr 479 . . . . . . . . 9 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) β†’ (𝑛 + 1) ≀ (β™―β€˜πΉ))
33 simpr 483 . . . . . . . . 9 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) β†’ 𝑦 ∈ 𝑉)
34 simplrr 776 . . . . . . . . 9 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) 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‘›)}))
3520, 21, 23, 25, 26, 27, 28, 30, 32, 33, 34eupth2lem3 30088 . . . . . . . 8 ((((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘›)}))) ∧ 𝑦 ∈ 𝑉) β†’ (Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘¦) ↔ 𝑦 ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})))
3635pm5.32da 577 . . . . . . 7 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ ((𝑦 ∈ 𝑉 ∧ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘¦)) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}))))
37 0elpw 5350 . . . . . . . . . . 11 βˆ… ∈ 𝒫 𝑉
3820wlkepvtx 29516 . . . . . . . . . . . . . . . 16 (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((π‘ƒβ€˜0) ∈ 𝑉 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) ∈ 𝑉))
3938simpld 493 . . . . . . . . . . . . . . 15 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (π‘ƒβ€˜0) ∈ 𝑉)
406, 7, 393syl 18 . . . . . . . . . . . . . 14 (πœ‘ β†’ (π‘ƒβ€˜0) ∈ 𝑉)
4140ad2antrr 724 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘ƒβ€˜0) ∈ 𝑉)
4220wlkp 29472 . . . . . . . . . . . . . . . 16 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
436, 7, 423syl 18 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
4443ad2antrr 724 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
45 peano2nn0 12540 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•0)
4645adantl 480 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + 1) ∈ β„•0)
4746adantr 479 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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„•0)
48 nn0uz 12892 . . . . . . . . . . . . . . . . 17 β„•0 = (β„€β‰₯β€˜0)
4947, 48eleqtrdi 2835 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‰₯β€˜0))
509ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ (β™―β€˜πΉ) ∈ β„•0)
5150nn0zd 12612 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ (β™―β€˜πΉ) ∈ β„€)
52 elfz5 13523 . . . . . . . . . . . . . . . 16 (((𝑛 + 1) ∈ (β„€β‰₯β€˜0) ∧ (β™―β€˜πΉ) ∈ β„€) β†’ ((𝑛 + 1) ∈ (0...(β™―β€˜πΉ)) ↔ (𝑛 + 1) ≀ (β™―β€˜πΉ)))
5349, 51, 52syl2anc 582 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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) ∈ (0...(β™―β€˜πΉ)) ↔ (𝑛 + 1) ≀ (β™―β€˜πΉ)))
5431, 53mpbird 256 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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) ∈ (0...(β™―β€˜πΉ)))
5544, 54ffvelcdmd 7089 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘ƒβ€˜(𝑛 + 1)) ∈ 𝑉)
5641, 55prssd 4821 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))} βŠ† 𝑉)
57 prex 5428 . . . . . . . . . . . . 13 {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))} ∈ V
5857elpw 4602 . . . . . . . . . . . 12 ({(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))} ∈ 𝒫 𝑉 ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))} βŠ† 𝑉)
5956, 58sylibr 233 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))} ∈ 𝒫 𝑉)
60 ifcl 4569 . . . . . . . . . . 11 ((βˆ… ∈ 𝒫 𝑉 ∧ {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))} ∈ 𝒫 𝑉) β†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}) ∈ 𝒫 𝑉)
6137, 59, 60sylancr 585 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}) ∈ 𝒫 𝑉)
6261elpwid 4607 . . . . . . . . 9 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}) βŠ† 𝑉)
6362sseld 3971 . . . . . . . 8 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ (𝑦 ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}) β†’ 𝑦 ∈ 𝑉))
6463pm4.71rd 561 . . . . . . 7 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ (𝑦 ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))}))))
6536, 64bitr4d 281 . . . . . 6 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ ((𝑦 ∈ 𝑉 ∧ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘¦)) ↔ 𝑦 ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})))
6619, 65bitrid 282 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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†’ (𝑦 ∈ {π‘₯ ∈ 𝑉 ∣ Β¬ 2 βˆ₯ ((VtxDegβ€˜βŸ¨π‘‰, (𝐼 β†Ύ (𝐹 β€œ (0..^(𝑛 + 1))))⟩)β€˜π‘₯)} ↔ 𝑦 ∈ if((π‘ƒβ€˜0) = (π‘ƒβ€˜(𝑛 + 1)), βˆ…, {(π‘ƒβ€˜0), (π‘ƒβ€˜(𝑛 + 1))})))
6766eqrdv 2723 . . . 4 (((πœ‘ ∧ 𝑛 ∈ β„•0) ∧ ((𝑛 + 1) ≀ (β™―β€˜πΉ) ∧ {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‘₯ ∈ 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))}))
6867exp32 419 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + 1) ≀ (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‘₯ ∈ 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))}))))
6968a2d 29 . 2 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (((𝑛 + 1) ≀ (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))}))))
7015, 69syld 47 1 ((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))}))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3419   βŠ† wss 3940  βˆ…c0 4318  ifcif 4524  π’« cpw 4598  {cpr 4626  βŸ¨cop 4630   class class class wbr 5143   β†Ύ cres 5674   β€œ cima 5675  Fun wfun 6536  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415  β„cr 11135  0cc0 11136  1c1 11137   + caddc 11139   ≀ cle 11277  2c2 12295  β„•0cn0 12500  β„€cz 12586  β„€β‰₯cuz 12850  ...cfz 13514  ..^cfzo 13657  β™―chash 14319   βˆ₯ cdvds 16228  Vtxcvtx 28851  iEdgciedg 28852  UPGraphcupgr 28935  VtxDegcvtxdg 29321  Walkscwlks 29452  EulerPathsceupth 30049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-oadd 8487  df-er 8721  df-map 8843  df-pm 8844  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-sup 9463  df-inf 9464  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-rp 13005  df-xadd 13123  df-fz 13515  df-fzo 13658  df-seq 13997  df-exp 14057  df-hash 14320  df-word 14495  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-dvds 16229  df-vtx 28853  df-iedg 28854  df-edg 28903  df-uhgr 28913  df-ushgr 28914  df-upgr 28937  df-uspgr 29005  df-vtxdg 29322  df-wlks 29455  df-trls 29548  df-eupth 30050
This theorem is referenced by:  eupth2  30091
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