Theorem eupth2lems 30498

Description: Lemma for eupth2 30499 (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 30499. (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	⊢ (𝜑 → 𝐺 ∈ UPGraph)
eupth2.f	⊢ (𝜑 → Fun 𝐼)
eupth2.p	⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)

Assertion

Ref	Expression
eupth2lems	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐹   𝑥,𝐼   𝑥,𝑉   𝑥,𝑛

Allowed substitution hints:   𝜑(𝑛)   𝑃(𝑥,𝑛)   𝐹(𝑛)   𝐺(𝑥,𝑛)   𝐼(𝑛)   𝑉(𝑛)

Proof of Theorem eupth2lems

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0re 12504	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
2	1	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
3	2	lep1d 12137	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ≤ (𝑛 + 1))
4		peano2re 11371	. . . . . 6 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
5	2, 4	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℝ)
6		eupth2.p	. . . . . . . 8 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
7		eupthiswlk 30472	. . . . . . . 8 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
8		wlkcl 29874	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
9	6, 7, 8	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ0)
10	9	nn0red 12557	. . . . . 6 ⊢ (𝜑 → (♯‘𝐹) ∈ ℝ)
11	10	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (♯‘𝐹) ∈ ℝ)
12		letr 11292	. . . . 5 ⊢ ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (♯‘𝐹) ∈ ℝ) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
13	2, 5, 11, 12	syl3anc 1394	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
14	3, 13	mpand 707	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) ≤ (♯‘𝐹) → 𝑛 ≤ (♯‘𝐹)))
15	14	imim1d 83	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))))
16		fveq2 6871	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦))
17	16	breq2d 5117	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
18	17	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
19	18	elrab 3653	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} ↔ (𝑦 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
20		eupth2.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
21		eupth2.i	. . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺)
22		eupth2.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
23	22	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → 𝐺 ∈ UPGraph)
24		eupth2.f	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐼)
25	24	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → Fun 𝐼)
26	6	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → 𝐹(EulerPaths‘𝐺)𝑃)
27		eqid 2765	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩
28		eqid 2765	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩
29		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
30	29	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → 𝑛 ∈ ℕ0)
31		simprl 782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑛 + 1) ≤ (♯‘𝐹))
32	31	adantr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → (𝑛 + 1) ≤ (♯‘𝐹))
33		simpr 489	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
34		simplrr 789	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))
35	20, 21, 23, 25, 26, 27, 28, 30, 32, 33, 34	eupth2lem3 30496	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) ∧ 𝑦 ∈ 𝑉) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦) ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
36	35	pm5.32da 589	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → ((𝑦 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
37		0elpw 5317	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑉
38	20	wlkepvtx 29917	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(♯‘𝐹)) ∈ 𝑉))
39	38	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘0) ∈ 𝑉)
40	6, 7, 39	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃‘0) ∈ 𝑉)
41	40	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑃‘0) ∈ 𝑉)
42	20	wlkp 29875	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
43	6, 7, 42	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
44	43	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
45		peano2nn0 12535	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
46	45	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
47	46	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑛 + 1) ∈ ℕ0)
48		nn0uz 12891	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 = (ℤ≥‘0)
49	47, 48	eleqtrdi 2875	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑛 + 1) ∈ (ℤ≥‘0))
50	9	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (♯‘𝐹) ∈ ℕ0)
51	50	nn0zd 12607	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (♯‘𝐹) ∈ ℤ)
52		elfz5 13535	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) ∈ (ℤ≥‘0) ∧ (♯‘𝐹) ∈ ℤ) → ((𝑛 + 1) ∈ (0...(♯‘𝐹)) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
53	49, 51, 52	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → ((𝑛 + 1) ∈ (0...(♯‘𝐹)) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
54	31, 53	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑛 + 1) ∈ (0...(♯‘𝐹)))
55	44, 54	ffvelcdmd 7070	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑃‘(𝑛 + 1)) ∈ 𝑉)
56	41, 55	prssd 4783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
57		prex 5400	. . . . . . . . . . . . 13 ⊢ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ V
58	57	elpw 4562	. . . . . . . . . . . 12 ⊢ ({(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉 ↔ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
59	56, 58	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉)
60		ifcl 4529	. . . . . . . . . . 11 ⊢ ((∅ ∈ 𝒫 𝑉 ∧ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ∈ 𝒫 𝑉)
61	37, 59, 60	sylancr 598	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ∈ 𝒫 𝑉)
62	61	elpwid 4567	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ⊆ 𝑉)
63	62	sseld 3938	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) → 𝑦 ∈ 𝑉))
64	63	pm4.71rd 571	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
65	36, 64	bitr4d 285	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → ((𝑦 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)) ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
66	19, 65	bitrid 286	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝑦 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
67	66	eqrdv 2763	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (♯‘𝐹) ∧ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
68	67	exp32 425	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) ≤ (♯‘𝐹) → ({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
69	68	a2d 30	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
70	15, 69	syld 48	1 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {crab 3417   ⊆ wss 3907  ∅c0 4288  ifcif 4483  𝒫 cpw 4558  {cpr 4587  ⟨cop 4591   class class class wbr 5105   ↾ cres 5654   “ cima 5655  Fun wfun 6519  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℝcr 11087  0cc0 11088  1c1 11089   + caddc 11091   ≤ cle 11232  2c2 12286  ℕ₀cn0 12495  ℤcz 12582  ℤ_≥cuz 12853  ...cfz 13526  ..^cfzo 13673  ♯chash 14357   ∥ cdvds 16300  Vtxcvtx 29255  iEdgciedg 29256  UPGraphcupgr 29339  VtxDegcvtxdg 29724  Walkscwlks 29855  EulerPathsceupth 30457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-dju 9875  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-xnn0 12569  df-z 12583  df-uz 12854  df-rp 13008  df-xadd 13129  df-fz 13527  df-fzo 13674  df-seq 14029  df-exp 14089  df-hash 14358  df-word 14541  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-dvds 16301  df-vtx 29257  df-iedg 29258  df-edg 29307  df-uhgr 29317  df-ushgr 29318  df-upgr 29341  df-uspgr 29409  df-vtxdg 29725  df-wlks 29858  df-trls 29949  df-eupth 30458

This theorem is referenced by:  eupth2  30499