Theorem eupth2lems 30174

Description: Lemma for eupth2 30175 (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 30175. (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 12458	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
2	1	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
3	2	lep1d 12121	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ≤ (𝑛 + 1))
4		peano2re 11354	. . . . . 6 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
5	2, 4	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℝ)
6		eupth2.p	. . . . . . . 8 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
7		eupthiswlk 30148	. . . . . . . 8 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
8		wlkcl 29550	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
9	6, 7, 8	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ0)
10	9	nn0red 12511	. . . . . 6 ⊢ (𝜑 → (♯‘𝐹) ∈ ℝ)
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (♯‘𝐹) ∈ ℝ)
12		letr 11275	. . . . 5 ⊢ ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (♯‘𝐹) ∈ ℝ) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
13	2, 5, 11, 12	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
14	3, 13	mpand 695	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) ≤ (♯‘𝐹) → 𝑛 ≤ (♯‘𝐹)))
15	14	imim1d 82	. 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 6861	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦))
17	16	breq2d 5122	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
18	17	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
19	18	elrab 3662	. . . . . 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 730	. . . . . . . . 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 730	. . . . . . . . 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 730	. . . . . . . . 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 2730	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩
28		eqid 2730	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩
29		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
30	29	ad2antrr 726	. . . . . . . . 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 770	. . . . . . . . . 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 480	. . . . . . . . 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 484	. . . . . . . . 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 777	. . . . . . . . 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 30172	. . . . . . . 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 579	. . . . . . 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 5314	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑉
38	20	wlkepvtx 29595	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(♯‘𝐹)) ∈ 𝑉))
39	38	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘0) ∈ 𝑉)
40	6, 7, 39	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃‘0) ∈ 𝑉)
41	40	ad2antrr 726	. . . . . . . . . . . . 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 29551	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
43	6, 7, 42	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
44	43	ad2antrr 726	. . . . . . . . . . . . . 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 12489	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
46	45	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
47	46	adantr 480	. . . . . . . . . . . . . . . . 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 12842	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 = (ℤ≥‘0)
49	47, 48	eleqtrdi 2839	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 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 12562	. . . . . . . . . . . . . . . 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 13484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) ∈ (ℤ≥‘0) ∧ (♯‘𝐹) ∈ ℤ) → ((𝑛 + 1) ∈ (0...(♯‘𝐹)) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
53	49, 51, 52	syl2anc 584	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . 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 4789	. . . . . . . . . . . 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 5395	. . . . . . . . . . . . 13 ⊢ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ V
58	57	elpw 4570	. . . . . . . . . . . 12 ⊢ ({(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉 ↔ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
59	56, 58	sylibr 234	. . . . . . . . . . 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 4537	. . . . . . . . . . 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 587	. . . . . . . . . 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 4575	. . . . . . . . 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 3948	. . . . . . . 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 562	. . . . . . 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 282	. . . . . 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 283	. . . . 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 2728	. . . 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 420	. . 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 29	. 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 47	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408   ⊆ wss 3917  ∅c0 4299  ifcif 4491  𝒫 cpw 4566  {cpr 4594  ⟨cop 4598   class class class wbr 5110   ↾ cres 5643   “ cima 5644  Fun wfun 6508  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   ≤ cle 11216  2c2 12248  ℕ₀cn0 12449  ℤcz 12536  ℤ_≥cuz 12800  ...cfz 13475  ..^cfzo 13622  ♯chash 14302   ∥ cdvds 16229  Vtxcvtx 28930  iEdgciedg 28931  UPGraphcupgr 29014  VtxDegcvtxdg 29400  Walkscwlks 29531  EulerPathsceupth 30133

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-rp 12959  df-xadd 13080  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-word 14486  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16230  df-vtx 28932  df-iedg 28933  df-edg 28982  df-uhgr 28992  df-ushgr 28993  df-upgr 29016  df-uspgr 29084  df-vtxdg 29401  df-wlks 29534  df-trls 29627  df-eupth 30134

This theorem is referenced by:  eupth2  30175