Theorem eupth2lems 30219

Description: Lemma for eupth2 30220 (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 30220. (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 12510	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
2	1	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
3	2	lep1d 12173	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ≤ (𝑛 + 1))
4		peano2re 11408	. . . . . 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 30193	. . . . . . . 8 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
8		wlkcl 29595	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
9	6, 7, 8	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ0)
10	9	nn0red 12563	. . . . . 6 ⊢ (𝜑 → (♯‘𝐹) ∈ ℝ)
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (♯‘𝐹) ∈ ℝ)
12		letr 11329	. . . . 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 6876	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦))
17	16	breq2d 5131	. . . . . . . 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 3671	. . . . . 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 2735	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩
28		eqid 2735	. . . . . . . . 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 30217	. . . . . . . 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 5326	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑉
38	20	wlkepvtx 29640	. . . . . . . . . . . . . . . 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 29596	. . . . . . . . . . . . . . . 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 12541	. . . . . . . . . . . . . . . . . . 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 12894	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 = (ℤ≥‘0)
49	47, 48	eleqtrdi 2844	. . . . . . . . . . . . . . . 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 12614	. . . . . . . . . . . . . . . 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 13533	. . . . . . . . . . . . . . . 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 7075	. . . . . . . . . . . . 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 4798	. . . . . . . . . . . 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 5407	. . . . . . . . . . . . 13 ⊢ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ V
58	57	elpw 4579	. . . . . . . . . . . 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 4546	. . . . . . . . . . 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 4584	. . . . . . . . 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 3957	. . . . . . . 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 2733	. . . 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 2108  {crab 3415   ⊆ wss 3926  ∅c0 4308  ifcif 4500  𝒫 cpw 4575  {cpr 4603  ⟨cop 4607   class class class wbr 5119   ↾ cres 5656   “ cima 5657  Fun wfun 6525  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  ℝcr 11128  0cc0 11129  1c1 11130   + caddc 11132   ≤ cle 11270  2c2 12295  ℕ₀cn0 12501  ℤcz 12588  ℤ_≥cuz 12852  ...cfz 13524  ..^cfzo 13671  ♯chash 14348   ∥ cdvds 16272  Vtxcvtx 28975  iEdgciedg 28976  UPGraphcupgr 29059  VtxDegcvtxdg 29445  Walkscwlks 29576  EulerPathsceupth 30178

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8719  df-map 8842  df-pm 8843  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-n0 12502  df-xnn0 12575  df-z 12589  df-uz 12853  df-rp 13009  df-xadd 13129  df-fz 13525  df-fzo 13672  df-seq 14020  df-exp 14080  df-hash 14349  df-word 14532  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-dvds 16273  df-vtx 28977  df-iedg 28978  df-edg 29027  df-uhgr 29037  df-ushgr 29038  df-upgr 29061  df-uspgr 29129  df-vtxdg 29446  df-wlks 29579  df-trls 29672  df-eupth 30179

This theorem is referenced by:  eupth2  30220