Theorem eupth2lems 27659

Description: Lemma for eupth2 27660 (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 27660. (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 11657	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
2	1	adantl 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
3	2	lep1d 11312	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ≤ (𝑛 + 1))
4		peano2re 10551	. . . . . 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 27632	. . . . . . . 8 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
8		wlkcl 26980	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
9	6, 7, 8	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ0)
10	9	nn0red 11708	. . . . . 6 ⊢ (𝜑 → (♯‘𝐹) ∈ ℝ)
11	10	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (♯‘𝐹) ∈ ℝ)
12		letr 10472	. . . . 5 ⊢ ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (♯‘𝐹) ∈ ℝ) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
13	2, 5, 11, 12	syl3anc 1439	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
14	3, 13	mpand 685	. . 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 6448	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦))
17	16	breq2d 4900	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
18	17	notbid 310	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
19	18	elrab 3572	. . . . . 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 720	. . . . . . . . 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 720	. . . . . . . . 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 720	. . . . . . . . 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 2778	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩
28		eqid 2778	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩
29		simpr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
30	29	ad2antrr 716	. . . . . . . . 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 761	. . . . . . . . . 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 474	. . . . . . . . 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 479	. . . . . . . . 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 768	. . . . . . . . 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 27657	. . . . . . . 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 574	. . . . . . 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 5070	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑉
38	20	wlkepvtx 27024	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(♯‘𝐹)) ∈ 𝑉))
39	38	simpld 490	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘0) ∈ 𝑉)
40	6, 7, 39	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃‘0) ∈ 𝑉)
41	40	ad2antrr 716	. . . . . . . . . . . . 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 26981	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
43	6, 7, 42	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
44	43	ad2antrr 716	. . . . . . . . . . . . . 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 11689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
46	45	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
47	46	adantr 474	. . . . . . . . . . . . . . . . 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 12033	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 = (ℤ≥‘0)
49	47, 48	syl6eleq 2869	. . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . 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 11837	. . . . . . . . . . . . . . . 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 12656	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) ∈ (ℤ≥‘0) ∧ (♯‘𝐹) ∈ ℤ) → ((𝑛 + 1) ∈ (0...(♯‘𝐹)) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
53	49, 51, 52	syl2anc 579	. . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . 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	ffvelrnd 6626	. . . . . . . . . . . . 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 4586	. . . . . . . . . . . 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 5143	. . . . . . . . . . . . 13 ⊢ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ V
58	57	elpw 4385	. . . . . . . . . . . 12 ⊢ ({(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉 ↔ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
59	56, 58	sylibr 226	. . . . . . . . . . 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 4351	. . . . . . . . . . 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 581	. . . . . . . . . 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 4391	. . . . . . . . 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 3820	. . . . . . . 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 558	. . . . . . 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 274	. . . . . 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	syl5bb 275	. . . . 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 2776	. . . 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 413	. . 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 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {crab 3094   ⊆ wss 3792  ∅c0 4141  ifcif 4307  𝒫 cpw 4379  {cpr 4400  ⟨cop 4404   class class class wbr 4888   ↾ cres 5359   “ cima 5360  Fun wfun 6131  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  ℝcr 10273  0cc0 10274  1c1 10275   + caddc 10277   ≤ cle 10414  2c2 11435  ℕ₀cn0 11647  ℤcz 11733  ℤ_≥cuz 11997  ...cfz 12648  ..^cfzo 12789  ♯chash 13441   ∥ cdvds 15396  Vtxcvtx 26361  iEdgciedg 26362  UPGraphcupgr 26445  VtxDegcvtxdg 26830  Walkscwlks 26961  EulerPathsceupth 27617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-inf 8639  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-xnn0 11720  df-z 11734  df-uz 11998  df-rp 12143  df-xadd 12263  df-fz 12649  df-fzo 12790  df-seq 13125  df-exp 13184  df-hash 13442  df-word 13606  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-dvds 15397  df-vtx 26363  df-iedg 26364  df-edg 26413  df-uhgr 26423  df-ushgr 26424  df-upgr 26447  df-uspgr 26516  df-vtxdg 26831  df-wlks 26964  df-trls 27060  df-eupth 27618

This theorem is referenced by:  eupth2  27660