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Theorem eupth2lems 30000

Description: Lemma for eupth2 30001 (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 30001. (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 12485	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℝ)
2	1	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℝ)
3	2	lep1d 12149	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ≤ (𝑛 + 1))
4		peano2re 11391	. . . . . 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 29974	. . . . . . . 8 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
8		wlkcl 29381	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
9	6, 7, 8	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
10	9	nn0red 12537	. . . . . 6 ⊢ (𝜑 → (♯‘𝐹) ∈ ℝ)
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (♯‘𝐹) ∈ ℝ)
12		letr 11312	. . . . 5 ⊢ ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (♯‘𝐹) ∈ ℝ) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
13	2, 5, 11, 12	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (♯‘𝐹)) → 𝑛 ≤ (♯‘𝐹)))
14	3, 13	mpand 692	. . 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 6885	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦))
17	16	breq2d 5153	. . . . . . . 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 3678	. . . . . 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 727	. . . . . . . . 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 727	. . . . . . . . 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 727	. . . . . . . . 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 2726	. . . . . . . . 9 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩
28		eqid 2726	. . . . . . . . 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 723	. . . . . . . . 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 768	. . . . . . . . . 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 775	. . . . . . . . 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 29998	. . . . . . . 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 578	. . . . . . 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 5347	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑉
38	20	wlkepvtx 29426	. . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . 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 29382	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
43	6, 7, 42	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
44	43	ad2antrr 723	. . . . . . . . . . . . . 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 12516	. . . . . . . . . . . . . . . . . . 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 12868	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
49	47, 48	eleqtrdi 2837	. . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 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 12588	. . . . . . . . . . . . . . . 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 13499	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) ∈ (ℤ_≥‘0) ∧ (♯‘𝐹) ∈ ℤ) → ((𝑛 + 1) ∈ (0...(♯‘𝐹)) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
53	49, 51, 52	syl2anc 583	. . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . 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 4820	. . . . . . . . . . . 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 5425	. . . . . . . . . . . . 13 ⊢ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ V
58	57	elpw 4601	. . . . . . . . . . . 12 ⊢ ({(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉 ↔ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
59	56, 58	sylibr 233	. . . . . . . . . . 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 4568	. . . . . . . . . . 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 586	. . . . . . . . . 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 4606	. . . . . . . . 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 3976	. . . . . . . 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 2724	. . . 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))}))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 ⊆ wss 3943 ∅c0 4317 ifcif 4523 𝒫 cpw 4597 {cpr 4625 ⟨cop 4629 class class class wbr 5141 ↾ cres 5671 “ cima 5672 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 ≤ cle 11253 2c2 12271 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13490 ..^cfzo 13633 ♯chash 14295 ∥ cdvds 16204 Vtxcvtx 28764 iEdgciedg 28765 UPGraphcupgr 28848 VtxDegcvtxdg 29231 Walkscwlks 29362 EulerPathsceupth 29959

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12981 df-xadd 13099 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-word 14471 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-vtx 28766 df-iedg 28767 df-edg 28816 df-uhgr 28826 df-ushgr 28827 df-upgr 28850 df-uspgr 28918 df-vtxdg 29232 df-wlks 29365 df-trls 29458 df-eupth 29960

This theorem is referenced by: eupth2 30001

W3C validator