Theorem eupth2 29246

Description: The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (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
eupth2	⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))

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

Allowed substitution hints:   𝑃(𝑥)   𝐺(𝑥)

Proof of Theorem eupth2

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupth2.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
2		eupth2.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
3		eupth2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
4		eupth2.f	. . . . . . 7 ⊢ (𝜑 → Fun 𝐼)
5		eupth2.p	. . . . . . 7 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
6		eqid 2731	. . . . . . 7 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩
7	1, 2, 3, 4, 5, 6	eupthvdres 29242	. . . . . 6 ⊢ (𝜑 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = (VtxDeg‘𝐺))
8	7	fveq1d 6849	. . . . 5 ⊢ (𝜑 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) = ((VtxDeg‘𝐺)‘𝑥))
9	8	breq2d 5122	. . . 4 ⊢ (𝜑 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
10	9	notbid 317	. . 3 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
11	10	rabbidv 3413	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
12		eupthiswlk 29219	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13		wlkcl 28626	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
14	5, 12, 13	3syl 18	. . 3 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ0)
15		nn0re 12431	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℝ)
16	15	leidd 11730	. . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ≤ (♯‘𝐹))
17		breq1 5113	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 ≤ (♯‘𝐹) ↔ 0 ≤ (♯‘𝐹)))
18		oveq2 7370	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (0..^𝑚) = (0..^0))
19	18	imaeq2d 6018	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^0)))
20	19	reseq2d 5942	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^0))))
21	20	opeq2d 4842	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)
22	21	fveq2d 6851	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩))
23	22	fveq1d 6849	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥))
24	23	breq2d 5122	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
25	24	notbid 317	. . . . . . . . 9 ⊢ (𝑚 = 0 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
26	25	rabbidv 3413	. . . . . . . 8 ⊢ (𝑚 = 0 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)})
27		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑃‘𝑚) = (𝑃‘0))
28	27	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘0)))
29	27	preq2d 4706	. . . . . . . . 9 ⊢ (𝑚 = 0 → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘0)})
30	28, 29	ifbieq2d 4517	. . . . . . . 8 ⊢ (𝑚 = 0 → if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
31	26, 30	eqeq12d 2747	. . . . . . 7 ⊢ (𝑚 = 0 → ({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) ↔ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)})))
32	17, 31	imbi12d 344	. . . . . 6 ⊢ (𝑚 = 0 → ((𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)})) ↔ (0 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))))
33	32	imbi2d 340	. . . . 5 ⊢ (𝑚 = 0 → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}))) ↔ (𝜑 → (0 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)})))))
34		breq1 5113	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ (♯‘𝐹) ↔ 𝑛 ≤ (♯‘𝐹)))
35		oveq2 7370	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (0..^𝑚) = (0..^𝑛))
36	35	imaeq2d 6018	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^𝑛)))
37	36	reseq2d 5942	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^𝑛))))
38	37	opeq2d 4842	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)
39	38	fveq2d 6851	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩))
40	39	fveq1d 6849	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥))
41	40	breq2d 5122	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
42	41	notbid 317	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
43	42	rabbidv 3413	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)})
44		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑃‘𝑚) = (𝑃‘𝑛))
45	44	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘𝑛)))
46	44	preq2d 4706	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘𝑛)})
47	45, 46	ifbieq2d 4517	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))
48	43, 47	eqeq12d 2747	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) ↔ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})))
49	34, 48	imbi12d 344	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)})) ↔ (𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))))
50	49	imbi2d 340	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}))) ↔ (𝜑 → (𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})))))
51		breq1 5113	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ (♯‘𝐹) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
52		oveq2 7370	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 + 1) → (0..^𝑚) = (0..^(𝑛 + 1)))
53	52	imaeq2d 6018	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 + 1) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(𝑛 + 1))))
54	53	reseq2d 5942	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1)))))
55	54	opeq2d 4842	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)
56	55	fveq2d 6851	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩))
57	56	fveq1d 6849	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥))
58	57	breq2d 5122	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
59	58	notbid 317	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
60	59	rabbidv 3413	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)})
61		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝑃‘𝑚) = (𝑃‘(𝑛 + 1)))
62	61	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘(𝑛 + 1))))
63	61	preq2d 4706	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘(𝑛 + 1))})
64	62, 63	ifbieq2d 4517	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
65	60, 64	eqeq12d 2747	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → ({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) ↔ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
66	51, 65	imbi12d 344	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)})) ↔ ((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))}))))
67	66	imbi2d 340	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}))) ↔ (𝜑 → ((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))})))))
68		breq1 5113	. . . . . . 7 ⊢ (𝑚 = (♯‘𝐹) → (𝑚 ≤ (♯‘𝐹) ↔ (♯‘𝐹) ≤ (♯‘𝐹)))
69		oveq2 7370	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (♯‘𝐹) → (0..^𝑚) = (0..^(♯‘𝐹)))
70	69	imaeq2d 6018	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (♯‘𝐹) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(♯‘𝐹))))
71	70	reseq2d 5942	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (♯‘𝐹) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
72	71	opeq2d 4842	. . . . . . . . . . . . 13 ⊢ (𝑚 = (♯‘𝐹) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
73	72	fveq2d 6851	. . . . . . . . . . . 12 ⊢ (𝑚 = (♯‘𝐹) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩))
74	73	fveq1d 6849	. . . . . . . . . . 11 ⊢ (𝑚 = (♯‘𝐹) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥))
75	74	breq2d 5122	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘𝐹) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
76	75	notbid 317	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
77	76	rabbidv 3413	. . . . . . . 8 ⊢ (𝑚 = (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)})
78		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘𝐹) → (𝑃‘𝑚) = (𝑃‘(♯‘𝐹)))
79	78	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
80	78	preq2d 4706	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
81	79, 80	ifbieq2d 4517	. . . . . . . 8 ⊢ (𝑚 = (♯‘𝐹) → if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
82	77, 81	eqeq12d 2747	. . . . . . 7 ⊢ (𝑚 = (♯‘𝐹) → ({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}) ↔ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})))
83	68, 82	imbi12d 344	. . . . . 6 ⊢ (𝑚 = (♯‘𝐹) → ((𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)})) ↔ ((♯‘𝐹) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))))
84	83	imbi2d 340	. . . . 5 ⊢ (𝑚 = (♯‘𝐹) → ((𝜑 → (𝑚 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑚), ∅, {(𝑃‘0), (𝑃‘𝑚)}))) ↔ (𝜑 → ((♯‘𝐹) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})))))
85	1, 2, 3, 4, 5	eupth2lemb 29244	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)
86		eqid 2731	. . . . . . . 8 ⊢ (𝑃‘0) = (𝑃‘0)
87	86	iftruei 4498	. . . . . . 7 ⊢ if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}) = ∅
88	85, 87	eqtr4di 2789	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
89	88	a1d 25	. . . . 5 ⊢ (𝜑 → (0 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)})))
90	1, 2, 3, 4, 5	eupth2lems 29245	. . . . . . 7 ⊢ ((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))}))))
91	90	expcom 414	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝜑 → ((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))})))))
92	91	a2d 29	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝜑 → (𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)}))) → (𝜑 → ((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))})))))
93	33, 50, 67, 84, 89, 92	nn0ind 12607	. . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (𝜑 → ((♯‘𝐹) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))))
94	16, 93	mpid 44	. . 3 ⊢ ((♯‘𝐹) ∈ ℕ0 → (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})))
95	14, 94	mpcom 38	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
96	11, 95	eqtr3d 2773	1 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2106  {crab 3405  ∅c0 4287  ifcif 4491  {cpr 4593  ⟨cop 4597   class class class wbr 5110   ↾ cres 5640   “ cima 5641  Fun wfun 6495  ‘cfv 6501  (class class class)co 7362  0cc0 11060  1c1 11061   + caddc 11063   ≤ cle 11199  2c2 12217  ℕ₀cn0 12422  ..^cfzo 13577  ♯chash 14240   ∥ cdvds 16147  Vtxcvtx 28010  iEdgciedg 28011  UPGraphcupgr 28094  VtxDegcvtxdg 28476  Walkscwlks 28607  EulerPathsceupth 29204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9387  df-inf 9388  df-dju 9846  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-xnn0 12495  df-z 12509  df-uz 12773  df-rp 12925  df-xadd 13043  df-fz 13435  df-fzo 13578  df-seq 13917  df-exp 13978  df-hash 14241  df-word 14415  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-dvds 16148  df-vtx 28012  df-iedg 28013  df-edg 28062  df-uhgr 28072  df-ushgr 28073  df-upgr 28096  df-uspgr 28164  df-vtxdg 28477  df-wlks 28610  df-trls 28703  df-eupth 29205

This theorem is referenced by:  eulerpathpr  29247  eulercrct  29249