Theorem eupth2 30571

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 2763	. . . . . . 7 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩
7	1, 2, 3, 4, 5, 6	eupthvdres 30567	. . . . . 6 ⊢ (𝜑 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = (VtxDeg‘𝐺))
8	7	fveq1d 6885	. . . . 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 321	. . 3 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
11	10	rabbidv 3423	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
12		eupthiswlk 30544	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13		wlkcl 29946	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
14	5, 12, 13	3syl 19	. . 3 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ0)
15		nn0re 12514	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℝ)
16	15	leidd 11781	. . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ≤ (♯‘𝐹))
17		breq1 5113	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 ≤ (♯‘𝐹) ↔ 0 ≤ (♯‘𝐹)))
18		oveq2 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (0..^𝑚) = (0..^0))
19	18	imaeq2d 6064	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^0)))
20	19	reseq2d 5980	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^0))))
21	20	opeq2d 4846	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)
22	21	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩))
23	22	fveq1d 6885	. . . . . . . . . . 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 321	. . . . . . . . 9 ⊢ (𝑚 = 0 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
26	25	rabbidv 3423	. . . . . . . 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 6883	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑃‘𝑚) = (𝑃‘0))
28	27	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘0)))
29	27	preq2d 4707	. . . . . . . . 9 ⊢ (𝑚 = 0 → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘0)})
30	28, 29	ifbieq2d 4515	. . . . . . . 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 2779	. . . . . . 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 347	. . . . . 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 343	. . . . 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 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (0..^𝑚) = (0..^𝑛))
36	35	imaeq2d 6064	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^𝑛)))
37	36	reseq2d 5980	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^𝑛))))
38	37	opeq2d 4846	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)
39	38	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩))
40	39	fveq1d 6885	. . . . . . . . . . 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 321	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
43	42	rabbidv 3423	. . . . . . . 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 6883	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑃‘𝑚) = (𝑃‘𝑛))
45	44	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘𝑛)))
46	44	preq2d 4707	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘𝑛)})
47	45, 46	ifbieq2d 4515	. . . . . . . 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 2779	. . . . . . 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 347	. . . . . 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 343	. . . . 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 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 + 1) → (0..^𝑚) = (0..^(𝑛 + 1)))
53	52	imaeq2d 6064	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 + 1) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(𝑛 + 1))))
54	53	reseq2d 5980	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1)))))
55	54	opeq2d 4846	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)
56	55	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩))
57	56	fveq1d 6885	. . . . . . . . . . 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 321	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
60	59	rabbidv 3423	. . . . . . . 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 6883	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝑃‘𝑚) = (𝑃‘(𝑛 + 1)))
62	61	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘(𝑛 + 1))))
63	61	preq2d 4707	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘(𝑛 + 1))})
64	62, 63	ifbieq2d 4515	. . . . . . . 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 2779	. . . . . . 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 347	. . . . . 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 343	. . . . 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 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (♯‘𝐹) → (0..^𝑚) = (0..^(♯‘𝐹)))
70	69	imaeq2d 6064	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (♯‘𝐹) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(♯‘𝐹))))
71	70	reseq2d 5980	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (♯‘𝐹) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
72	71	opeq2d 4846	. . . . . . . . . . . . 13 ⊢ (𝑚 = (♯‘𝐹) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
73	72	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑚 = (♯‘𝐹) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩))
74	73	fveq1d 6885	. . . . . . . . . . 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 321	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
77	76	rabbidv 3423	. . . . . . . 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 6883	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘𝐹) → (𝑃‘𝑚) = (𝑃‘(♯‘𝐹)))
79	78	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
80	78	preq2d 4707	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
81	79, 80	ifbieq2d 4515	. . . . . . . 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 2779	. . . . . . 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 347	. . . . . 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 343	. . . . 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 30569	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)
86		eqid 2763	. . . . . . . 8 ⊢ (𝑃‘0) = (𝑃‘0)
87	86	iftruei 4495	. . . . . . 7 ⊢ if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}) = ∅
88	85, 87	eqtr4di 2816	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
89	88	a1d 26	. . . . 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 30570	. . . . . . 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 418	. . . . . 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 30	. . . . 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 12692	. . . 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 45	. . 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 39	. 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 2800	1 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  {crab 3416  ∅c0 4287  ifcif 4488  {cpr 4592  ⟨cop 4596   class class class wbr 5110   ↾ cres 5665   “ cima 5666  Fun wfun 6532  ‘cfv 6538  (class class class)co 7412  0cc0 11101  1c1 11102   + caddc 11104   ≤ cle 11245  2c2 12296  ℕ₀cn0 12505  ..^cfzo 13684  ♯chash 14368   ∥ cdvds 16311  Vtxcvtx 29327  iEdgciedg 29328  UPGraphcupgr 29411  VtxDegcvtxdg 29796  Walkscwlks 29927  EulerPathsceupth 30529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-rp 13018  df-xadd 13139  df-fz 13537  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-word 14553  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-dvds 16312  df-vtx 29329  df-iedg 29330  df-edg 29379  df-uhgr 29389  df-ushgr 29390  df-upgr 29413  df-uspgr 29481  df-vtxdg 29797  df-wlks 29930  df-trls 30021  df-eupth 30530

This theorem is referenced by:  eulerpathpr  30572  eulercrct  30574