Theorem eupth2fi 16600

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)
eupth2fi.g	⊢ (𝜑 → 𝐺 ∈ UMGraph)
eupth2.f	⊢ (𝜑 → Fun 𝐼)
eupth2.p	⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
eupth2fi.fi	⊢ (𝜑 → 𝑉 ∈ Fin)

Assertion

Ref	Expression
eupth2fi	⊢ (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 eupth2fi

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		eupth2fi.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ UMGraph)
4		eupth2.f	. . . . . . 7 ⊢ (𝜑 → Fun 𝐼)
5		eupth2.p	. . . . . . 7 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
6		eqid 2234	. . . . . . 7 ⊢ ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩
7	1, 2, 3, 4, 5, 6	eupthvdres 16596	. . . . . 6 ⊢ (𝜑 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = (VtxDeg‘𝐺))
8	7	fveq1d 5677	. . . . 5 ⊢ (𝜑 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) = ((VtxDeg‘𝐺)‘𝑥))
9	8	breq2d 4126	. . . 4 ⊢ (𝜑 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
10	9	notbid 673	. . 3 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)))
11	10	rabbidv 2804	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
12		eupthiswlk 16576	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13		wlkcl 16453	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
14	5, 12, 13	3syl 17	. . 3 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ0)
15		nn0re 9522	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℝ)
16	15	leidd 8805	. . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ≤ (♯‘𝐹))
17		breq1 4117	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 ≤ (♯‘𝐹) ↔ 0 ≤ (♯‘𝐹)))
18		oveq2 6066	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (0..^𝑚) = (0..^0))
19	18	imaeq2d 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^0)))
20	19	reseq2d 5043	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^0))))
21	20	opeq2d 3895	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)
22	21	fveq2d 5679	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩))
23	22	fveq1d 5677	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥))
24	23	breq2d 4126	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
25	24	notbid 673	. . . . . . . . 9 ⊢ (𝑚 = 0 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)))
26	25	rabbidv 2804	. . . . . . . 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 5675	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑃‘𝑚) = (𝑃‘0))
28	27	eqeq2d 2246	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘0)))
29	27	preq2d 3780	. . . . . . . . 9 ⊢ (𝑚 = 0 → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘0)})
30	28, 29	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 234	. . . . . 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 230	. . . . 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 4117	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ (♯‘𝐹) ↔ 𝑛 ≤ (♯‘𝐹)))
35		oveq2 6066	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (0..^𝑚) = (0..^𝑛))
36	35	imaeq2d 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^𝑛)))
37	36	reseq2d 5043	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^𝑛))))
38	37	opeq2d 3895	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)
39	38	fveq2d 5679	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩))
40	39	fveq1d 5677	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥))
41	40	breq2d 4126	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
42	41	notbid 673	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)))
43	42	rabbidv 2804	. . . . . . . 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 5675	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑃‘𝑚) = (𝑃‘𝑛))
45	44	eqeq2d 2246	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘𝑛)))
46	44	preq2d 3780	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘𝑛)})
47	45, 46	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 234	. . . . . 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 230	. . . . 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 4117	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ (♯‘𝐹) ↔ (𝑛 + 1) ≤ (♯‘𝐹)))
52		oveq2 6066	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 + 1) → (0..^𝑚) = (0..^(𝑛 + 1)))
53	52	imaeq2d 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 + 1) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(𝑛 + 1))))
54	53	reseq2d 5043	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1)))))
55	54	opeq2d 3895	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)
56	55	fveq2d 5679	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩))
57	56	fveq1d 5677	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥))
58	57	breq2d 4126	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
59	58	notbid 673	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)))
60	59	rabbidv 2804	. . . . . . . 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 5675	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝑃‘𝑚) = (𝑃‘(𝑛 + 1)))
62	61	eqeq2d 2246	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘(𝑛 + 1))))
63	61	preq2d 3780	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘(𝑛 + 1))})
64	62, 63	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 234	. . . . . 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 230	. . . . 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 4117	. . . . . . 7 ⊢ (𝑚 = (♯‘𝐹) → (𝑚 ≤ (♯‘𝐹) ↔ (♯‘𝐹) ≤ (♯‘𝐹)))
69		oveq2 6066	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (♯‘𝐹) → (0..^𝑚) = (0..^(♯‘𝐹)))
70	69	imaeq2d 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (♯‘𝐹) → (𝐹 “ (0..^𝑚)) = (𝐹 “ (0..^(♯‘𝐹))))
71	70	reseq2d 5043	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (♯‘𝐹) → (𝐼 ↾ (𝐹 “ (0..^𝑚))) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
72	71	opeq2d 3895	. . . . . . . . . . . . 13 ⊢ (𝑚 = (♯‘𝐹) → ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
73	72	fveq2d 5679	. . . . . . . . . . . 12 ⊢ (𝑚 = (♯‘𝐹) → (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩) = (VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩))
74	73	fveq1d 5677	. . . . . . . . . . 11 ⊢ (𝑚 = (♯‘𝐹) → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥))
75	74	breq2d 4126	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘𝐹) → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
76	75	notbid 673	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑚)))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)))
77	76	rabbidv 2804	. . . . . . . 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 5675	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘𝐹) → (𝑃‘𝑚) = (𝑃‘(♯‘𝐹)))
79	78	eqeq2d 2246	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → ((𝑃‘0) = (𝑃‘𝑚) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
80	78	preq2d 3780	. . . . . . . . 9 ⊢ (𝑚 = (♯‘𝐹) → {(𝑃‘0), (𝑃‘𝑚)} = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
81	79, 80	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 234	. . . . . 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 230	. . . . 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		eupth2fi.fi	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ Fin)
86	1, 2, 3, 4, 5, 85	eupth2lembfi 16598	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)
87		eqid 2234	. . . . . . . 8 ⊢ (𝑃‘0) = (𝑃‘0)
88	87	iftruei 3632	. . . . . . 7 ⊢ if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}) = ∅
89	86, 88	eqtr4di 2285	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)}))
90	89	a1d 22	. . . . 5 ⊢ (𝜑 → (0 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘0), ∅, {(𝑃‘0), (𝑃‘0)})))
91	1, 2, 3, 4, 5, 85	eupth2lemsfi 16599	. . . . . . 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))}))))
92	91	expcom 116	. . . . . 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))})))))
93	92	a2d 26	. . . . 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))})))))
94	33, 50, 67, 84, 90, 93	nn0ind 9710	. . . 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))}))))
95	16, 94	mpid 42	. . 3 ⊢ ((♯‘𝐹) ∈ ℕ0 → (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})))
96	14, 95	mpcom 36	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))
97	11, 96	eqtr3d 2269	1 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ∈ wcel 2205  {crab 2526  ∅c0 3512  ifcif 3624  {cpr 3695  ⟨cop 3697   class class class wbr 4114   ↾ cres 4756   “ cima 4757  Fun wfun 5351  ‘cfv 5357  (class class class)co 6058  Fincfn 6988  0cc0 8143  1c1 8144   + caddc 8146   ≤ cle 8325  2c2 9305  ℕ₀cn0 9513  ..^cfzo 10498  ♯chash 11163   ∥ cdvds 12498  Vtxcvtx 16133  iEdgciedg 16134  UMGraphcumgr 16213  VtxDegcvtxdg 16407  Walkscwlks 16438  EulerPathsceupth 16563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-q 9970  df-rp 10005  df-xadd 10125  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-word 11250  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-ushgrm 16191  df-upgren 16214  df-umgren 16215  df-uspgren 16276  df-subgr 16375  df-vtxdg 16408  df-wlks 16439  df-trls 16502  df-eupth 16564

This theorem is referenced by:  eulerpathprum  16601