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Theorem eupth2lems 26964
Description: Lemma for eupth2 26965 (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 26965. (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 (𝜑𝐺 ∈ UPGraph )
eupth2.f (𝜑 → Fun 𝐼)
eupth2.p (𝜑𝐹(EulerPaths‘𝐺)𝑃)
Assertion
Ref Expression
eupth2lems ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 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.
StepHypRef Expression
1 nn0re 11245 . . . . . 6 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
21adantl 482 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
32lep1d 10899 . . . 4 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ≤ (𝑛 + 1))
4 peano2re 10153 . . . . . 6 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
52, 4syl 17 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℝ)
6 eupth2.p . . . . . . . 8 (𝜑𝐹(EulerPaths‘𝐺)𝑃)
7 eupthiswlk 26938 . . . . . . . 8 (𝐹(EulerPaths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
8 wlkcl 26381 . . . . . . . 8 (𝐹(Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)
96, 7, 83syl 18 . . . . . . 7 (𝜑 → (#‘𝐹) ∈ ℕ0)
109nn0red 11296 . . . . . 6 (𝜑 → (#‘𝐹) ∈ ℝ)
1110adantr 481 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → (#‘𝐹) ∈ ℝ)
12 letr 10075 . . . . 5 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (#‘𝐹) ∈ ℝ) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (#‘𝐹)) → 𝑛 ≤ (#‘𝐹)))
132, 5, 11, 12syl3anc 1323 . . . 4 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑛 + 1) ∧ (𝑛 + 1) ≤ (#‘𝐹)) → 𝑛 ≤ (#‘𝐹)))
143, 13mpand 710 . . 3 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) ≤ (#‘𝐹) → 𝑛 ≤ (#‘𝐹)))
1514imim1d 82 . 2 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))))
16 fveq2 6148 . . . . . . . . 9 (𝑥 = 𝑦 → ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) = ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦))
1716breq2d 4625 . . . . . . . 8 (𝑥 = 𝑦 → (2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
1817notbid 308 . . . . . . 7 (𝑥 = 𝑦 → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)))
1918elrab 3346 . . . . . 6 (𝑦 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} ↔ (𝑦𝑉 ∧ ¬ 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 (𝜑𝐺 ∈ UPGraph )
2322ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝐺 ∈ UPGraph )
24 eupth2.f . . . . . . . . . 10 (𝜑 → Fun 𝐼)
2524ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → Fun 𝐼)
266ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝐹(EulerPaths‘𝐺)𝑃)
27 eqid 2621 . . . . . . . . 9 𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩
28 eqid 2621 . . . . . . . . 9 𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩ = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩
29 simpr 477 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
3029ad2antrr 761 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝑛 ∈ ℕ0)
31 simprl 793 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ≤ (#‘𝐹))
3231adantr 481 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → (𝑛 + 1) ≤ (#‘𝐹))
33 simpr 477 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → 𝑦𝑉)
34 simplrr 800 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))
3520, 21, 23, 25, 26, 27, 28, 30, 32, 33, 34eupth2lem3 26962 . . . . . . . 8 ((((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) ∧ 𝑦𝑉) → (¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦) ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
3635pm5.32da 672 . . . . . . 7 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → ((𝑦𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)) ↔ (𝑦𝑉𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
37 0elpw 4794 . . . . . . . . . . 11 ∅ ∈ 𝒫 𝑉
3820wlkepvtx 26425 . . . . . . . . . . . . . . . 16 (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(#‘𝐹)) ∈ 𝑉))
3938simpld 475 . . . . . . . . . . . . . . 15 (𝐹(Walks‘𝐺)𝑃 → (𝑃‘0) ∈ 𝑉)
406, 7, 393syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑃‘0) ∈ 𝑉)
4140ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑃‘0) ∈ 𝑉)
4220wlkp 26382 . . . . . . . . . . . . . . . 16 (𝐹(Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
436, 7, 423syl 18 . . . . . . . . . . . . . . 15 (𝜑𝑃:(0...(#‘𝐹))⟶𝑉)
4443ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → 𝑃:(0...(#‘𝐹))⟶𝑉)
45 peano2nn0 11277 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
4645adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
4746adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ∈ ℕ0)
48 nn0uz 11666 . . . . . . . . . . . . . . . . 17 0 = (ℤ‘0)
4947, 48syl6eleq 2708 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ∈ (ℤ‘0))
509ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (#‘𝐹) ∈ ℕ0)
5150nn0zd 11424 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (#‘𝐹) ∈ ℤ)
52 elfz5 12276 . . . . . . . . . . . . . . . 16 (((𝑛 + 1) ∈ (ℤ‘0) ∧ (#‘𝐹) ∈ ℤ) → ((𝑛 + 1) ∈ (0...(#‘𝐹)) ↔ (𝑛 + 1) ≤ (#‘𝐹)))
5349, 51, 52syl2anc 692 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → ((𝑛 + 1) ∈ (0...(#‘𝐹)) ↔ (𝑛 + 1) ≤ (#‘𝐹)))
5431, 53mpbird 247 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑛 + 1) ∈ (0...(#‘𝐹)))
5544, 54ffvelrnd 6316 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑃‘(𝑛 + 1)) ∈ 𝑉)
5641, 55prssd 4322 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
57 prex 4870 . . . . . . . . . . . . 13 {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ V
5857elpw 4136 . . . . . . . . . . . 12 ({(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉 ↔ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ⊆ 𝑉)
5956, 58sylibr 224 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉)
60 ifcl 4102 . . . . . . . . . . 11 ((∅ ∈ 𝒫 𝑉 ∧ {(𝑃‘0), (𝑃‘(𝑛 + 1))} ∈ 𝒫 𝑉) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ∈ 𝒫 𝑉)
6137, 59, 60sylancr 694 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ∈ 𝒫 𝑉)
6261elpwid 4141 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ⊆ 𝑉)
6362sseld 3582 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) → 𝑦𝑉))
6463pm4.71rd 666 . . . . . . 7 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}) ↔ (𝑦𝑉𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
6536, 64bitr4d 271 . . . . . 6 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → ((𝑦𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑦)) ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
6619, 65syl5bb 272 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → (𝑦 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} ↔ 𝑦 ∈ if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))
6766eqrdv 2619 . . . 4 (((𝜑𝑛 ∈ ℕ0) ∧ ((𝑛 + 1) ≤ (#‘𝐹) ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}))) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))
6867exp32 630 . . 3 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) ≤ (#‘𝐹) → ({𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)}) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
6968a2d 29 . 2 ((𝜑𝑛 ∈ ℕ0) → (((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))
7015, 69syld 47 1 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 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 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3555  c0 3891  ifcif 4058  𝒫 cpw 4130  {cpr 4150  cop 4154   class class class wbr 4613  cres 5076  cima 5077  Fun wfun 5841  wf 5843  cfv 5847  (class class class)co 6604  cr 9879  0cc0 9880  1c1 9881   + caddc 9883  cle 10019  2c2 11014  0cn0 11236  cz 11321  cuz 11631  ...cfz 12268  ..^cfzo 12406  #chash 13057  cdvds 14907  Vtxcvtx 25774  iEdgciedg 25775   UPGraph cupgr 25871  VtxDegcvtxdg 26248  Walkscwlks 26362  EulerPathsceupth 26923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-rp 11777  df-xadd 11891  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-word 13238  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-dvds 14908  df-vtx 25776  df-iedg 25777  df-edg 25840  df-uhgr 25849  df-ushgr 25850  df-upgr 25873  df-uspgr 25938  df-vtxdg 26249  df-wlks 26365  df-trls 26458  df-eupth 26924
This theorem is referenced by:  eupth2  26965
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