Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eupth2lem3lem3 Structured version   Visualization version   GIF version

Theorem eupth2lem3lem3 41398
 Description: Lemma for eupth2lem3 41404, formerly part of proof of eupath2lem3 26506: If a loop {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)
Hypotheses
Ref Expression
trlsegvdeg.v 𝑉 = (Vtx‘𝐺)
trlsegvdeg.i 𝐼 = (iEdg‘𝐺)
trlsegvdeg.f (𝜑 → Fun 𝐼)
trlsegvdeg.n (𝜑𝑁 ∈ (0..^(#‘𝐹)))
trlsegvdeg.u (𝜑𝑈𝑉)
trlsegvdeg.w (𝜑𝐹(TrailS‘𝐺)𝑃)
trlsegvdeg.vx (𝜑 → (Vtx‘𝑋) = 𝑉)
trlsegvdeg.vy (𝜑 → (Vtx‘𝑌) = 𝑉)
trlsegvdeg.vz (𝜑 → (Vtx‘𝑍) = 𝑉)
trlsegvdeg.ix (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
trlsegvdeg.iy (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
trlsegvdeg.iz (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
eupth2lem3.o (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))
eupth2lem3lem3.e (𝜑 → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))
Assertion
Ref Expression
eupth2lem3lem3 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))
Distinct variable groups:   𝑥,𝑈   𝑥,𝑉   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝑃(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝐼(𝑥)   𝑁(𝑥)   𝑌(𝑥)   𝑍(𝑥)

Proof of Theorem eupth2lem3lem3
StepHypRef Expression
1 trlsegvdeg.u . . . . 5 (𝜑𝑈𝑉)
2 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑈 → ((VtxDeg‘𝑋)‘𝑥) = ((VtxDeg‘𝑋)‘𝑈))
32breq2d 4595 . . . . . . 7 (𝑥 = 𝑈 → (2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
43notbid 307 . . . . . 6 (𝑥 = 𝑈 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
54elrab3 3332 . . . . 5 (𝑈𝑉 → (𝑈 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
61, 5syl 17 . . . 4 (𝜑 → (𝑈 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
7 eupth2lem3.o . . . . 5 (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))
87eleq2d 2673 . . . 4 (𝜑 → (𝑈 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
96, 8bitr3d 269 . . 3 (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
109adantr 480 . 2 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
11 2z 11286 . . . . . 6 2 ∈ ℤ
1211a1i 11 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → 2 ∈ ℤ)
13 trlsegvdeg.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
14 trlsegvdeg.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
15 trlsegvdeg.f . . . . . . . 8 (𝜑 → Fun 𝐼)
16 trlsegvdeg.n . . . . . . . 8 (𝜑𝑁 ∈ (0..^(#‘𝐹)))
17 trlsegvdeg.w . . . . . . . 8 (𝜑𝐹(TrailS‘𝐺)𝑃)
18 trlsegvdeg.vx . . . . . . . 8 (𝜑 → (Vtx‘𝑋) = 𝑉)
19 trlsegvdeg.vy . . . . . . . 8 (𝜑 → (Vtx‘𝑌) = 𝑉)
20 trlsegvdeg.vz . . . . . . . 8 (𝜑 → (Vtx‘𝑍) = 𝑉)
21 trlsegvdeg.ix . . . . . . . 8 (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
22 trlsegvdeg.iy . . . . . . . 8 (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
23 trlsegvdeg.iz . . . . . . . 8 (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
2413, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23eupth2lem3lem1 41396 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0)
2524nn0zd 11356 . . . . . 6 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ)
2625adantr 480 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ)
2713, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23eupth2lem3lem2 41397 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ0)
2827nn0zd 11356 . . . . . 6 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ)
2928adantr 480 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ)
30 iddvds 14833 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 2)
3111, 30ax-mp 5 . . . . . . 7 2 ∥ 2
3219ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (Vtx‘𝑌) = 𝑉)
33 fvex 6113 . . . . . . . . 9 (𝐹𝑁) ∈ V
3433a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (𝐹𝑁) ∈ V)
351ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → 𝑈𝑉)
3622ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
37 eupth2lem3lem3.e . . . . . . . . . . . . . 14 (𝜑 → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))
3837adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))
39 ifptru 1017 . . . . . . . . . . . . . 14 ((𝑃𝑁) = (𝑃‘(𝑁 + 1)) → (if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))) ↔ (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}))
4039adantl 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))) ↔ (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}))
4138, 40mpbid 221 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)})
42 sneq 4135 . . . . . . . . . . . . 13 ((𝑃𝑁) = 𝑈 → {(𝑃𝑁)} = {𝑈})
4342eqcoms 2618 . . . . . . . . . . . 12 (𝑈 = (𝑃𝑁) → {(𝑃𝑁)} = {𝑈})
4441, 43sylan9eq 2664 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (𝐼‘(𝐹𝑁)) = {𝑈})
4544opeq2d 4347 . . . . . . . . . 10 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → ⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩ = ⟨(𝐹𝑁), {𝑈}⟩)
4645sneqd 4137 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {⟨(𝐹𝑁), {𝑈}⟩})
4736, 46eqtrd 2644 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {𝑈}⟩})
4832, 34, 35, 471loopgrvd2 40718 . . . . . . 7 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 2)
4931, 48syl5breqr 4621 . . . . . 6 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
50 dvds0 14835 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 0)
5111, 50ax-mp 5 . . . . . . 7 2 ∥ 0
5219ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (Vtx‘𝑌) = 𝑉)
5333a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝐹𝑁) ∈ V)
5413, 14, 15, 16, 1, 17trlsegvdeglem1 41388 . . . . . . . . . 10 (𝜑 → ((𝑃𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉))
5554simpld 474 . . . . . . . . 9 (𝜑 → (𝑃𝑁) ∈ 𝑉)
5655ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝑃𝑁) ∈ 𝑉)
5722adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
5841opeq2d 4347 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩ = ⟨(𝐹𝑁), {(𝑃𝑁)}⟩)
5958sneqd 4137 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
6057, 59eqtrd 2644 . . . . . . . . 9 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
6160adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
621adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → 𝑈𝑉)
6362anim1i 590 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝑈𝑉𝑈 ≠ (𝑃𝑁)))
64 eldifsn 4260 . . . . . . . . 9 (𝑈 ∈ (𝑉 ∖ {(𝑃𝑁)}) ↔ (𝑈𝑉𝑈 ≠ (𝑃𝑁)))
6563, 64sylibr 223 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → 𝑈 ∈ (𝑉 ∖ {(𝑃𝑁)}))
6652, 53, 56, 61, 651loopgrvd0 40719 . . . . . . 7 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 0)
6751, 66syl5breqr 4621 . . . . . 6 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
6849, 67pm2.61dane 2869 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
69 dvdsadd2b 14866 . . . . 5 ((2 ∈ ℤ ∧ ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ ∧ (((VtxDeg‘𝑌)‘𝑈) ∈ ℤ ∧ 2 ∥ ((VtxDeg‘𝑌)‘𝑈))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈))))
7012, 26, 29, 68, 69syl112anc 1322 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈))))
7127nn0cnd 11230 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℂ)
7224nn0cnd 11230 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℂ)
7371, 72addcomd 10117 . . . . . 6 (𝜑 → (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
7473breq2d 4595 . . . . 5 (𝜑 → (2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7574adantr 480 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7670, 75bitrd 267 . . 3 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7776notbid 307 . 2 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ ¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
78 simpr 476 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (𝑃𝑁) = (𝑃‘(𝑁 + 1)))
7978eqeq2d 2620 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((𝑃‘0) = (𝑃𝑁) ↔ (𝑃‘0) = (𝑃‘(𝑁 + 1))))
8078preq2d 4219 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → {(𝑃‘0), (𝑃𝑁)} = {(𝑃‘0), (𝑃‘(𝑁 + 1))})
8179, 80ifbieq2d 4061 . . 3 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}) = if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))
8281eleq2d 2673 . 2 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))
8310, 77, 823bitr3d 297 1 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979   ∥ cdvds 14821  Vtxcvtx 25673  iEdgciedg 25674  VtxDegcvtxdg 40681  TrailSctrls 40899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-dvds 14822  df-uhgr 25724  df-ushgr 25725  df-edga 25793  df-uspgr 40380  df-vtxdg 40682  df-1wlks 40800  df-trls 40901 This theorem is referenced by:  eupth2lem3lem7  41402
 Copyright terms: Public domain W3C validator