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Theorem eupth2lem3lem3 28015
Description: Lemma for eupth2lem3 28021, formerly part of proof of eupth2lem3 28021: If a loop {(𝑃𝑁), (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 6645 . . . . . . . 8 (𝑥 = 𝑈 → ((VtxDeg‘𝑋)‘𝑥) = ((VtxDeg‘𝑋)‘𝑈))
32breq2d 5042 . . . . . . 7 (𝑥 = 𝑈 → (2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
43notbid 321 . . . . . 6 (𝑥 = 𝑈 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
54elrab3 3629 . . . . 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 2875 . . . 4 (𝜑 → (𝑈 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
96, 8bitr3d 284 . . 3 (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
109adantr 484 . 2 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
11 2z 12002 . . . . . 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 28013 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0)
2524nn0zd 12073 . . . . . 6 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ)
2625adantr 484 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ)
2713, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23eupth2lem3lem2 28014 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ0)
2827nn0zd 12073 . . . . . 6 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ)
2928adantr 484 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ)
30 z2even 15711 . . . . . . 7 2 ∥ 2
3119ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (Vtx‘𝑌) = 𝑉)
32 fvexd 6660 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (𝐹𝑁) ∈ V)
331ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → 𝑈𝑉)
3422ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
35 eupth2lem3lem3.e . . . . . . . . . . . . . 14 (𝜑 → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))
3635adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))
37 ifptru 1071 . . . . . . . . . . . . . 14 ((𝑃𝑁) = (𝑃‘(𝑁 + 1)) → (if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))) ↔ (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}))
3837adantl 485 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))) ↔ (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}))
3936, 38mpbid 235 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)})
40 sneq 4535 . . . . . . . . . . . . 13 ((𝑃𝑁) = 𝑈 → {(𝑃𝑁)} = {𝑈})
4140eqcoms 2806 . . . . . . . . . . . 12 (𝑈 = (𝑃𝑁) → {(𝑃𝑁)} = {𝑈})
4239, 41sylan9eq 2853 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (𝐼‘(𝐹𝑁)) = {𝑈})
4342opeq2d 4772 . . . . . . . . . 10 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → ⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩ = ⟨(𝐹𝑁), {𝑈}⟩)
4443sneqd 4537 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {⟨(𝐹𝑁), {𝑈}⟩})
4534, 44eqtrd 2833 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {𝑈}⟩})
4631, 32, 33, 451loopgrvd2 27293 . . . . . . 7 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 2)
4730, 46breqtrrid 5068 . . . . . 6 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
48 z0even 15708 . . . . . . 7 2 ∥ 0
4919ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (Vtx‘𝑌) = 𝑉)
50 fvexd 6660 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝐹𝑁) ∈ V)
5113, 14, 15, 16, 1, 17trlsegvdeglem1 28005 . . . . . . . . . 10 (𝜑 → ((𝑃𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉))
5251simpld 498 . . . . . . . . 9 (𝜑 → (𝑃𝑁) ∈ 𝑉)
5352ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝑃𝑁) ∈ 𝑉)
5422adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
5539opeq2d 4772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩ = ⟨(𝐹𝑁), {(𝑃𝑁)}⟩)
5655sneqd 4537 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
5754, 56eqtrd 2833 . . . . . . . . 9 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
5857adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
591adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → 𝑈𝑉)
6059anim1i 617 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝑈𝑉𝑈 ≠ (𝑃𝑁)))
61 eldifsn 4680 . . . . . . . . 9 (𝑈 ∈ (𝑉 ∖ {(𝑃𝑁)}) ↔ (𝑈𝑉𝑈 ≠ (𝑃𝑁)))
6260, 61sylibr 237 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → 𝑈 ∈ (𝑉 ∖ {(𝑃𝑁)}))
6349, 50, 53, 58, 621loopgrvd0 27294 . . . . . . 7 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 0)
6448, 63breqtrrid 5068 . . . . . 6 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
6547, 64pm2.61dane 3074 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
66 dvdsadd2b 15648 . . . . 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))))
6712, 26, 29, 65, 66syl112anc 1371 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈))))
6827nn0cnd 11945 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℂ)
6924nn0cnd 11945 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℂ)
7068, 69addcomd 10831 . . . . . 6 (𝜑 → (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
7170breq2d 5042 . . . . 5 (𝜑 → (2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7271adantr 484 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7367, 72bitrd 282 . . 3 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7473notbid 321 . 2 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ ¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
75 simpr 488 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (𝑃𝑁) = (𝑃‘(𝑁 + 1)))
7675eqeq2d 2809 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((𝑃‘0) = (𝑃𝑁) ↔ (𝑃‘0) = (𝑃‘(𝑁 + 1))))
7775preq2d 4636 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → {(𝑃‘0), (𝑃𝑁)} = {(𝑃‘0), (𝑃‘(𝑁 + 1))})
7876, 77ifbieq2d 4450 . . 3 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}) = if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))
7978eleq2d 2875 . 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))})))
8010, 74, 793bitr3d 312 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 209  wa 399  if-wif 1058   = wceq 1538  wcel 2111  wne 2987  {crab 3110  Vcvv 3441  cdif 3878  wss 3881  c0 4243  ifcif 4425  {csn 4525  {cpr 4527  cop 4531   class class class wbr 5030  cres 5521  cima 5522  Fun wfun 6318  cfv 6324  (class class class)co 7135  0cc0 10526  1c1 10527   + caddc 10529  2c2 11680  cz 11969  ...cfz 12885  ..^cfzo 13028  chash 13686  cdvds 15599  Vtxcvtx 26789  iEdgciedg 26790  VtxDegcvtxdg 27255  Trailsctrls 27480
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-xadd 12496  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-dvds 15600  df-edg 26841  df-uhgr 26851  df-ushgr 26852  df-uspgr 26943  df-vtxdg 27256  df-wlks 27389  df-trls 27482
This theorem is referenced by:  eupth2lem3lem7  28019
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