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Theorem eupth2lem3lem3 26950
 Description: Lemma for eupth2lem3 26956, formerly part of proof of eupth2lem3 26956: 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 6150 . . . . . . . 8 (𝑥 = 𝑈 → ((VtxDeg‘𝑋)‘𝑥) = ((VtxDeg‘𝑋)‘𝑈))
32breq2d 4630 . . . . . . 7 (𝑥 = 𝑈 → (2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
43notbid 308 . . . . . 6 (𝑥 = 𝑈 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
54elrab3 3352 . . . . 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 2689 . . . 4 (𝜑 → (𝑈 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
96, 8bitr3d 270 . . 3 (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
109adantr 481 . 2 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)})))
11 2z 11354 . . . . . 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 26948 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0)
2524nn0zd 11424 . . . . . 6 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ)
2625adantr 481 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑋)‘𝑈) ∈ ℤ)
2713, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23eupth2lem3lem2 26949 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ0)
2827nn0zd 11424 . . . . . 6 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ)
2928adantr 481 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((VtxDeg‘𝑌)‘𝑈) ∈ ℤ)
30 iddvds 14914 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 2)
3111, 30ax-mp 5 . . . . . . 7 2 ∥ 2
3219ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (Vtx‘𝑌) = 𝑉)
33 fvex 6160 . . . . . . . . 9 (𝐹𝑁) ∈ V
3433a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (𝐹𝑁) ∈ V)
351ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → 𝑈𝑉)
3622ad2antrr 761 . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))
39 ifptru 1022 . . . . . . . . . . . . . 14 ((𝑃𝑁) = (𝑃‘(𝑁 + 1)) → (if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))) ↔ (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}))
4039adantl 482 . . . . . . . . . . . . 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 222 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)})
42 sneq 4163 . . . . . . . . . . . . 13 ((𝑃𝑁) = 𝑈 → {(𝑃𝑁)} = {𝑈})
4342eqcoms 2634 . . . . . . . . . . . 12 (𝑈 = (𝑃𝑁) → {(𝑃𝑁)} = {𝑈})
4441, 43sylan9eq 2680 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (𝐼‘(𝐹𝑁)) = {𝑈})
4544opeq2d 4382 . . . . . . . . . 10 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → ⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩ = ⟨(𝐹𝑁), {𝑈}⟩)
4645sneqd 4165 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {⟨(𝐹𝑁), {𝑈}⟩})
4736, 46eqtrd 2660 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {𝑈}⟩})
4832, 34, 35, 471loopgrvd2 26279 . . . . . . 7 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 2)
4931, 48syl5breqr 4656 . . . . . 6 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 = (𝑃𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
50 dvds0 14916 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 0)
5111, 50ax-mp 5 . . . . . . 7 2 ∥ 0
5219ad2antrr 761 . . . . . . . 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 26940 . . . . . . . . . 10 (𝜑 → ((𝑃𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉))
5554simpld 475 . . . . . . . . 9 (𝜑 → (𝑃𝑁) ∈ 𝑉)
5655ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝑃𝑁) ∈ 𝑉)
5722adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
5841opeq2d 4382 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩ = ⟨(𝐹𝑁), {(𝑃𝑁)}⟩)
5958sneqd 4165 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
6057, 59eqtrd 2660 . . . . . . . . 9 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
6160adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (iEdg‘𝑌) = {⟨(𝐹𝑁), {(𝑃𝑁)}⟩})
621adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → 𝑈𝑉)
6362anim1i 591 . . . . . . . . 9 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → (𝑈𝑉𝑈 ≠ (𝑃𝑁)))
64 eldifsn 4292 . . . . . . . . 9 (𝑈 ∈ (𝑉 ∖ {(𝑃𝑁)}) ↔ (𝑈𝑉𝑈 ≠ (𝑃𝑁)))
6563, 64sylibr 224 . . . . . . . 8 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → 𝑈 ∈ (𝑉 ∖ {(𝑃𝑁)}))
6652, 53, 56, 61, 651loopgrvd0 26280 . . . . . . 7 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → ((VtxDeg‘𝑌)‘𝑈) = 0)
6751, 66syl5breqr 4656 . . . . . 6 (((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) ∧ 𝑈 ≠ (𝑃𝑁)) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
6849, 67pm2.61dane 2883 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → 2 ∥ ((VtxDeg‘𝑌)‘𝑈))
69 dvdsadd2b 14947 . . . . 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 1327 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈))))
7127nn0cnd 11298 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℂ)
7224nn0cnd 11298 . . . . . . 7 (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℂ)
7371, 72addcomd 10183 . . . . . 6 (𝜑 → (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
7473breq2d 4630 . . . . 5 (𝜑 → (2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7574adantr 481 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ (((VtxDeg‘𝑌)‘𝑈) + ((VtxDeg‘𝑋)‘𝑈)) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7670, 75bitrd 268 . . 3 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
7776notbid 308 . 2 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ ¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))))
78 simpr 477 . . . . 5 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (𝑃𝑁) = (𝑃‘(𝑁 + 1)))
7978eqeq2d 2636 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → ((𝑃‘0) = (𝑃𝑁) ↔ (𝑃‘0) = (𝑃‘(𝑁 + 1))))
8078preq2d 4250 . . . 4 ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → {(𝑃‘0), (𝑃𝑁)} = {(𝑃‘0), (𝑃‘(𝑁 + 1))})
8179, 80ifbieq2d 4088 . . 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 2689 . 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 298 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 196   ∧ wa 384  if-wif 1011   = wceq 1480   ∈ wcel 1992   ≠ wne 2796  {crab 2916  Vcvv 3191   ∖ cdif 3557   ⊆ wss 3560  ∅c0 3896  ifcif 4063  {csn 4153  {cpr 4155  ⟨cop 4159   class class class wbr 4618   ↾ cres 5081   “ cima 5082  Fun wfun 5844  ‘cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884  2c2 11015  ℤcz 11322  ...cfz 12265  ..^cfzo 12403  #chash 13054   ∥ cdvds 14902  Vtxcvtx 25769  iEdgciedg 25770  VtxDegcvtxdg 26242  Trailsctrls 26450 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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-xadd 11891  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-dvds 14903  df-edg 25835  df-uhgr 25844  df-ushgr 25845  df-uspgr 25933  df-vtxdg 26243  df-wlks 26359  df-trls 26452 This theorem is referenced by:  eupth2lem3lem7  26954
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