eupth2lem3lem6fi - Intuitionistic Logic Explorer

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Theorem eupth2lem3lem6fi 16592

Description: If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than (𝑃‘0) and (𝑃‘𝑁): if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)

**Hypotheses**
Ref	Expression
trlsegvdeg.v	⊢ 𝑉 = (Vtx‘𝐺)
trlsegvdeg.i	⊢ 𝐼 = (iEdg‘𝐺)
trlsegvdeg.f	⊢ (𝜑 → Fun 𝐼)
trlsegvdeg.n	⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
trlsegvdeg.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
trlsegvdeg.w	⊢ (𝜑 → 𝐹(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) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
trlsegvdeg.iz	⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
eupth2lem3lem6fi.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
eupth2lem3lem6fi.v	⊢ (𝜑 → 𝑉 ∈ Fin)
eupth2lem3lem6fi.o	⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))
eupth2lem3lem6fi.e	⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})

**Assertion**
Ref	Expression
eupth2lem3lem6fi	⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (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 eupth2lem3lem6fi**
Step	Hyp	Ref	Expression
1		trlsegvdeg.iy	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
2	1	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
3		trlsegvdeg.vy	. . . . . . . 8 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)
4	3	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (Vtx‘𝑌) = 𝑉)
5		trlsegvdeg.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
6		trlsegvdeg.i	. . . . . . . . . . 11 ⊢ 𝐼 = (iEdg‘𝐺)
7	6	trlf1 16509	. . . . . . . . . 10 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
8		f1f 5578	. . . . . . . . . 10 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
9	5, 7, 8	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
10		trlsegvdeg.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
11	9, 10	ffvelcdmd 5818	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼)
12	11	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝐹‘𝑁) ∈ dom 𝐼)
13		trlsegvdeg.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
14	13	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ∈ 𝑉)
15		eupth2lem3lem6fi.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ Fin)
16	15	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑉 ∈ Fin)
17		trlsegvdeg.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
18	13, 3	eleqtrrd 2314	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑌))
19		df-vtx 16135	. . . . . . . . . . 11 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
20	19	mptrcl 5765	. . . . . . . . . 10 ⊢ (𝑈 ∈ (Vtx‘𝑌) → 𝑌 ∈ V)
21	18, 20	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ V)
22		trlsegvdeg.f	. . . . . . . . . . . 12 ⊢ (𝜑 → Fun 𝐼)
23	22	funfnd 5388	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 Fn dom 𝐼)
24		fnressn 5875	. . . . . . . . . . 11 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
25	23, 11, 24	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
26	1, 25	eqtr4d 2270	. . . . . . . . 9 ⊢ (𝜑 → (iEdg‘𝑌) = (𝐼 ↾ {(𝐹‘𝑁)}))
27		eupth2lem3lem6fi.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
28	17, 6, 21, 3, 26, 27	upgrspan 16400	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ UPGraph)
29	28	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑌 ∈ UPGraph)
30		funfvex 5692	. . . . . . . . 9 ⊢ ((Fun 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼‘(𝐹‘𝑁)) ∈ V)
31	22, 11, 30	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ V)
32	31	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝐼‘(𝐹‘𝑁)) ∈ V)
33		eupth2lem3lem6fi.e	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
34		simpl 109	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1))) → 𝑈 ≠ (𝑃‘𝑁))
35	34	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ≠ (𝑃‘𝑁))
36		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1))) → 𝑈 ≠ (𝑃‘(𝑁 + 1)))
37	36	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ≠ (𝑃‘(𝑁 + 1)))
38	35, 37	nelprd 3720	. . . . . . . . . . . 12 ⊢ (((𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → ¬ 𝑈 ∈ {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
39		df-nel 2510	. . . . . . . . . . . 12 ⊢ (𝑈 ∉ {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ↔ ¬ 𝑈 ∈ {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
40	38, 39	sylibr 134	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ∉ {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
41		neleq2 2514	. . . . . . . . . . 11 ⊢ ((𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} → (𝑈 ∉ (𝐼‘(𝐹‘𝑁)) ↔ 𝑈 ∉ {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}))
42	40, 41	imbitrrid 156	. . . . . . . . . 10 ⊢ ((𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} → (((𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ∉ (𝐼‘(𝐹‘𝑁))))
43	42	expd 258	. . . . . . . . 9 ⊢ ((𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} → ((𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) → ((𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1))) → 𝑈 ∉ (𝐼‘(𝐹‘𝑁)))))
44	33, 43	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) → ((𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1))) → 𝑈 ∉ (𝐼‘(𝐹‘𝑁)))))
45	44	3imp 1220	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ∉ (𝐼‘(𝐹‘𝑁)))
46	2, 4, 12, 14, 16, 29, 32, 45	1hevtxdg0fi 16428	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → ((VtxDeg‘𝑌)‘𝑈) = 0)
47	46	oveq2d 6074	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) = (((VtxDeg‘𝑋)‘𝑈) + 0))
48		trlsegvdeg.vx	. . . . . . . . 9 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
49		trlsegvdeg.vz	. . . . . . . . 9 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
50		trlsegvdeg.ix	. . . . . . . . 9 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
51		trlsegvdeg.iz	. . . . . . . . 9 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
52	17, 6, 22, 10, 13, 5, 48, 3, 49, 50, 1, 51, 27, 15	eupth2lem3lem1fi 16589	. . . . . . . 8 ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ₀)
53	52	nn0cnd 9572	. . . . . . 7 ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℂ)
54	53	addridd 8438	. . . . . 6 ⊢ (𝜑 → (((VtxDeg‘𝑋)‘𝑈) + 0) = ((VtxDeg‘𝑋)‘𝑈))
55	54	3ad2ant1 1045	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (((VtxDeg‘𝑋)‘𝑈) + 0) = ((VtxDeg‘𝑋)‘𝑈))
56	47, 55	eqtrd 2267	. . . 4 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) = ((VtxDeg‘𝑋)‘𝑈))
57	56	breq2d 4126	. . 3 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
58	57	notbid 673	. 2 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
59		fveq2 5675	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → ((VtxDeg‘𝑋)‘𝑥) = ((VtxDeg‘𝑋)‘𝑈))
60	59	breq2d 4126	. . . . . . 7 ⊢ (𝑥 = 𝑈 → (2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
61	60	notbid 673	. . . . . 6 ⊢ (𝑥 = 𝑈 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
62	61	elrab3 2977	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
63	13, 62	syl 14	. . . 4 ⊢ (𝜑 → (𝑈 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈)))
64		eupth2lem3lem6fi.o	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))
65	64	eleq2d 2304	. . . 4 ⊢ (𝜑 → (𝑈 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})))
66	63, 65	bitr3d 190	. . 3 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})))
67	66	3ad2ant1 1045	. 2 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})))
68	34	3ad2ant3 1047	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ≠ (𝑃‘𝑁))
69	36	3ad2ant3 1047	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → 𝑈 ≠ (𝑃‘(𝑁 + 1)))
70	68, 69	2thd 175	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 ≠ (𝑃‘𝑁) ↔ 𝑈 ≠ (𝑃‘(𝑁 + 1))))
71		neeq1 2427	. . . . . . 7 ⊢ (𝑈 = (𝑃‘0) → (𝑈 ≠ (𝑃‘𝑁) ↔ (𝑃‘0) ≠ (𝑃‘𝑁)))
72		neeq1 2427	. . . . . . 7 ⊢ (𝑈 = (𝑃‘0) → (𝑈 ≠ (𝑃‘(𝑁 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘(𝑁 + 1))))
73	71, 72	bibi12d 235	. . . . . 6 ⊢ (𝑈 = (𝑃‘0) → ((𝑈 ≠ (𝑃‘𝑁) ↔ 𝑈 ≠ (𝑃‘(𝑁 + 1))) ↔ ((𝑃‘0) ≠ (𝑃‘𝑁) ↔ (𝑃‘0) ≠ (𝑃‘(𝑁 + 1)))))
74	70, 73	syl5ibcom 155	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 = (𝑃‘0) → ((𝑃‘0) ≠ (𝑃‘𝑁) ↔ (𝑃‘0) ≠ (𝑃‘(𝑁 + 1)))))
75	74	pm5.32rd 451	. . . 4 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (((𝑃‘0) ≠ (𝑃‘𝑁) ∧ 𝑈 = (𝑃‘0)) ↔ ((𝑃‘0) ≠ (𝑃‘(𝑁 + 1)) ∧ 𝑈 = (𝑃‘0))))
76	68	neneqd 2435	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → ¬ 𝑈 = (𝑃‘𝑁))
77		biorf 752	. . . . . . 7 ⊢ (¬ 𝑈 = (𝑃‘𝑁) → (𝑈 = (𝑃‘0) ↔ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘0))))
78	76, 77	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 = (𝑃‘0) ↔ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘0))))
79		orcom 736	. . . . . 6 ⊢ ((𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘0)) ↔ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘𝑁)))
80	78, 79	bitrdi 196	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 = (𝑃‘0) ↔ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘𝑁))))
81	80	anbi2d 464	. . . 4 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (((𝑃‘0) ≠ (𝑃‘𝑁) ∧ 𝑈 = (𝑃‘0)) ↔ ((𝑃‘0) ≠ (𝑃‘𝑁) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘𝑁)))))
82	69	neneqd 2435	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → ¬ 𝑈 = (𝑃‘(𝑁 + 1)))
83		biorf 752	. . . . . . 7 ⊢ (¬ 𝑈 = (𝑃‘(𝑁 + 1)) → (𝑈 = (𝑃‘0) ↔ (𝑈 = (𝑃‘(𝑁 + 1)) ∨ 𝑈 = (𝑃‘0))))
84	82, 83	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 = (𝑃‘0) ↔ (𝑈 = (𝑃‘(𝑁 + 1)) ∨ 𝑈 = (𝑃‘0))))
85		orcom 736	. . . . . 6 ⊢ ((𝑈 = (𝑃‘(𝑁 + 1)) ∨ 𝑈 = (𝑃‘0)) ↔ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘(𝑁 + 1))))
86	84, 85	bitrdi 196	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 = (𝑃‘0) ↔ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))))
87	86	anbi2d 464	. . . 4 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (((𝑃‘0) ≠ (𝑃‘(𝑁 + 1)) ∧ 𝑈 = (𝑃‘0)) ↔ ((𝑃‘0) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘(𝑁 + 1))))))
88	75, 81, 87	3bitr3d 218	. . 3 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (((𝑃‘0) ≠ (𝑃‘𝑁) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘𝑁))) ↔ ((𝑃‘0) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘(𝑁 + 1))))))
89		eupth2lem1 16579	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}) ↔ ((𝑃‘0) ≠ (𝑃‘𝑁) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘𝑁)))))
90	14, 89	syl 14	. . 3 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}) ↔ ((𝑃‘0) ≠ (𝑃‘𝑁) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘𝑁)))))
91		eupth2lem1 16579	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}) ↔ ((𝑃‘0) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘(𝑁 + 1))))))
92	14, 91	syl 14	. . 3 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}) ↔ ((𝑃‘0) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘0) ∨ 𝑈 = (𝑃‘(𝑁 + 1))))))
93	88, 90, 92	3bitr4d 220	. 2 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (𝑈 ∈ if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))
94	58, 67, 93	3bitrd 214	1 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (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 set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 ∉ wnel 2509 {crab 2526 Vcvv 2815 ∅c0 3512 ifcif 3624 {csn 3694 {cpr 3695 ⟨cop 3697 class class class wbr 4114 × cxp 4752 dom cdm 4754 ↾ cres 4756 “ cima 4757 Fun wfun 5351 Fn wfn 5352 ⟶wf 5353 –1-1→wf1 5354 ‘cfv 5357 (class class class)co 6058 1^st c1st 6345 Fincfn 6988 0cc0 8143 1c1 8144 + caddc 8146 2c2 9305 ...cfz 10361 ..^cfzo 10498 ♯chash 11163 ∥ cdvds 12498 Basecbs 13296 Vtxcvtx 16133 iEdgciedg 16134 UPGraphcupgr 16212 VtxDegcvtxdg 16407 Trailsctrls 16501

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-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-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-ifp 987 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 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-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-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-frec 6635 df-1o 6660 df-2o 6661 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-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-xadd 10125 df-fz 10362 df-fzo 10499 df-ihash 11164 df-word 11250 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-upgren 16214 df-subgr 16375 df-vtxdg 16408 df-wlks 16439 df-trls 16502

This theorem is referenced by: eupth2lem3lem7fi 16595

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