trlsegvdeg

Description: Formerly part of proof of eupth2lem3 : If a trail in a graph G induces a subgraph Z with the vertices V of G and the edges being the edges of the walk, and a subgraph X with the vertices V of G and the edges being the edges of the walk except the last one, and a subgraph Y with the vertices V of G and one edges being the last edge of the walk, then the vertex degree of any vertex U of G within Z is the sum of the vertex degree of U within X and the vertex degree of U within Y . Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015) (Revised by AV, 20-Feb-2021)

	Ref	Expression
Hypotheses	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 ) ) ) )
Assertion	trlsegvdeg	⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑍 ) ‘ 𝑈 ) = ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) )

Proof

Step	Hyp	Ref	Expression
1		trlsegvdeg.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		trlsegvdeg.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		trlsegvdeg.f	⊢ ( 𝜑 → Fun 𝐼 )
4		trlsegvdeg.n	⊢ ( 𝜑 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
5		trlsegvdeg.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
6		trlsegvdeg.w	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
7		trlsegvdeg.vx	⊢ ( 𝜑 → ( Vtx ‘ 𝑋 ) = 𝑉 )
8		trlsegvdeg.vy	⊢ ( 𝜑 → ( Vtx ‘ 𝑌 ) = 𝑉 )
9		trlsegvdeg.vz	⊢ ( 𝜑 → ( Vtx ‘ 𝑍 ) = 𝑉 )
10		trlsegvdeg.ix	⊢ ( 𝜑 → ( iEdg ‘ 𝑋 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) )
11		trlsegvdeg.iy	⊢ ( 𝜑 → ( iEdg ‘ 𝑌 ) = { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } )
12		trlsegvdeg.iz	⊢ ( 𝜑 → ( iEdg ‘ 𝑍 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ... 𝑁 ) ) ) )
13		eqid	⊢ ( iEdg ‘ 𝑋 ) = ( iEdg ‘ 𝑋 )
14		eqid	⊢ ( iEdg ‘ 𝑌 ) = ( iEdg ‘ 𝑌 )
15		eqid	⊢ ( Vtx ‘ 𝑋 ) = ( Vtx ‘ 𝑋 )
16	8 7	eqtr4d	⊢ ( 𝜑 → ( Vtx ‘ 𝑌 ) = ( Vtx ‘ 𝑋 ) )
17	9 7	eqtr4d	⊢ ( 𝜑 → ( Vtx ‘ 𝑍 ) = ( Vtx ‘ 𝑋 ) )
18	1 2 3 4 5 6 7 8 9 10 11 12	trlsegvdeglem4	⊢ ( 𝜑 → dom ( iEdg ‘ 𝑋 ) = ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) )
19	1 2 3 4 5 6 7 8 9 10 11 12	trlsegvdeglem5	⊢ ( 𝜑 → dom ( iEdg ‘ 𝑌 ) = { ( 𝐹 ‘ 𝑁 ) } )
20	18 19	ineq12d	⊢ ( 𝜑 → ( dom ( iEdg ‘ 𝑋 ) ∩ dom ( iEdg ‘ 𝑌 ) ) = ( ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ∩ { ( 𝐹 ‘ 𝑁 ) } ) )
21		fzonel	⊢ ¬ 𝑁 ∈ ( 0 ..^ 𝑁 )
22	2	trlf1	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 )
23	6 22	syl	⊢ ( 𝜑 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 )
24		elfzouz2	⊢ ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ♯ ‘ 𝐹 ) ∈ ( ℤ_≥ ‘ 𝑁 ) )
25		fzoss2	⊢ ( ( ♯ ‘ 𝐹 ) ∈ ( ℤ_≥ ‘ 𝑁 ) → ( 0 ..^ 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
26	4 24 25	3syl	⊢ ( 𝜑 → ( 0 ..^ 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
27		f1elima	⊢ ( ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ ( 0 ..^ 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ↔ 𝑁 ∈ ( 0 ..^ 𝑁 ) ) )
28	23 4 26 27	syl3anc	⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ↔ 𝑁 ∈ ( 0 ..^ 𝑁 ) ) )
29	21 28	mtbiri	⊢ ( 𝜑 → ¬ ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) )
30	29	orcd	⊢ ( 𝜑 → ( ¬ ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∨ ¬ ( 𝐹 ‘ 𝑁 ) ∈ dom 𝐼 ) )
31		ianor	⊢ ( ¬ ( ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∧ ( 𝐹 ‘ 𝑁 ) ∈ dom 𝐼 ) ↔ ( ¬ ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∨ ¬ ( 𝐹 ‘ 𝑁 ) ∈ dom 𝐼 ) )
32		elin	⊢ ( ( 𝐹 ‘ 𝑁 ) ∈ ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ↔ ( ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∧ ( 𝐹 ‘ 𝑁 ) ∈ dom 𝐼 ) )
33	31 32	xchnxbir	⊢ ( ¬ ( 𝐹 ‘ 𝑁 ) ∈ ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ↔ ( ¬ ( 𝐹 ‘ 𝑁 ) ∈ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∨ ¬ ( 𝐹 ‘ 𝑁 ) ∈ dom 𝐼 ) )
34	30 33	sylibr	⊢ ( 𝜑 → ¬ ( 𝐹 ‘ 𝑁 ) ∈ ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) )
35		disjsn	⊢ ( ( ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ∩ { ( 𝐹 ‘ 𝑁 ) } ) = ∅ ↔ ¬ ( 𝐹 ‘ 𝑁 ) ∈ ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) )
36	34 35	sylibr	⊢ ( 𝜑 → ( ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ∩ { ( 𝐹 ‘ 𝑁 ) } ) = ∅ )
37	20 36	eqtrd	⊢ ( 𝜑 → ( dom ( iEdg ‘ 𝑋 ) ∩ dom ( iEdg ‘ 𝑌 ) ) = ∅ )
38	1 2 3 4 5 6 7 8 9 10 11 12	trlsegvdeglem2	⊢ ( 𝜑 → Fun ( iEdg ‘ 𝑋 ) )
39	1 2 3 4 5 6 7 8 9 10 11 12	trlsegvdeglem3	⊢ ( 𝜑 → Fun ( iEdg ‘ 𝑌 ) )
40	5 7	eleqtrrd	⊢ ( 𝜑 → 𝑈 ∈ ( Vtx ‘ 𝑋 ) )
41		f1f	⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 )
42	6 22 41	3syl	⊢ ( 𝜑 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 )
43	3 42 4	resunimafz0	⊢ ( 𝜑 → ( 𝐼 ↾ ( 𝐹 “ ( 0 ... 𝑁 ) ) ) = ( ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ∪ { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } ) )
44	10 11	uneq12d	⊢ ( 𝜑 → ( ( iEdg ‘ 𝑋 ) ∪ ( iEdg ‘ 𝑌 ) ) = ( ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ∪ { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } ) )
45	43 12 44	3eqtr4d	⊢ ( 𝜑 → ( iEdg ‘ 𝑍 ) = ( ( iEdg ‘ 𝑋 ) ∪ ( iEdg ‘ 𝑌 ) ) )
46	1 2 3 4 5 6 7 8 9 10 11 12	trlsegvdeglem6	⊢ ( 𝜑 → dom ( iEdg ‘ 𝑋 ) ∈ Fin )
47	1 2 3 4 5 6 7 8 9 10 11 12	trlsegvdeglem7	⊢ ( 𝜑 → dom ( iEdg ‘ 𝑌 ) ∈ Fin )
48	13 14 15 16 17 37 38 39 40 45 46 47	vtxdfiun	⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑍 ) ‘ 𝑈 ) = ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) )

Metamath Proof Explorer

Theorem trlsegvdeg

Proof