Metamath Proof Explorer

Theorem eupth2lem3lem3

Description: Lemma for eupth2lem3 , formerly part of proof of eupth2lem3 : 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)

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 ) ) ) )
eupth2lem3.o ⊢ ( 𝜑 → { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) } = if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) )
eupth2lem3lem3.e ⊢ ( 𝜑 → if- ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } , { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
Assertion eupth2lem3lem3 ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ¬ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ↔ 𝑈 ∈ if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ) ) )

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 eupth2lem3.o ⊢ ( 𝜑 → { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) } = if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) )
14 eupth2lem3lem3.e ⊢ ( 𝜑 → if- ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } , { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
15 fveq2 ⊢ ( 𝑥 = 𝑈 → ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) = ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) )
16 15 breq2d ⊢ ( 𝑥 = 𝑈 → ( 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) ↔ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) )
17 16 notbid ⊢ ( 𝑥 = 𝑈 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) ↔ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) )
18 17 elrab3 ⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) } ↔ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) )
19 5 18 syl ⊢ ( 𝜑 → ( 𝑈 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) } ↔ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) )
20 13 eleq2d ⊢ ( 𝜑 → ( 𝑈 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) } ↔ 𝑈 ∈ if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) ) )
21 19 20 bitr3d ⊢ ( 𝜑 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ↔ 𝑈 ∈ if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) ) )
22 21 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ↔ 𝑈 ∈ if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) ) )
23 2z ⊢ 2 ∈ ℤ
24 23 a1i ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → 2 ∈ ℤ )
25 1 2 3 4 5 6 7 8 9 10 11 12 eupth2lem3lem1 ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ∈ ℕ0 )
26 25 nn0zd ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ∈ ℤ )
27 26 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ∈ ℤ )
28 1 2 3 4 5 6 7 8 9 10 11 12 eupth2lem3lem2 ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ∈ ℕ0 )
29 28 nn0zd ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ∈ ℤ )
30 29 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ∈ ℤ )
31 z2even ⊢ 2 ∥ 2
32 8 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → ( Vtx ‘ 𝑌 ) = 𝑉 )
33 fvexd ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → ( 𝐹 ‘ 𝑁 ) ∈ V )
34 5 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → 𝑈 ∈ 𝑉 )
35 11 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → ( iEdg ‘ 𝑌 ) = { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } )
36 14 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → if- ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } , { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
37 ifptru ⊢ ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) → ( if- ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } , { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) ↔ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } ) )
38 37 adantl ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } , { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) ↔ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } ) )
39 36 38 mpbid ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } )
40 sneq ⊢ ( ( 𝑃 ‘ 𝑁 ) = 𝑈 → { ( 𝑃 ‘ 𝑁 ) } = { 𝑈 } )
41 40 eqcoms ⊢ ( 𝑈 = ( 𝑃 ‘ 𝑁 ) → { ( 𝑃 ‘ 𝑁 ) } = { 𝑈 } )
42 39 41 sylan9eq ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { 𝑈 } )
43 42 opeq2d ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ = ⟨ ( 𝐹 ‘ 𝑁 ) , { 𝑈 } ⟩ )
44 43 sneqd ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } = { ⟨ ( 𝐹 ‘ 𝑁 ) , { 𝑈 } ⟩ } )
45 35 44 eqtrd ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → ( iEdg ‘ 𝑌 ) = { ⟨ ( 𝐹 ‘ 𝑁 ) , { 𝑈 } ⟩ } )
46 32 33 34 45 1loopgrvd2 ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) = 2 )
47 31 46 breqtrrid ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 = ( 𝑃 ‘ 𝑁 ) ) → 2 ∥ ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) )
48 z0even ⊢ 2 ∥ 0
49 8 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → ( Vtx ‘ 𝑌 ) = 𝑉 )
50 fvexd ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → ( 𝐹 ‘ 𝑁 ) ∈ V )
51 1 2 3 4 5 6 trlsegvdeglem1 ⊢ ( 𝜑 → ( ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ∧ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) )
52 51 simpld ⊢ ( 𝜑 → ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 )
53 52 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 )
54 11 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( iEdg ‘ 𝑌 ) = { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } )
55 39 opeq2d ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ = ⟨ ( 𝐹 ‘ 𝑁 ) , { ( 𝑃 ‘ 𝑁 ) } ⟩ )
56 55 sneqd ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } = { ⟨ ( 𝐹 ‘ 𝑁 ) , { ( 𝑃 ‘ 𝑁 ) } ⟩ } )
57 54 56 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( iEdg ‘ 𝑌 ) = { ⟨ ( 𝐹 ‘ 𝑁 ) , { ( 𝑃 ‘ 𝑁 ) } ⟩ } )
58 57 adantr ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → ( iEdg ‘ 𝑌 ) = { ⟨ ( 𝐹 ‘ 𝑁 ) , { ( 𝑃 ‘ 𝑁 ) } ⟩ } )
59 5 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → 𝑈 ∈ 𝑉 )
60 59 anim1i ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → ( 𝑈 ∈ 𝑉 ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) )
61 eldifsn ⊢ ( 𝑈 ∈ ( 𝑉 ∖ { ( 𝑃 ‘ 𝑁 ) } ) ↔ ( 𝑈 ∈ 𝑉 ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) )
62 60 61 sylibr ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → 𝑈 ∈ ( 𝑉 ∖ { ( 𝑃 ‘ 𝑁 ) } ) )
63 49 50 53 58 62 1loopgrvd0 ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) = 0 )
64 48 63 breqtrrid ⊢ ( ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ∧ 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ) → 2 ∥ ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) )
65 47 64 pm2.61dane ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → 2 ∥ ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) )
66 dvdsadd2b ⊢ ( ( 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 ) ) ) )
67 24 27 30 65 66 syl112anc ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ↔ 2 ∥ ( ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) ) )
68 28 nn0cnd ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ∈ ℂ )
69 25 nn0cnd ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ∈ ℂ )
70 68 69 addcomd ⊢ ( 𝜑 → ( ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) = ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) )
71 70 breq2d ⊢ ( 𝜑 → ( 2 ∥ ( ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) ↔ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ) )
72 71 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( 2 ∥ ( ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ) ↔ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ) )
73 67 72 bitrd ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ↔ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ) )
74 73 notbid ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ↔ ¬ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ) )
75 simpr ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) )
76 75 eqeq2d ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) ↔ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) )
77 75 preq2d ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } )
78 76 77 ifbieq2d ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) = if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ) )
79 78 eleq2d ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 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 ) ) } ) ) )
80 22 74 79 3bitr3d ⊢ ( ( 𝜑 ∧ ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ¬ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ↔ 𝑈 ∈ if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ) ) )