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Metamath Proof Explorer

Theorem eupth2lem3lem7

Description: Lemma for eupth2lem3 : Combining trlsegvdeg , eupth2lem3lem3 , eupth2lem3lem4 and eupth2lem3lem6 . (Contributed by Mario Carneiro, 8-Apr-2015) (Revised by AV, 27-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 ) } ) )
eupth2lem3.e ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } )
Assertion eupth2lem3lem7 ( 𝜑 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝑍 ) ‘ 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 eupth2lem3.e ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } )
15 1 2 3 4 5 6 7 8 9 10 11 12 trlsegvdeg ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑍 ) ‘ 𝑈 ) = ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) )
16 15 breq2d ⊢ ( 𝜑 → ( 2 ∥ ( ( VtxDeg ‘ 𝑍 ) ‘ 𝑈 ) ↔ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ) )
17 16 notbid ⊢ ( 𝜑 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝑍 ) ‘ 𝑈 ) ↔ ¬ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ) )
18 ifpprsnss ⊢ ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } → if- ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } , { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
19 14 18 syl ⊢ ( 𝜑 → if- ( ( 𝑃 ‘ 𝑁 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) } , { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 13 19 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 ) ) } ) ) )
21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 eupth2lem3lem5 ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ 𝒫 𝑉 )
22 1 2 3 4 5 6 7 8 9 10 11 12 13 19 21 eupth2lem3lem4 ⊢ ( ( 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 ) ) } ) ) )
23 22 3expa ⊢ ( ( ( 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 ) ) } ) ) )
24 23 expcom ⊢ ( ( 𝑈 = ( 𝑃 ‘ 𝑁 ) ∨ 𝑈 = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ( 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 ) ) } ) ) ) )
25 neanior ⊢ ( ( 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ∧ 𝑈 ≠ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) ↔ ¬ ( 𝑈 = ( 𝑃 ‘ 𝑁 ) ∨ 𝑈 = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) )
26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 eupth2lem3lem6 ⊢ ( ( 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 ) ) } ) ) )
27 26 3expa ⊢ ( ( ( 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 ) ) } ) ) )
28 27 expcom ⊢ ( ( 𝑈 ≠ ( 𝑃 ‘ 𝑁 ) ∧ 𝑈 ≠ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ( 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 ) ) } ) ) ) )
29 25 28 sylbir ⊢ ( ¬ ( 𝑈 = ( 𝑃 ‘ 𝑁 ) ∨ 𝑈 = ( 𝑃 ‘ ( 𝑁 + 1 ) ) ) → ( ( 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 ) ) } ) ) ) )
30 24 29 pm2.61i ⊢ ( ( 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 ) ) } ) ) )
31 20 30 pm2.61dane ⊢ ( 𝜑 → ( ¬ 2 ∥ ( ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) + ( ( VtxDeg ‘ 𝑌 ) ‘ 𝑈 ) ) ↔ 𝑈 ∈ if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ) ) )
32 17 31 bitrd ⊢ ( 𝜑 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝑍 ) ‘ 𝑈 ) ↔ 𝑈 ∈ if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ) ) )