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Metamath Proof Explorer

Theorem wlkp1lem2

Description: Lemma for wlkp1 . (Contributed by AV, 6-Mar-2021)

Ref Expression
Hypotheses wlkp1.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
wlkp1.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
wlkp1.f ⊢ ( 𝜑 → Fun 𝐼 )
wlkp1.a ⊢ ( 𝜑 → 𝐼 ∈ Fin )
wlkp1.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
wlkp1.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
wlkp1.d ⊢ ( 𝜑 → ¬ 𝐵 ∈ dom 𝐼 )
wlkp1.w ⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
wlkp1.n ⊢ 𝑁 = ( ♯ ‘ 𝐹 )
wlkp1.e ⊢ ( 𝜑 → 𝐸 ∈ ( Edg ‘ 𝐺 ) )
wlkp1.x ⊢ ( 𝜑 → { ( 𝑃 ‘ 𝑁 ) , 𝐶 } ⊆ 𝐸 )
wlkp1.u ⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐼 ∪ { ⟨ 𝐵 , 𝐸 ⟩ } ) )
wlkp1.h ⊢ 𝐻 = ( 𝐹 ∪ { ⟨ 𝑁 , 𝐵 ⟩ } )
Assertion wlkp1lem2 ( 𝜑 → ( ♯ ‘ 𝐻 ) = ( 𝑁 + 1 ) )

Proof

Step Hyp Ref Expression
1 wlkp1.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 wlkp1.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3 wlkp1.f ⊢ ( 𝜑 → Fun 𝐼 )
4 wlkp1.a ⊢ ( 𝜑 → 𝐼 ∈ Fin )
5 wlkp1.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
6 wlkp1.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
7 wlkp1.d ⊢ ( 𝜑 → ¬ 𝐵 ∈ dom 𝐼 )
8 wlkp1.w ⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
9 wlkp1.n ⊢ 𝑁 = ( ♯ ‘ 𝐹 )
10 wlkp1.e ⊢ ( 𝜑 → 𝐸 ∈ ( Edg ‘ 𝐺 ) )
11 wlkp1.x ⊢ ( 𝜑 → { ( 𝑃 ‘ 𝑁 ) , 𝐶 } ⊆ 𝐸 )
12 wlkp1.u ⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐼 ∪ { ⟨ 𝐵 , 𝐸 ⟩ } ) )
13 wlkp1.h ⊢ 𝐻 = ( 𝐹 ∪ { ⟨ 𝑁 , 𝐵 ⟩ } )
14 13 fveq2i ⊢ ( ♯ ‘ 𝐻 ) = ( ♯ ‘ ( 𝐹 ∪ { ⟨ 𝑁 , 𝐵 ⟩ } ) )
15 14 a1i ⊢ ( 𝜑 → ( ♯ ‘ 𝐻 ) = ( ♯ ‘ ( 𝐹 ∪ { ⟨ 𝑁 , 𝐵 ⟩ } ) ) )
16 opex ⊢ ⟨ 𝑁 , 𝐵 ⟩ ∈ V
17 2 wlkf ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom 𝐼 )
18 wrdfin ⊢ ( 𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin )
19 8 17 18 3syl ⊢ ( 𝜑 → 𝐹 ∈ Fin )
20 fzonel ⊢ ¬ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) )
21 20 a1i ⊢ ( 𝜑 → ¬ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
22 eleq1 ⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
23 22 notbid ⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( ¬ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ¬ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
24 21 23 imbitrrid ⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 𝜑 → ¬ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
25 9 24 ax-mp ⊢ ( 𝜑 → ¬ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
26 wrdfn ⊢ ( 𝐹 ∈ Word dom 𝐼 → 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
27 fnop ⊢ ( ( 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ ⟨ 𝑁 , 𝐵 ⟩ ∈ 𝐹 ) → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
28 27 ex ⊢ ( 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ⟨ 𝑁 , 𝐵 ⟩ ∈ 𝐹 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
29 8 17 26 28 4syl ⊢ ( 𝜑 → ( ⟨ 𝑁 , 𝐵 ⟩ ∈ 𝐹 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
30 25 29 mtod ⊢ ( 𝜑 → ¬ ⟨ 𝑁 , 𝐵 ⟩ ∈ 𝐹 )
31 19 30 jca ⊢ ( 𝜑 → ( 𝐹 ∈ Fin ∧ ¬ ⟨ 𝑁 , 𝐵 ⟩ ∈ 𝐹 ) )
32 hashunsng ⊢ ( ⟨ 𝑁 , 𝐵 ⟩ ∈ V → ( ( 𝐹 ∈ Fin ∧ ¬ ⟨ 𝑁 , 𝐵 ⟩ ∈ 𝐹 ) → ( ♯ ‘ ( 𝐹 ∪ { ⟨ 𝑁 , 𝐵 ⟩ } ) ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) )
33 16 31 32 mpsyl ⊢ ( 𝜑 → ( ♯ ‘ ( 𝐹 ∪ { ⟨ 𝑁 , 𝐵 ⟩ } ) ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
34 9 eqcomi ⊢ ( ♯ ‘ 𝐹 ) = 𝑁
35 34 a1i ⊢ ( 𝜑 → ( ♯ ‘ 𝐹 ) = 𝑁 )
36 35 oveq1d ⊢ ( 𝜑 → ( ( ♯ ‘ 𝐹 ) + 1 ) = ( 𝑁 + 1 ) )
37 15 33 36 3eqtrd ⊢ ( 𝜑 → ( ♯ ‘ 𝐻 ) = ( 𝑁 + 1 ) )