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Metamath Proof Explorer

Theorem 3wlkdlem5

Description: Lemma 5 for 3wlkd . (Contributed by Alexander van der Vekens, 11-Nov-2017) (Revised by AV, 7-Feb-2021)

Ref Expression
Hypotheses 3wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
3wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3wlkd.s ⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
Assertion 3wlkdlem5 ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )

Proof

Step Hyp Ref Expression
1 3wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2 3wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3 3wlkd.s ⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
4 3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
5 simpl ⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) → 𝐴 ≠ 𝐵 )
6 simpl ⊢ ( ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) → 𝐵 ≠ 𝐶 )
7 id ⊢ ( 𝐶 ≠ 𝐷 → 𝐶 ≠ 𝐷 )
8 5 6 7 3anim123i ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐷 ) )
9 4 8 syl ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐷 ) )
10 1 2 3 3wlkdlem3 ⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) )
11 simpl ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 0 ) = 𝐴 )
12 simpr ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 1 ) = 𝐵 )
13 11 12 neeq12d ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ↔ 𝐴 ≠ 𝐵 ) )
14 13 adantr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ↔ 𝐴 ≠ 𝐵 ) )
15 12 adantr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 1 ) = 𝐵 )
16 simpl ⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( 𝑃 ‘ 2 ) = 𝐶 )
17 16 adantl ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 2 ) = 𝐶 )
18 15 17 neeq12d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐵 ≠ 𝐶 ) )
19 simpr ⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( 𝑃 ‘ 3 ) = 𝐷 )
20 16 19 neeq12d ⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ↔ 𝐶 ≠ 𝐷 ) )
21 20 adantl ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ↔ 𝐶 ≠ 𝐷 ) )
22 14 18 21 3anbi123d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ) ↔ ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐷 ) ) )
23 10 22 syl ⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ) ↔ ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐷 ) ) )
24 9 23 mpbird ⊢ ( 𝜑 → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ) )
25 1 2 3wlkdlem2 ⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = { 0 , 1 , 2 }
26 25 raleqi ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ↔ ∀ 𝑘 ∈ { 0 , 1 , 2 } ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
27 c0ex ⊢ 0 ∈ V
28 1ex ⊢ 1 ∈ V
29 2ex ⊢ 2 ∈ V
30 fveq2 ⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 0 ) )
31 fv0p1e1 ⊢ ( 𝑘 = 0 → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ 1 ) )
32 30 31 neeq12d ⊢ ( 𝑘 = 0 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
33 fveq2 ⊢ ( 𝑘 = 1 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 1 ) )
34 oveq1 ⊢ ( 𝑘 = 1 → ( 𝑘 + 1 ) = ( 1 + 1 ) )
35 1p1e2 ⊢ ( 1 + 1 ) = 2
36 34 35 eqtrdi ⊢ ( 𝑘 = 1 → ( 𝑘 + 1 ) = 2 )
37 36 fveq2d ⊢ ( 𝑘 = 1 → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ 2 ) )
38 33 37 neeq12d ⊢ ( 𝑘 = 1 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) )
39 fveq2 ⊢ ( 𝑘 = 2 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 2 ) )
40 oveq1 ⊢ ( 𝑘 = 2 → ( 𝑘 + 1 ) = ( 2 + 1 ) )
41 2p1e3 ⊢ ( 2 + 1 ) = 3
42 40 41 eqtrdi ⊢ ( 𝑘 = 2 → ( 𝑘 + 1 ) = 3 )
43 42 fveq2d ⊢ ( 𝑘 = 2 → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ 3 ) )
44 39 43 neeq12d ⊢ ( 𝑘 = 2 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ) )
45 27 28 29 32 38 44 raltp ⊢ ( ∀ 𝑘 ∈ { 0 , 1 , 2 } ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ↔ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ) )
46 26 45 bitri ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ↔ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 3 ) ) )
47 24 46 sylibr ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )