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Metamath Proof Explorer

Theorem 3wlkdlem10

Description: Lemma 10 for 3wlkd . (Contributed by Alexander van der Vekens, 12-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 ) )
3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
Assertion 3wlkdlem10 ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )

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 3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
6 1 2 3 4 5 3wlkdlem9 ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )
7 1 2 3 3wlkdlem3 ⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) )
8 preq12 ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { 𝐴 , 𝐵 } )
9 8 adantr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { 𝐴 , 𝐵 } )
10 9 sseq1d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ) )
11 simplr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 1 ) = 𝐵 )
12 simprl ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 2 ) = 𝐶 )
13 11 12 preq12d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } = { 𝐵 , 𝐶 } )
14 13 sseq1d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ↔ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ) )
15 preq12 ⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } = { 𝐶 , 𝐷 } )
16 15 adantl ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } = { 𝐶 , 𝐷 } )
17 16 sseq1d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ↔ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )
18 10 14 17 3anbi123d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) ↔ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) ) )
19 7 18 syl ⊢ ( 𝜑 → ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) ↔ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) ) )
20 6 19 mpbird ⊢ ( 𝜑 → ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )
21 1 2 3wlkdlem2 ⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = { 0 , 1 , 2 }
22 21 raleqi ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ ∀ 𝑘 ∈ { 0 , 1 , 2 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )
23 c0ex ⊢ 0 ∈ V
24 1ex ⊢ 1 ∈ V
25 2ex ⊢ 2 ∈ V
26 fveq2 ⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 0 ) )
27 fv0p1e1 ⊢ ( 𝑘 = 0 → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ 1 ) )
28 26 27 preq12d ⊢ ( 𝑘 = 0 → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } )
29 2fveq3 ⊢ ( 𝑘 = 0 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) )
30 28 29 sseq12d ⊢ ( 𝑘 = 0 → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ) )
31 fveq2 ⊢ ( 𝑘 = 1 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 1 ) )
32 oveq1 ⊢ ( 𝑘 = 1 → ( 𝑘 + 1 ) = ( 1 + 1 ) )
33 1p1e2 ⊢ ( 1 + 1 ) = 2
34 32 33 eqtrdi ⊢ ( 𝑘 = 1 → ( 𝑘 + 1 ) = 2 )
35 34 fveq2d ⊢ ( 𝑘 = 1 → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ 2 ) )
36 31 35 preq12d ⊢ ( 𝑘 = 1 → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } )
37 2fveq3 ⊢ ( 𝑘 = 1 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) )
38 36 37 sseq12d ⊢ ( 𝑘 = 1 → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ) )
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 preq12d ⊢ ( 𝑘 = 2 → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } )
45 2fveq3 ⊢ ( 𝑘 = 2 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) )
46 44 45 sseq12d ⊢ ( 𝑘 = 2 → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )
47 23 24 25 30 38 46 raltp ⊢ ( ∀ 𝑘 ∈ { 0 , 1 , 2 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )
48 22 47 bitri ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )
49 20 48 sylibr ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )