Metamath Proof Explorer

Theorem 2pthdlem1

Description: Lemma 1 for 2pthd . (Contributed by AV, 14-Feb-2021)

Ref Expression
Hypotheses 2wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
2wlkd.s ⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
2wlkd.n ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
Assertion 2pthdlem1 ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) )

Proof

Step Hyp Ref Expression
1 2wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2 2wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
3 2wlkd.s ⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
4 2wlkd.n ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
5 1 2 3 2wlkdlem3 ⊢ ( 𝜑 → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) )
6 simpl ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 0 ) = 𝐴 )
7 simpr ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 1 ) = 𝐵 )
8 6 7 neeq12d ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ↔ 𝐴 ≠ 𝐵 ) )
9 8 bicomd ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
10 9 3adant3 ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
11 10 biimpcd ⊢ ( 𝐴 ≠ 𝐵 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
12 11 adantr ⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
13 12 imp ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) )
14 13 a1d ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) ) → ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
15 eqid ⊢ 1 = 1
16 eqneqall ⊢ ( 1 = 1 → ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) )
17 15 16 mp1i ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) ) → ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) )
18 simpr ⊢ ( ( ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 2 ) = 𝐶 )
19 simpl ⊢ ( ( ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 1 ) = 𝐵 )
20 18 19 neeq12d ⊢ ( ( ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ↔ 𝐶 ≠ 𝐵 ) )
21 necom ⊢ ( 𝐶 ≠ 𝐵 ↔ 𝐵 ≠ 𝐶 )
22 20 21 bitr2di ⊢ ( ( ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝐵 ≠ 𝐶 ↔ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) )
23 22 3adant1 ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝐵 ≠ 𝐶 ↔ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) )
24 23 biimpcd ⊢ ( 𝐵 ≠ 𝐶 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) )
25 24 adantl ⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) )
26 25 imp ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) ) → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) )
27 26 a1d ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) ) → ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) )
28 14 17 27 3jca ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) ) → ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
29 4 5 28 syl2anc ⊢ ( 𝜑 → ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
30 1 fveq2i ⊢ ( ♯ ‘ 𝑃 ) = ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
31 s3len ⊢ ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) = 3
32 30 31 eqtri ⊢ ( ♯ ‘ 𝑃 ) = 3
33 32 oveq2i ⊢ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) = ( 0 ..^ 3 )
34 fzo0to3tp ⊢ ( 0 ..^ 3 ) = { 0 , 1 , 2 }
35 33 34 eqtri ⊢ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) = { 0 , 1 , 2 }
36 35 raleqi ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ∀ 𝑘 ∈ { 0 , 1 , 2 } ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) )
37 c0ex ⊢ 0 ∈ V
38 1ex ⊢ 1 ∈ V
39 2ex ⊢ 2 ∈ V
40 neeq1 ⊢ ( 𝑘 = 0 → ( 𝑘 ≠ 1 ↔ 0 ≠ 1 ) )
41 fveq2 ⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 0 ) )
42 41 neeq1d ⊢ ( 𝑘 = 0 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
43 40 42 imbi12d ⊢ ( 𝑘 = 0 → ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
44 neeq1 ⊢ ( 𝑘 = 1 → ( 𝑘 ≠ 1 ↔ 1 ≠ 1 ) )
45 fveq2 ⊢ ( 𝑘 = 1 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 1 ) )
46 45 neeq1d ⊢ ( 𝑘 = 1 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ↔ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) )
47 44 46 imbi12d ⊢ ( 𝑘 = 1 → ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
48 neeq1 ⊢ ( 𝑘 = 2 → ( 𝑘 ≠ 1 ↔ 2 ≠ 1 ) )
49 fveq2 ⊢ ( 𝑘 = 2 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 2 ) )
50 49 neeq1d ⊢ ( 𝑘 = 2 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ↔ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) )
51 48 50 imbi12d ⊢ ( 𝑘 = 2 → ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
52 37 38 39 43 47 51 raltp ⊢ ( ∀ 𝑘 ∈ { 0 , 1 , 2 } ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
53 36 52 bitri ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
54 29 53 sylibr ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) )
55 2 fveq2i ⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ )
56 s2len ⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ ) = 2
57 55 56 eqtri ⊢ ( ♯ ‘ 𝐹 ) = 2
58 57 oveq2i ⊢ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) = ( 1 ..^ 2 )
59 fzo12sn ⊢ ( 1 ..^ 2 ) = { 1 }
60 58 59 eqtri ⊢ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) = { 1 }
61 60 raleqi ⊢ ( ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ∀ 𝑗 ∈ { 1 } ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) )
62 neeq2 ⊢ ( 𝑗 = 1 → ( 𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 1 ) )
63 fveq2 ⊢ ( 𝑗 = 1 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) )
64 63 neeq2d ⊢ ( 𝑗 = 1 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ↔ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) )
65 62 64 imbi12d ⊢ ( 𝑗 = 1 → ( ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ) )
66 38 65 ralsn ⊢ ( ∀ 𝑗 ∈ { 1 } ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) )
67 61 66 bitri ⊢ ( ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) )
68 67 ralbii ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) )
69 54 68 sylibr ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) )