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Metamath Proof Explorer

Theorem ccat2s1fvwALT

Description: Alternate proof of ccat2s1fvw using words of length 2, see df-s2 . A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018) (Revised by AV, 28-Jan-2024) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion ccat2s1fvwALT ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) ‘ 𝐼 ) = ( 𝑊 ‘ 𝐼 ) )

Proof

Step Hyp Ref Expression
1 ccatw2s1ccatws2 ⊢ ( 𝑊 ∈ Word 𝑉 → ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) = ( 𝑊 ++ ⟨“ 𝑋 𝑌 ”⟩ ) )
2 1 fveq1d ⊢ ( 𝑊 ∈ Word 𝑉 → ( ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) ‘ 𝐼 ) = ( ( 𝑊 ++ ⟨“ 𝑋 𝑌 ”⟩ ) ‘ 𝐼 ) )
3 2 3ad2ant1 ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) ‘ 𝐼 ) = ( ( 𝑊 ++ ⟨“ 𝑋 𝑌 ”⟩ ) ‘ 𝐼 ) )
4 simp1 ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝑊 ∈ Word 𝑉 )
5 s2cli ⊢ ⟨“ 𝑋 𝑌 ”⟩ ∈ Word V
6 5 a1i ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → ⟨“ 𝑋 𝑌 ”⟩ ∈ Word V )
7 simp2 ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝐼 ∈ ℕ0 )
8 lencl ⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 )
9 8 nn0zd ⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℤ )
10 9 3ad2ant1 ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℤ )
11 simp3 ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝐼 < ( ♯ ‘ 𝑊 ) )
12 elfzo0z ⊢ ( 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ ( 𝐼 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℤ ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) )
13 7 10 11 12 syl3anbrc ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
14 ccatval1 ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ⟨“ 𝑋 𝑌 ”⟩ ∈ Word V ∧ 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 ++ ⟨“ 𝑋 𝑌 ”⟩ ) ‘ 𝐼 ) = ( 𝑊 ‘ 𝐼 ) )
15 4 6 13 14 syl3anc ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → ( ( 𝑊 ++ ⟨“ 𝑋 𝑌 ”⟩ ) ‘ 𝐼 ) = ( 𝑊 ‘ 𝐼 ) )
16 3 15 eqtrd ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) ‘ 𝐼 ) = ( 𝑊 ‘ 𝐼 ) )