Theorem swrdfv

Description: A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015)

	Ref	Expression
Assertion	swrdfv	⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ( 0 ... 𝐿 ) ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ∧ 𝑋 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ) → ( ( 𝑆 substr ⟨ 𝐹 , 𝐿 ⟩ ) ‘ 𝑋 ) = ( 𝑆 ‘ ( 𝑋 + 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		swrdval2	⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ( 0 ... 𝐿 ) ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑆 substr ⟨ 𝐹 , 𝐿 ⟩ ) = ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( 𝑆 ‘ ( 𝑥 + 𝐹 ) ) ) )
2	1	fveq1d	⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ( 0 ... 𝐿 ) ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ( 𝑆 substr ⟨ 𝐹 , 𝐿 ⟩ ) ‘ 𝑋 ) = ( ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( 𝑆 ‘ ( 𝑥 + 𝐹 ) ) ) ‘ 𝑋 ) )
3		fvoveq1	⊢ ( 𝑥 = 𝑋 → ( 𝑆 ‘ ( 𝑥 + 𝐹 ) ) = ( 𝑆 ‘ ( 𝑋 + 𝐹 ) ) )
4		eqid	⊢ ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( 𝑆 ‘ ( 𝑥 + 𝐹 ) ) ) = ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( 𝑆 ‘ ( 𝑥 + 𝐹 ) ) )
5		fvex	⊢ ( 𝑆 ‘ ( 𝑋 + 𝐹 ) ) ∈ V
6	3 4 5	fvmpt	⊢ ( 𝑋 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) → ( ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( 𝑆 ‘ ( 𝑥 + 𝐹 ) ) ) ‘ 𝑋 ) = ( 𝑆 ‘ ( 𝑋 + 𝐹 ) ) )
7	2 6	sylan9eq	⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ( 0 ... 𝐿 ) ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ∧ 𝑋 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ) → ( ( 𝑆 substr ⟨ 𝐹 , 𝐿 ⟩ ) ‘ 𝑋 ) = ( 𝑆 ‘ ( 𝑋 + 𝐹 ) ) )