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Metamath Proof Explorer

Theorem cats1cli

Description: Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016)

Ref Expression
Hypotheses cats1cld.1 ⊢ 𝑇 = ( 𝑆 ++ ⟨“ 𝑋 ”⟩ )
cats1cli.2 ⊢ 𝑆 ∈ Word V
Assertion cats1cli 𝑇 ∈ Word V

Proof

Step Hyp Ref Expression
1 cats1cld.1 ⊢ 𝑇 = ( 𝑆 ++ ⟨“ 𝑋 ”⟩ )
2 cats1cli.2 ⊢ 𝑆 ∈ Word V
3 s1cli ⊢ ⟨“ 𝑋 ”⟩ ∈ Word V
4 ccatcl ⊢ ( ( 𝑆 ∈ Word V ∧ ⟨“ 𝑋 ”⟩ ∈ Word V ) → ( 𝑆 ++ ⟨“ 𝑋 ”⟩ ) ∈ Word V )
5 2 3 4 mp2an ⊢ ( 𝑆 ++ ⟨“ 𝑋 ”⟩ ) ∈ Word V
6 1 5 eqeltri ⊢ 𝑇 ∈ Word V