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Metamath Proof Explorer

Theorem cats1len

Description: The length 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
cats1fvn.3 ⊢ ( ♯ ‘ 𝑆 ) = 𝑀
cats1len.4 ⊢ ( 𝑀 + 1 ) = 𝑁
Assertion cats1len ( ♯ ‘ 𝑇 ) = 𝑁

Proof

Step Hyp Ref Expression
1 cats1cld.1 ⊢ 𝑇 = ( 𝑆 ++ ⟨“ 𝑋 ”⟩ )
2 cats1cli.2 ⊢ 𝑆 ∈ Word V
3 cats1fvn.3 ⊢ ( ♯ ‘ 𝑆 ) = 𝑀
4 cats1len.4 ⊢ ( 𝑀 + 1 ) = 𝑁
5 1 fveq2i ⊢ ( ♯ ‘ 𝑇 ) = ( ♯ ‘ ( 𝑆 ++ ⟨“ 𝑋 ”⟩ ) )
6 s1cli ⊢ ⟨“ 𝑋 ”⟩ ∈ Word V
7 ccatlen ⊢ ( ( 𝑆 ∈ Word V ∧ ⟨“ 𝑋 ”⟩ ∈ Word V ) → ( ♯ ‘ ( 𝑆 ++ ⟨“ 𝑋 ”⟩ ) ) = ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ ⟨“ 𝑋 ”⟩ ) ) )
8 2 6 7 mp2an ⊢ ( ♯ ‘ ( 𝑆 ++ ⟨“ 𝑋 ”⟩ ) ) = ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ ⟨“ 𝑋 ”⟩ ) )
9 s1len ⊢ ( ♯ ‘ ⟨“ 𝑋 ”⟩ ) = 1
10 3 9 oveq12i ⊢ ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ ⟨“ 𝑋 ”⟩ ) ) = ( 𝑀 + 1 )
11 8 10 eqtri ⊢ ( ♯ ‘ ( 𝑆 ++ ⟨“ 𝑋 ”⟩ ) ) = ( 𝑀 + 1 )
12 11 4 eqtri ⊢ ( ♯ ‘ ( 𝑆 ++ ⟨“ 𝑋 ”⟩ ) ) = 𝑁
13 5 12 eqtri ⊢ ( ♯ ‘ 𝑇 ) = 𝑁