Metamath Proof Explorer


Theorem climeldmeqf

Description: Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses climeldmeqf.p 𝑘 𝜑
climeldmeqf.n 𝑘 𝐹
climeldmeqf.o 𝑘 𝐺
climeldmeqf.z 𝑍 = ( ℤ𝑀 )
climeldmeqf.f ( 𝜑𝐹𝑉 )
climeldmeqf.g ( 𝜑𝐺𝑊 )
climeldmeqf.m ( 𝜑𝑀 ∈ ℤ )
climeldmeqf.e ( ( 𝜑𝑘𝑍 ) → ( 𝐹𝑘 ) = ( 𝐺𝑘 ) )
Assertion climeldmeqf ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) )

Proof

Step Hyp Ref Expression
1 climeldmeqf.p 𝑘 𝜑
2 climeldmeqf.n 𝑘 𝐹
3 climeldmeqf.o 𝑘 𝐺
4 climeldmeqf.z 𝑍 = ( ℤ𝑀 )
5 climeldmeqf.f ( 𝜑𝐹𝑉 )
6 climeldmeqf.g ( 𝜑𝐺𝑊 )
7 climeldmeqf.m ( 𝜑𝑀 ∈ ℤ )
8 climeldmeqf.e ( ( 𝜑𝑘𝑍 ) → ( 𝐹𝑘 ) = ( 𝐺𝑘 ) )
9 nfv 𝑘 𝑗𝑍
10 1 9 nfan 𝑘 ( 𝜑𝑗𝑍 )
11 nfcv 𝑘 𝑗
12 2 11 nffv 𝑘 ( 𝐹𝑗 )
13 3 11 nffv 𝑘 ( 𝐺𝑗 )
14 12 13 nfeq 𝑘 ( 𝐹𝑗 ) = ( 𝐺𝑗 )
15 10 14 nfim 𝑘 ( ( 𝜑𝑗𝑍 ) → ( 𝐹𝑗 ) = ( 𝐺𝑗 ) )
16 eleq1w ( 𝑘 = 𝑗 → ( 𝑘𝑍𝑗𝑍 ) )
17 16 anbi2d ( 𝑘 = 𝑗 → ( ( 𝜑𝑘𝑍 ) ↔ ( 𝜑𝑗𝑍 ) ) )
18 fveq2 ( 𝑘 = 𝑗 → ( 𝐹𝑘 ) = ( 𝐹𝑗 ) )
19 fveq2 ( 𝑘 = 𝑗 → ( 𝐺𝑘 ) = ( 𝐺𝑗 ) )
20 18 19 eqeq12d ( 𝑘 = 𝑗 → ( ( 𝐹𝑘 ) = ( 𝐺𝑘 ) ↔ ( 𝐹𝑗 ) = ( 𝐺𝑗 ) ) )
21 17 20 imbi12d ( 𝑘 = 𝑗 → ( ( ( 𝜑𝑘𝑍 ) → ( 𝐹𝑘 ) = ( 𝐺𝑘 ) ) ↔ ( ( 𝜑𝑗𝑍 ) → ( 𝐹𝑗 ) = ( 𝐺𝑗 ) ) ) )
22 15 21 8 chvarfv ( ( 𝜑𝑗𝑍 ) → ( 𝐹𝑗 ) = ( 𝐺𝑗 ) )
23 4 5 6 7 22 climeldmeq ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) )