Metamath Proof Explorer

Theorem climfveqf

Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses climfveqf.p ⊢ Ⅎ 𝑘 𝜑
climfveqf.n ⊢ Ⅎ 𝑘 𝐹
climfveqf.o ⊢ Ⅎ 𝑘 𝐺
climfveqf.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
climfveqf.f ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
climfveqf.g ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
climfveqf.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
climfveqf.e ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) )
Assertion climfveqf ( 𝜑 → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 climfveqf.p ⊢ Ⅎ 𝑘 𝜑
2 climfveqf.n ⊢ Ⅎ 𝑘 𝐹
3 climfveqf.o ⊢ Ⅎ 𝑘 𝐺
4 climfveqf.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
5 climfveqf.f ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
6 climfveqf.g ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
7 climfveqf.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
8 climfveqf.e ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) )
9 climdm ⊢ ( 𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) )
10 9 bilani ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) )
11 10 9 sylibr ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ )
12 nfcv ⊢ Ⅎ 𝑘 𝑗
13 12 nfel1 ⊢ Ⅎ 𝑘 𝑗 ∈ 𝑍
14 1 13 nfan ⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑍 )
15 2 12 nffv ⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 )
16 3 12 nffv ⊢ Ⅎ 𝑘 ( 𝐺 ‘ 𝑗 )
17 15 16 nfeq ⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 )
18 14 17 nfim ⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) )
19 eleq1w ⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍 ) )
20 19 anbi2d ⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ) )
21 fveq2 ⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) )
22 fveq2 ⊢ ( 𝑘 = 𝑗 → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑗 ) )
23 21 22 eqeq12d ⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ↔ ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) )
24 20 23 imbi12d ⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) ) )
25 18 24 8 chvarfv ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) )
26 4 5 6 7 25 climeldmeq ⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) )
27 26 adantr ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) )
28 11 27 mpbid ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ )
29 climdm ⊢ ( 𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) )
30 28 29 sylib ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) )
31 6 adantr ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊 )
32 5 adantr ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉 )
33 7 adantr ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ )
34 25 eqcomd ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) )
35 34 adantlr ⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑗 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) )
36 4 31 32 33 35 climeq ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) )
37 30 36 mpbid ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) )
38 climuni ⊢ ( ( 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ∧ 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) )
39 10 37 38 syl2anc ⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) )
40 ndmfv ⊢ ( ¬ 𝐹 ∈ dom ⇝ → ( ⇝ ‘ 𝐹 ) = ∅ )
41 40 adantl ⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐹 ) = ∅ )
42 simpr ⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ )
43 26 adantr ⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) )
44 42 43 mtbid ⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐺 ∈ dom ⇝ )
45 ndmfv ⊢ ( ¬ 𝐺 ∈ dom ⇝ → ( ⇝ ‘ 𝐺 ) = ∅ )
46 44 45 syl ⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐺 ) = ∅ )
47 41 46 eqtr4d ⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) )
48 39 47 pm2.61dan ⊢ ( 𝜑 → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) )