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Metamath Proof Explorer

Theorem cvgcau

Description: A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025)

Ref Expression
Hypotheses cvgcau.1 ⊢ Ⅎ 𝑗 𝐹
cvgcau.2 ⊢ Ⅎ 𝑘 𝐹
cvgcau.3 ⊢ ( 𝜑 → 𝑀 ∈ 𝑍 )
cvgcau.4 ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
cvgcau.5 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
cvgcau.6 ⊢ ( 𝜑 → 𝐹 ∈ dom ⇝ )
cvgcau.7 ⊢ ( 𝜑 → 𝑋 ∈ ℝ+ )
Assertion cvgcau ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑋 ) )

Proof

Step Hyp Ref Expression
1 cvgcau.1 ⊢ Ⅎ 𝑗 𝐹
2 cvgcau.2 ⊢ Ⅎ 𝑘 𝐹
3 cvgcau.3 ⊢ ( 𝜑 → 𝑀 ∈ 𝑍 )
4 cvgcau.4 ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
5 cvgcau.5 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
6 cvgcau.6 ⊢ ( 𝜑 → 𝐹 ∈ dom ⇝ )
7 cvgcau.7 ⊢ ( 𝜑 → 𝑋 ∈ ℝ+ )
8 breq2 ⊢ ( 𝑥 = 𝑋 → ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ↔ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑋 ) )
9 8 anbi2d ⊢ ( 𝑥 = 𝑋 → ( ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑋 ) ) )
10 9 rexralbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑋 ) ) )
11 5 3 eluzelz2d ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
12 1 2 5 caucvgbf ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉 ) → ( 𝐹 ∈ dom ⇝ ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) )
13 11 4 12 syl2anc ⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) )
14 6 13 mpbid ⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) )
15 10 14 7 rspcdva ⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑋 ) )