Metamath Proof Explorer

Theorem cau3

Description: Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of j in the assertion, so it can be used with rexanuz and friends.) (Contributed by Mario Carneiro, 15-Feb-2014)

Ref Expression
Hypothesis cau3.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
Assertion cau3 ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) )

Proof

Step Hyp Ref Expression
1 cau3.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2 uzssz ⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ
3 1 2 eqsstri ⊢ 𝑍 ⊆ ℤ
4 id ⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
5 eleq1 ⊢ ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ↔ ( 𝐹 ‘ 𝑗 ) ∈ ℂ ) )
6 eleq1 ⊢ ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑚 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ↔ ( 𝐹 ‘ 𝑚 ) ∈ ℂ ) )
7 abssub ⊢ ( ( ( 𝐹 ‘ 𝑗 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) → ( abs ‘ ( ( 𝐹 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) )
8 7 3adant1 ⊢ ( ( ⊤ ∧ ( 𝐹 ‘ 𝑗 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) → ( abs ‘ ( ( 𝐹 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) )
9 abssub ⊢ ( ( ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑗 ) ∈ ℂ ) → ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐹 ‘ 𝑗 ) ) ) = ( abs ‘ ( ( 𝐹 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑚 ) ) ) )
10 9 3adant1 ⊢ ( ( ⊤ ∧ ( 𝐹 ‘ 𝑚 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑗 ) ∈ ℂ ) → ( abs ‘ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐹 ‘ 𝑗 ) ) ) = ( abs ‘ ( ( 𝐹 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑚 ) ) ) )
11 abs3lem ⊢ ( ( ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑚 ) ∈ ℂ ) ∧ ( ( 𝐹 ‘ 𝑗 ) ∈ ℂ ∧ 𝑥 ∈ ℝ ) ) → ( ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑚 ) ) ) < ( 𝑥 / 2 ) ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) )
12 11 3adant1 ⊢ ( ( ⊤ ∧ ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑚 ) ∈ ℂ ) ∧ ( ( 𝐹 ‘ 𝑗 ) ∈ ℂ ∧ 𝑥 ∈ ℝ ) ) → ( ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑗 ) − ( 𝐹 ‘ 𝑚 ) ) ) < ( 𝑥 / 2 ) ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) )
13 3 4 5 6 8 10 12 cau3lem ⊢ ( ⊤ → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) )
14 13 mptru ⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) )