Theorem xlimclim2lem

Description: Lemma for xlimclim2 . Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Hypotheses	xlimclim2lem.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	xlimclim2lem.f	⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* )
	xlimclim2lem.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	xlimclim2lem.r	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ )
Assertion	xlimclim2lem	⊢ ( 𝜑 → ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ⇝ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		xlimclim2lem.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		xlimclim2lem.f	⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* )
3		xlimclim2lem.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
4		xlimclim2lem.r	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ )
5	1 2	fuzxrpmcn	⊢ ( 𝜑 → 𝐹 ∈ ( ℝ* ↑pm ℂ ) )
6	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → 𝐹 ∈ ( ℝ* ↑pm ℂ ) )
7	1	eluzelz2	⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ )
8	7	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → 𝑗 ∈ ℤ )
9	6 8	xlimres	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → ( 𝐹 ~~>* 𝐴 ↔ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) ~~>* 𝐴 ) )
10		eqid	⊢ ( ℤ≥ ‘ 𝑗 ) = ( ℤ≥ ‘ 𝑗 )
11		simpr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ )
12	3	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → 𝐴 ∈ ℝ )
13	8 10 11 12	xlimclim	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) ~~>* 𝐴 ↔ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) ⇝ 𝐴 ) )
14	1	fvexi	⊢ 𝑍 ∈ V
15	14	a1i	⊢ ( 𝜑 → 𝑍 ∈ V )
16	2 15	fexd	⊢ ( 𝜑 → 𝐹 ∈ V )
17		climres	⊢ ( ( 𝑗 ∈ ℤ ∧ 𝐹 ∈ V ) → ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴 ) )
18	7 16 17	syl2anr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴 ) )
19	18	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴 ) )
20	9 13 19	3bitrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑗 ) ) : ( ℤ≥ ‘ 𝑗 ) ⟶ ℝ ) → ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ⇝ 𝐴 ) )
21	20 4	r19.29a	⊢ ( 𝜑 → ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ⇝ 𝐴 ) )