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Theorem lgamucov 25177

Description: The 𝑈 regions used in the proof of lgamgulm 25174 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)

**Hypotheses**
Ref	Expression
lgamucov.u	⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}
lgamucov.a	⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
lgamucov.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
lgamucov	⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))

Distinct variable groups: 𝑘,𝑟,𝑥,𝐴 𝜑,𝑘,𝑟,𝑥 Allowed substitution hints: 𝑈(𝑥,𝑘,𝑟) 𝐽(𝑥,𝑘,𝑟)

Proof of Theorem lgamucov Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnxmet 22946	. . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
2	1	a1i 11	. . 3 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
3		difss 3964	. . . . 5 ⊢ (ℤ ∖ ℕ) ⊆ ℤ
4		lgamucov.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	sszcld 22990	. . . . 5 ⊢ ((ℤ ∖ ℕ) ⊆ ℤ → (ℤ ∖ ℕ) ∈ (Clsd‘𝐽))
6	4	cnfldtopon 22956	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
7	6	toponunii 21091	. . . . . 6 ⊢ ℂ = ∪ 𝐽
8	7	cldopn 21206	. . . . 5 ⊢ ((ℤ ∖ ℕ) ∈ (Clsd‘𝐽) → (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽)
9	3, 5, 8	mp2b 10	. . . 4 ⊢ (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽
10	9	a1i 11	. . 3 ⊢ (𝜑 → (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽)
11		lgamucov.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
12	4	cnfldtopn 22955	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
13	12	mopni2 22668	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽 ∧ 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) → ∃𝑎 ∈ ℝ⁺ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
14	2, 10, 11, 13	syl3anc 1494	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ⁺ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
15	11	eldifad 3810	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
16	15	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝐴 ∈ ℂ)
17	16	abscld 14552	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (abs‘𝐴) ∈ ℝ)
18		simprl 787	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝑎 ∈ ℝ⁺)
19	18	rpred 12156	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝑎 ∈ ℝ)
20	17, 19	readdcld 10386	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ((abs‘𝐴) + 𝑎) ∈ ℝ)
21		2re 11425	. . . . . . 7 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 2 ∈ ℝ)
23	22, 18	rerpdivcld 12187	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (2 / 𝑎) ∈ ℝ)
24	20, 23	readdcld 10386	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ)
25		arch 11615	. . . 4 ⊢ ((((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ → ∃𝑟 ∈ ℕ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
26	24, 25	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ∃𝑟 ∈ ℕ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
27	4	cnfldtop 22957	. . . . . . . 8 ⊢ 𝐽 ∈ Top
28	27	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐽 ∈ Top)
29		lgamucov.u	. . . . . . . . 9 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}
30		ssrab2 3912	. . . . . . . . 9 ⊢ {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))} ⊆ ℂ
31	29, 30	eqsstri 3860	. . . . . . . 8 ⊢ 𝑈 ⊆ ℂ
32	31	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝑈 ⊆ ℂ)
33	1	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (abs ∘ − ) ∈ (∞Met‘ℂ))
34	16	ad2antrr 717	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐴 ∈ ℂ)
35	18	ad2antrr 717	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝑎 ∈ ℝ⁺)
36	35	rphalfcld 12168	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝑎 / 2) ∈ ℝ⁺)
37	36	rpxrd 12157	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝑎 / 2) ∈ ℝ^*)
38	12	blopn 22675	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ^*) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ∈ 𝐽)
39	33, 34, 37, 38	syl3anc 1494	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ∈ 𝐽)
40		simplr 785	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → 𝑥 ∈ ℂ)
41	40	abscld 14552	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘𝑥) ∈ ℝ)
42		simp-4r 803	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → 𝑟 ∈ ℕ)
43	42	nnred 11367	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → 𝑟 ∈ ℝ)
44	24	ad4antr 724	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ)
45	20	ad4antr 724	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ((abs‘𝐴) + 𝑎) ∈ ℝ)
46	17	ad4antr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘𝐴) ∈ ℝ)
47	41, 46	resubcld 10782	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ((abs‘𝑥) − (abs‘𝐴)) ∈ ℝ)
48	19	ad4antr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → 𝑎 ∈ ℝ)
49	48	rehalfcld 11605	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (𝑎 / 2) ∈ ℝ)
50	34	ad2antrr 717	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → 𝐴 ∈ ℂ)
51	40, 50	subcld 10713	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (𝑥 − 𝐴) ∈ ℂ)
52	51	abscld 14552	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘(𝑥 − 𝐴)) ∈ ℝ)
53	40, 50	abs2difd 14573	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ((abs‘𝑥) − (abs‘𝐴)) ≤ (abs‘(𝑥 − 𝐴)))
54		eqid 2825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs ∘ − ) = (abs ∘ − )
55	54	cnmetdval 22944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴(abs ∘ − )𝑥) = (abs‘(𝐴 − 𝑥)))
56	50, 40, 55	syl2anc 579	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (𝐴(abs ∘ − )𝑥) = (abs‘(𝐴 − 𝑥)))
57	50, 40	abssubd 14569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘(𝐴 − 𝑥)) = (abs‘(𝑥 − 𝐴)))
58	56, 57	eqtrd 2861	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (𝐴(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝐴)))
59		simpr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (𝐴(abs ∘ − )𝑥) < (𝑎 / 2))
60	58, 59	eqbrtrrd 4897	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘(𝑥 − 𝐴)) < (𝑎 / 2))
61	47, 52, 49, 53, 60	lelttrd 10514	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ((abs‘𝑥) − (abs‘𝐴)) < (𝑎 / 2))
62	35	ad2antrr 717	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → 𝑎 ∈ ℝ⁺)
63		rphalflt 12143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℝ⁺ → (𝑎 / 2) < 𝑎)
64	62, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (𝑎 / 2) < 𝑎)
65	47, 49, 48, 61, 64	lttrd 10517	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ((abs‘𝑥) − (abs‘𝐴)) < 𝑎)
66	41, 46, 48	ltsubadd2d 10950	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (((abs‘𝑥) − (abs‘𝐴)) < 𝑎 ↔ (abs‘𝑥) < ((abs‘𝐴) + 𝑎)))
67	65, 66	mpbid 224	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘𝑥) < ((abs‘𝐴) + 𝑎))
68		2rp 12117	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ⁺
69	68	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → 2 ∈ ℝ⁺)
70	69, 62	rpdivcld 12173	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (2 / 𝑎) ∈ ℝ⁺)
71	45, 70	ltaddrpd 12189	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ((abs‘𝐴) + 𝑎) < (((abs‘𝐴) + 𝑎) + (2 / 𝑎)))
72	41, 45, 44, 67, 71	lttrd 10517	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘𝑥) < (((abs‘𝐴) + 𝑎) + (2 / 𝑎)))
73		simpllr 793	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
74	41, 44, 43, 72, 73	lttrd 10517	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘𝑥) < 𝑟)
75	41, 43, 74	ltled 10504	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → (abs‘𝑥) ≤ 𝑟)
76	42	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑟 ∈ ℕ)
77	76	nnrecred 11402	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (1 / 𝑟) ∈ ℝ)
78		simpllr 793	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ ℂ)
79		simpr 479	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
80	79	nn0cnd 11680	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
81	78, 80	addcld 10376	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 + 𝑘) ∈ ℂ)
82	81	abscld 14552	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑥 + 𝑘)) ∈ ℝ)
83	49	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑎 / 2) ∈ ℝ)
84	23	ad5antr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (2 / 𝑎) ∈ ℝ)
85	44	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ)
86	43	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑟 ∈ ℝ)
87	50	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ ℂ)
88	11	ad6antr 732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
89	88	dmgmn0 25165	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ≠ 0)
90	87, 89	absrpcld 14564	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (abs‘𝐴) ∈ ℝ⁺)
91	62	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑎 ∈ ℝ⁺)
92	90, 91	rpaddcld 12171	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → ((abs‘𝐴) + 𝑎) ∈ ℝ⁺)
93	84, 92	ltaddrp2d 12190	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (2 / 𝑎) < (((abs‘𝐴) + 𝑎) + (2 / 𝑎)))
94		simp-4r 803	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
95	84, 85, 86, 93, 94	lttrd 10517	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (2 / 𝑎) < 𝑟)
96	70	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (2 / 𝑎) ∈ ℝ⁺)
97	76	nnrpd 12154	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑟 ∈ ℝ⁺)
98	96, 97	ltrecd 12174	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → ((2 / 𝑎) < 𝑟 ↔ (1 / 𝑟) < (1 / (2 / 𝑎))))
99	95, 98	mpbid 224	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (1 / 𝑟) < (1 / (2 / 𝑎)))
100		2cnd 11429	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 2 ∈ ℂ)
101	91	rpcnd 12158	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑎 ∈ ℂ)
102		2ne0 11462	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
103	102	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 2 ≠ 0)
104	91	rpne0d 12161	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑎 ≠ 0)
105	100, 101, 103, 104	recdivd 11144	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (1 / (2 / 𝑎)) = (𝑎 / 2))
106	99, 105	breqtrd 4899	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (1 / 𝑟) < (𝑎 / 2))
107		eldmgm 25161	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (-𝑘 ∈ ℂ ∧ ¬ --𝑘 ∈ ℕ₀))
108	107	simprbi 492	. . . . . . . . . . . . . . . 16 ⊢ (-𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → ¬ --𝑘 ∈ ℕ₀)
109	80	negnegd 10704	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → --𝑘 = 𝑘)
110	109, 79	eqeltrd 2906	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → --𝑘 ∈ ℕ₀)
111	108, 110	nsyl3 136	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → ¬ -𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
112	1	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (abs ∘ − ) ∈ (∞Met‘ℂ))
113	37	ad3antrrr 721	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑎 / 2) ∈ ℝ^*)
114	80	negcld 10700	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → -𝑘 ∈ ℂ)
115		elbl2 22565	. . . . . . . . . . . . . . . . . 18 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑥 ∈ ℂ ∧ -𝑘 ∈ ℂ)) → (-𝑘 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ↔ (𝑥(abs ∘ − )-𝑘) < (𝑎 / 2)))
116	112, 113, 78, 114, 115	syl22anc 872	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (-𝑘 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ↔ (𝑥(abs ∘ − )-𝑘) < (𝑎 / 2)))
117	54	cnmetdval 22944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ -𝑘 ∈ ℂ) → (𝑥(abs ∘ − )-𝑘) = (abs‘(𝑥 − -𝑘)))
118	78, 114, 117	syl2anc 579	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑥(abs ∘ − )-𝑘) = (abs‘(𝑥 − -𝑘)))
119	78, 80	subnegd 10720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 − -𝑘) = (𝑥 + 𝑘))
120	119	fveq2d 6437	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑥 − -𝑘)) = (abs‘(𝑥 + 𝑘)))
121	118, 120	eqtrd 2861	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑥(abs ∘ − )-𝑘) = (abs‘(𝑥 + 𝑘)))
122	121	breq1d 4883	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → ((𝑥(abs ∘ − )-𝑘) < (𝑎 / 2) ↔ (abs‘(𝑥 + 𝑘)) < (𝑎 / 2)))
123	82, 83	ltnled 10503	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → ((abs‘(𝑥 + 𝑘)) < (𝑎 / 2) ↔ ¬ (𝑎 / 2) ≤ (abs‘(𝑥 + 𝑘))))
124	116, 122, 123	3bitrd 297	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (-𝑘 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ↔ ¬ (𝑎 / 2) ≤ (abs‘(𝑥 + 𝑘))))
125	48	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝑎 ∈ ℝ)
126		simplr 785	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝐴(abs ∘ − )𝑥) < (𝑎 / 2))
127		elbl3 22567	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → (𝐴 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ↔ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)))
128	112, 113, 78, 87, 127	syl22anc 872	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝐴 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ↔ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)))
129	126, 128	mpbird 249	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)))
130		blhalf 22580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ) ∧ (𝑎 ∈ ℝ ∧ 𝐴 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)))) → (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ (𝐴(ball‘(abs ∘ − ))𝑎))
131	112, 78, 125, 129, 130	syl22anc 872	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ (𝐴(ball‘(abs ∘ − ))𝑎))
132		simprr 789	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
133	132	ad5antr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
134	131, 133	sstrd 3837	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
135	134	sseld 3826	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (-𝑘 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) → -𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ))))
136	124, 135	sylbird 252	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (¬ (𝑎 / 2) ≤ (abs‘(𝑥 + 𝑘)) → -𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ))))
137	111, 136	mt3d 143	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (𝑎 / 2) ≤ (abs‘(𝑥 + 𝑘)))
138	77, 83, 82, 106, 137	ltletrd 10516	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (1 / 𝑟) < (abs‘(𝑥 + 𝑘)))
139	77, 82, 138	ltled 10504	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) ∧ 𝑘 ∈ ℕ₀) → (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))
140	139	ralrimiva 3175	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))
141	75, 140	jca 507	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) ∧ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)) → ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘))))
142	141	ex 403	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) ∧ 𝑥 ∈ ℂ) → ((𝐴(abs ∘ − )𝑥) < (𝑎 / 2) → ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))))
143	142	ss2rabdv 3908	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → {𝑥 ∈ ℂ ∣ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)} ⊆ {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))})
144		blval 22561	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ^*) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) = {𝑥 ∈ ℂ ∣ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)})
145	33, 34, 37, 144	syl3anc 1494	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) = {𝑥 ∈ ℂ ∣ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)})
146	29	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))})
147	143, 145, 146	3sstr4d 3873	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ 𝑈)
148	7	ssntr 21233	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑈 ⊆ ℂ) ∧ ((𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ∈ 𝐽 ∧ (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ 𝑈)) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ ((int‘𝐽)‘𝑈))
149	28, 32, 39, 147, 148	syl22anc 872	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ ((int‘𝐽)‘𝑈))
150		blcntr 22588	. . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ⁺) → 𝐴 ∈ (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)))
151	33, 34, 36, 150	syl3anc 1494	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐴 ∈ (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)))
152	149, 151	sseldd 3828	. . . . 5 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐴 ∈ ((int‘𝐽)‘𝑈))
153	152	ex 403	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) → ((((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟 → 𝐴 ∈ ((int‘𝐽)‘𝑈)))
154	153	reximdva 3225	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (∃𝑟 ∈ ℕ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈)))
155	26, 154	mpd 15	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))
156	14, 155	rexlimddv 3245	1 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 {crab 3121 ∖ cdif 3795 ⊆ wss 3798 class class class wbr 4873 ∘ ccom 5346 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 1c1 10253 + caddc 10255 ℝ^*cxr 10390 < clt 10391 ≤ cle 10392 − cmin 10585 -cneg 10586 / cdiv 11009 ℕcn 11350 2c2 11406 ℕ₀cn0 11618 ℤcz 11704 ℝ⁺crp 12112 abscabs 14351 TopOpenctopn 16435 ∞Metcxmet 20091 ballcbl 20093 ℂ_fldccnfld 20106 Topctop 21068 Clsdccld 21191 intcnt 21192

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fi 8586 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-fz 12620 df-fl 12888 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-rest 16436 df-topn 16437 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-xms 22495 df-ms 22496

This theorem is referenced by: lgamucov2 25178

W3C validator