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Theorem lgamucov 27180

Description: The 𝑈 regions used in the proof of lgamgulm 27177 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 24910	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
2		difss 4091	. . . . 5 ⊢ (ℤ ∖ ℕ) ⊆ ℤ
3		lgamucov.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	3	sszcld 24956	. . . . 5 ⊢ ((ℤ ∖ ℕ) ⊆ ℤ → (ℤ ∖ ℕ) ∈ (Clsd‘𝐽))
5	3	cnfldtopon 24920	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
6	5	toponunii 23054	. . . . . 6 ⊢ ℂ = ∪ 𝐽
7	6	cldopn 23169	. . . . 5 ⊢ ((ℤ ∖ ℕ) ∈ (Clsd‘𝐽) → (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽)
8	2, 4, 7	mp2b 10	. . . 4 ⊢ (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽
9	8	a1i 11	. . 3 ⊢ (𝜑 → (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽)
10		lgamucov.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
11	3	cnfldtopn 24919	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
12	11	mopni2 24631	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ (ℤ ∖ ℕ)) ∈ 𝐽 ∧ 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) → ∃𝑎 ∈ ℝ⁺ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
13	1, 9, 10, 12	mp3an2i 1495	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ⁺ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
14	10	eldifad 3918	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
15	14	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝐴 ∈ ℂ)
16	15	abscld 15492	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (abs‘𝐴) ∈ ℝ)
17		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝑎 ∈ ℝ⁺)
18	17	rpred 13061	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝑎 ∈ ℝ)
19	16, 18	readdcld 11239	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ((abs‘𝐴) + 𝑎) ∈ ℝ)
20		2re 12316	. . . . . . 7 ⊢ 2 ∈ ℝ
21	20	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 2 ∈ ℝ)
22	21, 17	rerpdivcld 13092	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (2 / 𝑎) ∈ ℝ)
23	19, 22	readdcld 11239	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ)
24		arch 12502	. . . 4 ⊢ ((((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ → ∃𝑟 ∈ ℕ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
25	23, 24	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ∃𝑟 ∈ ℕ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
26	3	cnfldtop 24921	. . . . . . . 8 ⊢ 𝐽 ∈ Top
27	26	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐽 ∈ Top)
28		lgamucov.u	. . . . . . . . 9 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}
29	28	ssrab3 4037	. . . . . . . 8 ⊢ 𝑈 ⊆ ℂ
30	29	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝑈 ⊆ ℂ)
31	15	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐴 ∈ ℂ)
32	17	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝑎 ∈ ℝ⁺)
33	32	rphalfcld 13073	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝑎 / 2) ∈ ℝ⁺)
34	33	rpxrd 13062	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → (𝑎 / 2) ∈ ℝ^*)
35	11	blopn 24638	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ^*) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ∈ 𝐽)
36	1, 31, 34, 35	mp3an2i 1495	. . . . . . 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)
37		simplr 780	. . . . . . . . . . . . 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)
38	37	abscld 15492	. . . . . . . . . . . 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)
39		simp-4r 795	. . . . . . . . . . . . 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)
40	39	nnred 12249	. . . . . . . . . . . 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)
41	23	ad4antr 744	. . . . . . . . . . . . 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)
42	19	ad4antr 744	. . . . . . . . . . . . . 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)
43	16	ad4antr 744	. . . . . . . . . . . . . . . . 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)
44	38, 43	resubcld 11643	. . . . . . . . . . . . . . . 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)
45	18	ad4antr 744	. . . . . . . . . . . . . . . . 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)
46	45	rehalfcld 12492	. . . . . . . . . . . . . . . 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)
47	31	ad2antrr 738	. . . . . . . . . . . . . . . . . . 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)
48	37, 47	subcld 11570	. . . . . . . . . . . . . . . . . 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)
49	48	abscld 15492	. . . . . . . . . . . . . . . . 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)
50	37, 47	abs2difd 15513	. . . . . . . . . . . . . . . . 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)))
51		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs ∘ − ) = (abs ∘ − )
52	51	cnmetdval 24908	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴(abs ∘ − )𝑥) = (abs‘(𝐴 − 𝑥)))
53	47, 37, 52	syl2anc 595	. . . . . . . . . . . . . . . . . . 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)))
54	47, 37	abssubd 15509	. . . . . . . . . . . . . . . . . . 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)))
55	53, 54	eqtrd 2798	. . . . . . . . . . . . . . . . . 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)))
56		simpr 489	. . . . . . . . . . . . . . . . . 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))
57	55, 56	eqbrtrrd 5136	. . . . . . . . . . . . . . . . 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))
58	44, 49, 46, 50, 57	lelttrd 11369	. . . . . . . . . . . . . . . 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))
59	32	ad2antrr 738	. . . . . . . . . . . . . . . . 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⁺)
60		rphalflt 13048	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℝ⁺ → (𝑎 / 2) < 𝑎)
61	59, 60	syl 18	. . . . . . . . . . . . . . . 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)
62	44, 46, 45, 58, 61	lttrd 11372	. . . . . . . . . . . . . . 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)
63	38, 43, 45	ltsubadd2d 11813	. . . . . . . . . . . . . . 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)))
64	62, 63	mpbid 235	. . . . . . . . . . . . . 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))
65		2rp 13022	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ⁺
66	65	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⁺)
67	66, 59	rpdivcld 13078	. . . . . . . . . . . . . . 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⁺)
68	42, 67	ltaddrpd 13094	. . . . . . . . . . . . . 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)))
69	38, 42, 41, 64, 68	lttrd 11372	. . . . . . . . . . . . 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)))
70		simpllr 787	. . . . . . . . . . . . 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)
71	38, 41, 40, 69, 70	lttrd 11372	. . . . . . . . . . . 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)
72	38, 40, 71	ltled 11359	. . . . . . . . . . 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)
73	39	adantr 485	. . . . . . . . . . . . . 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)
74	73	nnrecred 12288	. . . . . . . . . . . . 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)
75		simpllr 787	. . . . . . . . . . . . . . 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)
76		simpr 489	. . . . . . . . . . . . . . . 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)
77	76	nn0cnd 12568	. . . . . . . . . . . . . . 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)
78	75, 77	addcld 11229	. . . . . . . . . . . . . 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)
79	78	abscld 15492	. . . . . . . . . . . . 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)
80	46	adantr 485	. . . . . . . . . . . . . 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)
81	22	ad5antr 746	. . . . . . . . . . . . . . . . 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)
82	41	adantr 485	. . . . . . . . . . . . . . . . 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)
83	40	adantr 485	. . . . . . . . . . . . . . . . 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)
84	47	adantr 485	. . . . . . . . . . . . . . . . . . . 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)
85	10	ad6antr 748	. . . . . . . . . . . . . . . . . . . . 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)))
86	85	dmgmn0 27168	. . . . . . . . . . . . . . . . . . . 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)
87	84, 86	absrpcld 15504	. . . . . . . . . . . . . . . . . . 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⁺)
88	59	adantr 485	. . . . . . . . . . . . . . . . . . 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⁺)
89	87, 88	rpaddcld 13076	. . . . . . . . . . . . . . . . . 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⁺)
90	81, 89	ltaddrp2d 13095	. . . . . . . . . . . . . . . . 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)))
91		simp-4r 795	. . . . . . . . . . . . . . . . 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)
92	81, 82, 83, 90, 91	lttrd 11372	. . . . . . . . . . . . . . . 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)
93	67	adantr 485	. . . . . . . . . . . . . . . . 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⁺)
94	73	nnrpd 13059	. . . . . . . . . . . . . . . . 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⁺)
95	93, 94	ltrecd 13079	. . . . . . . . . . . . . . . 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))))
96	92, 95	mpbid 235	. . . . . . . . . . . . . . 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)))
97		2cnd 12320	. . . . . . . . . . . . . . . 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)
98	88	rpcnd 13063	. . . . . . . . . . . . . . . 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)
99		2ne0 12348	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
100	99	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)
101	88	rpne0d 13066	. . . . . . . . . . . . . . . 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)
102	97, 98, 100, 101	recdivd 12009	. . . . . . . . . . . . . . 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))
103	96, 102	breqtrd 5138	. . . . . . . . . . . . . 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))
104		eldmgm 27164	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (-𝑘 ∈ ℂ ∧ ¬ --𝑘 ∈ ℕ₀))
105	104	simprbi 502	. . . . . . . . . . . . . . . 16 ⊢ (-𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → ¬ --𝑘 ∈ ℕ₀)
106	77	negnegd 11561	. . . . . . . . . . . . . . . . 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)
107	106, 76	eqeltrd 2863	. . . . . . . . . . . . . . . 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)
108	105, 107	nsyl3 139	. . . . . . . . . . . . . . 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)))
109	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))
110	34	ad3antrrr 742	. . . . . . . . . . . . . . . . . 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^*)
111	77	negcld 11557	. . . . . . . . . . . . . . . . . 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)
112		elbl2 24528	. . . . . . . . . . . . . . . . . 18 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑥 ∈ ℂ ∧ -𝑘 ∈ ℂ)) → (-𝑘 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ↔ (𝑥(abs ∘ − )-𝑘) < (𝑎 / 2)))
113	109, 110, 75, 111, 112	syl22anc 851	. . . . . . . . . . . . . . . . 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)))
114	51	cnmetdval 24908	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ -𝑘 ∈ ℂ) → (𝑥(abs ∘ − )-𝑘) = (abs‘(𝑥 − -𝑘)))
115	75, 111, 114	syl2anc 595	. . . . . . . . . . . . . . . . . . 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)))
116	75, 77	subnegd 11577	. . . . . . . . . . . . . . . . . . . 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))
117	116	fveq2d 6887	. . . . . . . . . . . . . . . . . . 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)))
118	115, 117	eqtrd 2798	. . . . . . . . . . . . . . . . . 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)))
119	118	breq1d 5120	. . . . . . . . . . . . . . . . 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)))
120	79, 80	ltnled 11358	. . . . . . . . . . . . . . . . 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))))
121	113, 119, 120	3bitrd 308	. . . . . . . . . . . . . . . 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))))
122	45	adantr 485	. . . . . . . . . . . . . . . . . . 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)
123		simplr 780	. . . . . . . . . . . . . . . . . . . 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))
124		elbl3 24530	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → (𝐴 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ↔ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)))
125	109, 110, 75, 84, 124	syl22anc 851	. . . . . . . . . . . . . . . . . . . 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)))
126	123, 125	mpbird 260	. . . . . . . . . . . . . . . . . . 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)))
127		blhalf 24543	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ) ∧ (𝑎 ∈ ℝ ∧ 𝐴 ∈ (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)))) → (𝑥(ball‘(abs ∘ − ))(𝑎 / 2)) ⊆ (𝐴(ball‘(abs ∘ − ))𝑎))
128	109, 75, 122, 126, 127	syl22anc 851	. . . . . . . . . . . . . . . . . 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))
129		simprr 784	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
130	129	ad5antr 746	. . . . . . . . . . . . . . . . . 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)))
131	128, 130	sstrd 3948	. . . . . . . . . . . . . . . . 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)))
132	131	sseld 3937	. . . . . . . . . . . . . . . 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))))
133	121, 132	sylbird 263	. . . . . . . . . . . . . . 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))))
134	108, 133	mt3d 149	. . . . . . . . . . . . . 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)))
135	74, 80, 79, 103, 134	ltletrd 11371	. . . . . . . . . . . . 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)))
136	74, 79, 135	ltled 11359	. . . . . . . . . . . 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)))
137	136	ralrimiva 3157	. . . . . . . . . . 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)))
138	72, 137	jca 520	. . . . . . . . . 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))))
139	138	ex 417	. . . . . . . . 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)))))
140	139	ss2rabdv 4030	. . . . . . . 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)))})
141		blval 24524	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ^*) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) = {𝑥 ∈ ℂ ∣ (𝐴(abs ∘ − )𝑥) < (𝑎 / 2)})
142	1, 31, 34, 141	mp3an2i 1495	. . . . . . . 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)})
143	28	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)))})
144	140, 142, 143	3sstr4d 3993	. . . . . . 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)
145	6	ssntr 23196	. . . . . . 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))
146	27, 30, 36, 144, 145	syl22anc 851	. . . . . 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))
147		blcntr 24551	. . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ⁺) → 𝐴 ∈ (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)))
148	1, 31, 33, 147	mp3an2i 1495	. . . . . 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)))
149	146, 148	sseldd 3939	. . . . 5 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐴 ∈ ((int‘𝐽)‘𝑈))
150	149	ex 417	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) → ((((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟 → 𝐴 ∈ ((int‘𝐽)‘𝑈)))
151	150	reximdva 3178	. . 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)))
152	25, 151	mpd 16	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))
153	13, 152	rexlimddv 3172	1 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 ∖ cdif 3903 ⊆ wss 3906 class class class wbr 5110 ∘ ccom 5667 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11442 -cneg 11443 / cdiv 11872 ℕcn 12234 2c2 12296 ℕ₀cn0 12505 ℤcz 12592 ℝ⁺crp 13017 abscabs 15287 TopOpenctopn 17475 ∞Metcxmet 21488 ballcbl 21490 ℂ_fldccnfld 21503 Topctop 23031 Clsdccld 23154 intcnt 23155

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fi 9372 df-sup 9403 df-inf 9404 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-fz 13537 df-fl 13827 df-seq 14040 df-exp 14100 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-struct 17208 df-slot 17243 df-ndx 17255 df-base 17271 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-rest 17476 df-topn 17477 df-topgen 17497 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-xms 24458 df-ms 24459

This theorem is referenced by: lgamucov2 27181

W3C validator