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Theorem lgamucov 26976

Description: The 𝑈 regions used in the proof of lgamgulm 26973 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 24688	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
2		difss 4085	. . . . 5 ⊢ (ℤ ∖ ℕ) ⊆ ℤ
3		lgamucov.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	3	sszcld 24734	. . . . 5 ⊢ ((ℤ ∖ ℕ) ⊆ ℤ → (ℤ ∖ ℕ) ∈ (Clsd‘𝐽))
5	3	cnfldtopon 24698	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
6	5	toponunii 22832	. . . . . 6 ⊢ ℂ = ∪ 𝐽
7	6	cldopn 22947	. . . . 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 24697	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
12	11	mopni2 24409	. . 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 1468	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ⁺ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
14	10	eldifad 3910	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝐴 ∈ ℂ)
16	15	abscld 15348	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (abs‘𝐴) ∈ ℝ)
17		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝑎 ∈ ℝ⁺)
18	17	rpred 12936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 𝑎 ∈ ℝ)
19	16, 18	readdcld 11148	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ((abs‘𝐴) + 𝑎) ∈ ℝ)
20		2re 12206	. . . . . . 7 ⊢ 2 ∈ ℝ
21	20	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → 2 ∈ ℝ)
22	21, 17	rerpdivcld 12967	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (2 / 𝑎) ∈ ℝ)
23	19, 22	readdcld 11148	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ)
24		arch 12385	. . . 4 ⊢ ((((abs‘𝐴) + 𝑎) + (2 / 𝑎)) ∈ ℝ → ∃𝑟 ∈ ℕ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
25	23, 24	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ∃𝑟 ∈ ℕ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟)
26	3	cnfldtop 24699	. . . . . . . 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 4031	. . . . . . . 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 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝐴 ∈ ℂ)
32	17	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) ∧ 𝑟 ∈ ℕ) ∧ (((abs‘𝐴) + 𝑎) + (2 / 𝑎)) < 𝑟) → 𝑎 ∈ ℝ⁺)
33	32	rphalfcld 12948	. . . . . . . . 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 12937	. . . . . . . 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 24416	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ^*) → (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)) ∈ 𝐽)
36	1, 31, 34, 35	mp3an2i 1468	. . . . . . 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 768	. . . . . . . . . . . . 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 15348	. . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 12147	. . . . . . . . . . . 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 732	. . . . . . . . . . . . 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 732	. . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . 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 11552	. . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . 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 12375	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . 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 11479	. . . . . . . . . . . . . . . . . 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 15348	. . . . . . . . . . . . . . . . 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 15369	. . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs ∘ − ) = (abs ∘ − )
52	51	cnmetdval 24686	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴(abs ∘ − )𝑥) = (abs‘(𝐴 − 𝑥)))
53	47, 37, 52	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 15365	. . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . 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 11278	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 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 12923	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℝ⁺ → (𝑎 / 2) < 𝑎)
61	59, 60	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)
62	44, 46, 45, 58, 61	lttrd 11281	. . . . . . . . . . . . . . 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 11722	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 12897	. . . . . . . . . . . . . . . . 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 12953	. . . . . . . . . . . . . . 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 12969	. . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 11281	. . . . . . . . . . . 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 11268	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 12183	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 12451	. . . . . . . . . . . . . . 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 11138	. . . . . . . . . . . . . 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 15348	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 26964	. . . . . . . . . . . . . . . . . . . 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 15360	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 12951	. . . . . . . . . . . . . . . . . 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 12970	. . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 12934	. . . . . . . . . . . . . . . . 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 12954	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 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 12210	. . . . . . . . . . . . . . . 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 12938	. . . . . . . . . . . . . . . 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 12236	. . . . . . . . . . . . . . . . 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 12941	. . . . . . . . . . . . . . . 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 11921	. . . . . . . . . . . . . . 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 5119	. . . . . . . . . . . . . 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 26960	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (-𝑘 ∈ ℂ ∧ ¬ --𝑘 ∈ ℕ₀))
105	104	simprbi 496	. . . . . . . . . . . . . . . 16 ⊢ (-𝑘 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → ¬ --𝑘 ∈ ℕ₀)
106	77	negnegd 11470	. . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . 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 138	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 11466	. . . . . . . . . . . . . . . . . 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 24306	. . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . 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 24686	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ -𝑘 ∈ ℂ) → (𝑥(abs ∘ − )-𝑘) = (abs‘(𝑥 − -𝑘)))
115	75, 111, 114	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 11486	. . . . . . . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . . 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 11267	. . . . . . . . . . . . . . . . 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 305	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . 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 24308	. . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . 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 24321	. . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
130	129	ad5antr 734	. . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . 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 148	. . . . . . . . . . . . . 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 11280	. . . . . . . . . . . . 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 11268	. . . . . . . . . . . 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 3125	. . . . . . . . . . 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 511	. . . . . . . . . 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 412	. . . . . . . . 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 4024	. . . . . . . 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 24302	. . . . . . . . 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 1468	. . . . . . . 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 3986	. . . . . . 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 22974	. . . . . . 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 838	. . . . . 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 24329	. . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝑎 / 2) ∈ ℝ⁺) → 𝐴 ∈ (𝐴(ball‘(abs ∘ − ))(𝑎 / 2)))
148	1, 31, 33, 147	mp3an2i 1468	. . . . . 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 3931	. . . . 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 412	. . . 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 3146	. . 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 15	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ (𝐴(ball‘(abs ∘ − ))𝑎) ⊆ (ℂ ∖ (ℤ ∖ ℕ)))) → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))
153	13, 152	rexlimddv 3140	1 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 ∃wrex 3057 {crab 3396 ∖ cdif 3895 ⊆ wss 3898 class class class wbr 5093 ∘ ccom 5623 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 ℝcr 11012 0cc0 11013 1c1 11014 + caddc 11016 ℝ^*cxr 11152 < clt 11153 ≤ cle 11154 − cmin 11351 -cneg 11352 / cdiv 11781 ℕcn 12132 2c2 12187 ℕ₀cn0 12388 ℤcz 12475 ℝ⁺crp 12892 abscabs 15143 TopOpenctopn 17327 ∞Metcxmet 21278 ballcbl 21280 ℂ_fldccnfld 21293 Topctop 22809 Clsdccld 22932 intcnt 22933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fi 9302 df-sup 9333 df-inf 9334 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-fz 13410 df-fl 13698 df-seq 13911 df-exp 13971 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-struct 17060 df-slot 17095 df-ndx 17107 df-base 17123 df-plusg 17176 df-mulr 17177 df-starv 17178 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-rest 17328 df-topn 17329 df-topgen 17349 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-xms 24236 df-ms 24237

This theorem is referenced by: lgamucov2 26977

W3C validator