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Theorem reccn2 15645

Description: The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)

**Hypothesis**
Ref	Expression
reccn2.t	⊢ 𝑇 = (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2))

**Assertion**
Ref	Expression
reccn2	⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))

Distinct variable groups: 𝑦,𝑧,𝐴 𝑦,𝐵,𝑧 𝑦,𝑇,𝑧

**Proof of Theorem reccn2**
Step	Hyp	Ref	Expression
1		reccn2.t	. . 3 ⊢ 𝑇 = (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2))
2		1rp 13063	. . . . 5 ⊢ 1 ∈ ℝ⁺
3		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → 𝐴 ∈ (ℂ ∖ {0}))
4		eldifsn 4811	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
5	3, 4	sylib 218	. . . . . . 7 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
6		absrpcl 15339	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ⁺)
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → (abs‘𝐴) ∈ ℝ⁺)
8		rpmulcl 13082	. . . . . 6 ⊢ (((abs‘𝐴) ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → ((abs‘𝐴) · 𝐵) ∈ ℝ⁺)
9	7, 8	sylancom 587	. . . . 5 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → ((abs‘𝐴) · 𝐵) ∈ ℝ⁺)
10		ifcl 4593	. . . . 5 ⊢ ((1 ∈ ℝ⁺ ∧ ((abs‘𝐴) · 𝐵) ∈ ℝ⁺) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ∈ ℝ⁺)
11	2, 9, 10	sylancr 586	. . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ∈ ℝ⁺)
12	7	rphalfcld 13113	. . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → ((abs‘𝐴) / 2) ∈ ℝ⁺)
13	11, 12	rpmulcld 13117	. . 3 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2)) ∈ ℝ⁺)
14	1, 13	eqeltrid 2848	. 2 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → 𝑇 ∈ ℝ⁺)
15	5	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
16	15	simpld 494	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝐴 ∈ ℂ)
17		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑧 ∈ (ℂ ∖ {0}))
18		eldifsn 4811	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↔ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
19	17, 18	sylib 218	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
20	19	simpld 494	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑧 ∈ ℂ)
21	16, 20	mulcld 11312	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 · 𝑧) ∈ ℂ)
22		mulne0 11934	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0)) → (𝐴 · 𝑧) ≠ 0)
23	15, 19, 22	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 · 𝑧) ≠ 0)
24	16, 20, 21, 23	divsubdird 12111	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 − 𝑧) / (𝐴 · 𝑧)) = ((𝐴 / (𝐴 · 𝑧)) − (𝑧 / (𝐴 · 𝑧))))
25	16	mulridd 11309	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 · 1) = 𝐴)
26	25	oveq1d 7465	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 · 1) / (𝐴 · 𝑧)) = (𝐴 / (𝐴 · 𝑧)))
27		1cnd 11287	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 1 ∈ ℂ)
28		divcan5 11998	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → ((𝐴 · 1) / (𝐴 · 𝑧)) = (1 / 𝑧))
29	27, 19, 15, 28	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 · 1) / (𝐴 · 𝑧)) = (1 / 𝑧))
30	26, 29	eqtr3d 2782	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 / (𝐴 · 𝑧)) = (1 / 𝑧))
31	20	mulridd 11309	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 · 1) = 𝑧)
32	20, 16	mulcomd 11313	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 · 𝐴) = (𝐴 · 𝑧))
33	31, 32	oveq12d 7468	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝑧 · 1) / (𝑧 · 𝐴)) = (𝑧 / (𝐴 · 𝑧)))
34		divcan5 11998	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0)) → ((𝑧 · 1) / (𝑧 · 𝐴)) = (1 / 𝐴))
35	27, 15, 19, 34	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝑧 · 1) / (𝑧 · 𝐴)) = (1 / 𝐴))
36	33, 35	eqtr3d 2782	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 / (𝐴 · 𝑧)) = (1 / 𝐴))
37	30, 36	oveq12d 7468	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 / (𝐴 · 𝑧)) − (𝑧 / (𝐴 · 𝑧))) = ((1 / 𝑧) − (1 / 𝐴)))
38	24, 37	eqtrd 2780	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 − 𝑧) / (𝐴 · 𝑧)) = ((1 / 𝑧) − (1 / 𝐴)))
39	38	fveq2d 6926	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘((𝐴 − 𝑧) / (𝐴 · 𝑧))) = (abs‘((1 / 𝑧) − (1 / 𝐴))))
40	16, 20	subcld 11649	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 − 𝑧) ∈ ℂ)
41	40, 21, 23	absdivd 15506	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘((𝐴 − 𝑧) / (𝐴 · 𝑧))) = ((abs‘(𝐴 − 𝑧)) / (abs‘(𝐴 · 𝑧))))
42	39, 41	eqtr3d 2782	. . . . 5 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘((1 / 𝑧) − (1 / 𝐴))) = ((abs‘(𝐴 − 𝑧)) / (abs‘(𝐴 · 𝑧))))
43	16, 20	abssubd 15504	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 − 𝑧)) = (abs‘(𝑧 − 𝐴)))
44	20, 16	subcld 11649	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 − 𝐴) ∈ ℂ)
45	44	abscld 15487	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝑧 − 𝐴)) ∈ ℝ)
46	43, 45	eqeltrd 2844	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 − 𝑧)) ∈ ℝ)
47	14	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ∈ ℝ⁺)
48	47	rpred 13101	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ∈ ℝ)
49	21	abscld 15487	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 · 𝑧)) ∈ ℝ)
50		rpre 13067	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
51	50	ad2antlr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝐵 ∈ ℝ)
52	49, 51	remulcld 11322	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘(𝐴 · 𝑧)) · 𝐵) ∈ ℝ)
53		simprr 772	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝑧 − 𝐴)) < 𝑇)
54	43, 53	eqbrtrd 5188	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 − 𝑧)) < 𝑇)
55	9	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) · 𝐵) ∈ ℝ⁺)
56	55	rpred 13101	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) · 𝐵) ∈ ℝ)
57	12	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) / 2) ∈ ℝ⁺)
58	57	rpred 13101	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) / 2) ∈ ℝ)
59	56, 58	remulcld 11322	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2)) ∈ ℝ)
60		1re 11292	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
61		min2 13254	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ ((abs‘𝐴) · 𝐵) ∈ ℝ) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ≤ ((abs‘𝐴) · 𝐵))
62	60, 56, 61	sylancr 586	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ≤ ((abs‘𝐴) · 𝐵))
63	11	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ∈ ℝ⁺)
64	63	rpred 13101	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ∈ ℝ)
65	64, 56, 57	lemul1d 13144	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ≤ ((abs‘𝐴) · 𝐵) ↔ (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2)) ≤ (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2))))
66	62, 65	mpbid 232	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2)) ≤ (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2)))
67	1, 66	eqbrtrid 5201	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ≤ (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2)))
68	20	abscld 15487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝑧) ∈ ℝ)
69	16	abscld 15487	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝐴) ∈ ℝ)
70	69	recnd 11320	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝐴) ∈ ℂ)
71	70	2halvesd 12541	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) / 2) + ((abs‘𝐴) / 2)) = (abs‘𝐴))
72	69, 68	resubcld 11720	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) − (abs‘𝑧)) ∈ ℝ)
73	16, 20	abs2difd 15508	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) − (abs‘𝑧)) ≤ (abs‘(𝐴 − 𝑧)))
74		min1 13253	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ ∧ ((abs‘𝐴) · 𝐵) ∈ ℝ) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ≤ 1)
75	60, 56, 74	sylancr 586	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ≤ 1)
76		1red 11293	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 1 ∈ ℝ)
77	64, 76, 57	lemul1d 13144	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) ≤ 1 ↔ (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2)) ≤ (1 · ((abs‘𝐴) / 2))))
78	75, 77	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2)) ≤ (1 · ((abs‘𝐴) / 2)))
79	1, 78	eqbrtrid 5201	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ≤ (1 · ((abs‘𝐴) / 2)))
80	58	recnd 11320	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) / 2) ∈ ℂ)
81	80	mullidd 11310	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (1 · ((abs‘𝐴) / 2)) = ((abs‘𝐴) / 2))
82	79, 81	breqtrd 5192	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ≤ ((abs‘𝐴) / 2))
83	46, 48, 58, 54, 82	ltletrd 11452	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 − 𝑧)) < ((abs‘𝐴) / 2))
84	72, 46, 58, 73, 83	lelttrd 11450	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) − (abs‘𝑧)) < ((abs‘𝐴) / 2))
85	69, 68, 58	ltsubadd2d 11890	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) − (abs‘𝑧)) < ((abs‘𝐴) / 2) ↔ (abs‘𝐴) < ((abs‘𝑧) + ((abs‘𝐴) / 2))))
86	84, 85	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝐴) < ((abs‘𝑧) + ((abs‘𝐴) / 2)))
87	71, 86	eqbrtrd 5188	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) / 2) + ((abs‘𝐴) / 2)) < ((abs‘𝑧) + ((abs‘𝐴) / 2)))
88	58, 68, 58	ltadd1d 11885	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) / 2) < (abs‘𝑧) ↔ (((abs‘𝐴) / 2) + ((abs‘𝐴) / 2)) < ((abs‘𝑧) + ((abs‘𝐴) / 2))))
89	87, 88	mpbird 257	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) / 2) < (abs‘𝑧))
90	58, 68, 55, 89	ltmul2dd 13157	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2)) < (((abs‘𝐴) · 𝐵) · (abs‘𝑧)))
91	16, 20	absmuld 15505	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 · 𝑧)) = ((abs‘𝐴) · (abs‘𝑧)))
92	91	oveq1d 7465	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘(𝐴 · 𝑧)) · 𝐵) = (((abs‘𝐴) · (abs‘𝑧)) · 𝐵))
93	68	recnd 11320	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝑧) ∈ ℂ)
94	51	recnd 11320	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝐵 ∈ ℂ)
95	70, 93, 94	mul32d 11502	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) · (abs‘𝑧)) · 𝐵) = (((abs‘𝐴) · 𝐵) · (abs‘𝑧)))
96	92, 95	eqtrd 2780	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘(𝐴 · 𝑧)) · 𝐵) = (((abs‘𝐴) · 𝐵) · (abs‘𝑧)))
97	90, 96	breqtrrd 5194	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2)) < ((abs‘(𝐴 · 𝑧)) · 𝐵))
98	48, 59, 52, 67, 97	lelttrd 11450	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 < ((abs‘(𝐴 · 𝑧)) · 𝐵))
99	46, 48, 52, 54, 98	lttrd 11453	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 − 𝑧)) < ((abs‘(𝐴 · 𝑧)) · 𝐵))
100	21, 23	absrpcld 15499	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 · 𝑧)) ∈ ℝ⁺)
101	46, 51, 100	ltdivmuld 13152	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘(𝐴 − 𝑧)) / (abs‘(𝐴 · 𝑧))) < 𝐵 ↔ (abs‘(𝐴 − 𝑧)) < ((abs‘(𝐴 · 𝑧)) · 𝐵)))
102	99, 101	mpbird 257	. . . . 5 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘(𝐴 − 𝑧)) / (abs‘(𝐴 · 𝑧))) < 𝐵)
103	42, 102	eqbrtrd 5188	. . . 4 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵)
104	103	expr 456	. . 3 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑧 ∈ (ℂ ∖ {0})) → ((abs‘(𝑧 − 𝐴)) < 𝑇 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
105	104	ralrimiva 3152	. 2 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑇 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
106		breq2 5170	. . 3 ⊢ (𝑦 = 𝑇 → ((abs‘(𝑧 − 𝐴)) < 𝑦 ↔ (abs‘(𝑧 − 𝐴)) < 𝑇))
107	106	rspceaimv 3641	. 2 ⊢ ((𝑇 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑇 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵)) → ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
108	14, 105, 107	syl2anc 583	1 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ∖ cdif 3973 ifcif 4548 {csn 4648 class class class wbr 5166 ‘cfv 6575 (class class class)co 7450 ℂcc 11184 ℝcr 11185 0cc0 11186 1c1 11187 + caddc 11189 · cmul 11191 < clt 11326 ≤ cle 11327 − cmin 11522 / cdiv 11949 2c2 12350 ℝ⁺crp 13059 abscabs 15285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-sup 9513 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-n0 12556 df-z 12642 df-uz 12906 df-rp 13060 df-seq 14055 df-exp 14115 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287

This theorem is referenced by: rlimdiv 15696 divcnOLD 24911 divcn 24913 climrec 45526

W3C validator