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Theorem reccn2 15545

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 12981	. . . . 5 ⊢ 1 ∈ ℝ⁺
3		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → 𝐴 ∈ (ℂ ∖ {0}))
4		eldifsn 4785	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
5	3, 4	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
6		absrpcl 15239	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ⁺)
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → (abs‘𝐴) ∈ ℝ⁺)
8		rpmulcl 13000	. . . . . 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 4568	. . . . 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 13031	. . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → ((abs‘𝐴) / 2) ∈ ℝ⁺)
13	11, 12	rpmulcld 13035	. . 3 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2)) ∈ ℝ⁺)
14	1, 13	eqeltrid 2831	. 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 768	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑧 ∈ (ℂ ∖ {0}))
18		eldifsn 4785	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↔ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
19	17, 18	sylib 217	. . . . . . . . . 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 11235	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 · 𝑧) ∈ ℂ)
22		mulne0 11857	. . . . . . . . . 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 12030	. . . . . . . 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 11232	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 · 1) = 𝐴)
26	25	oveq1d 7419	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 · 1) / (𝐴 · 𝑧)) = (𝐴 / (𝐴 · 𝑧)))
27		1cnd 11210	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 1 ∈ ℂ)
28		divcan5 11917	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → ((𝐴 · 1) / (𝐴 · 𝑧)) = (1 / 𝑧))
29	27, 19, 15, 28	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 · 1) / (𝐴 · 𝑧)) = (1 / 𝑧))
30	26, 29	eqtr3d 2768	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 / (𝐴 · 𝑧)) = (1 / 𝑧))
31	20	mulridd 11232	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 · 1) = 𝑧)
32	20, 16	mulcomd 11236	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 · 𝐴) = (𝐴 · 𝑧))
33	31, 32	oveq12d 7422	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝑧 · 1) / (𝑧 · 𝐴)) = (𝑧 / (𝐴 · 𝑧)))
34		divcan5 11917	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0)) → ((𝑧 · 1) / (𝑧 · 𝐴)) = (1 / 𝐴))
35	27, 15, 19, 34	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝑧 · 1) / (𝑧 · 𝐴)) = (1 / 𝐴))
36	33, 35	eqtr3d 2768	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 / (𝐴 · 𝑧)) = (1 / 𝐴))
37	30, 36	oveq12d 7422	. . . . . . . 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 2766	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((𝐴 − 𝑧) / (𝐴 · 𝑧)) = ((1 / 𝑧) − (1 / 𝐴)))
39	38	fveq2d 6888	. . . . . 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 11572	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝐴 − 𝑧) ∈ ℂ)
41	40, 21, 23	absdivd 15406	. . . . . 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 2768	. . . . 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 15404	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 − 𝑧)) = (abs‘(𝑧 − 𝐴)))
44	20, 16	subcld 11572	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (𝑧 − 𝐴) ∈ ℂ)
45	44	abscld 15387	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝑧 − 𝐴)) ∈ ℝ)
46	43, 45	eqeltrd 2827	. . . . . . 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 13019	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ∈ ℝ)
49	21	abscld 15387	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 · 𝑧)) ∈ ℝ)
50		rpre 12985	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
51	50	ad2antlr 724	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝐵 ∈ ℝ)
52	49, 51	remulcld 11245	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘(𝐴 · 𝑧)) · 𝐵) ∈ ℝ)
53		simprr 770	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝑧 − 𝐴)) < 𝑇)
54	43, 53	eqbrtrd 5163	. . . . . . 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 13019	. . . . . . . . 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 13019	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) / 2) ∈ ℝ)
59	56, 58	remulcld 11245	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2)) ∈ ℝ)
60		1re 11215	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
61		min2 13172	. . . . . . . . . . 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 13019	. . . . . . . . . . 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 13062	. . . . . . . . . 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 231	. . . . . . . . 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 5176	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ≤ (((abs‘𝐴) · 𝐵) · ((abs‘𝐴) / 2)))
68	20	abscld 15387	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝑧) ∈ ℝ)
69	16	abscld 15387	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝐴) ∈ ℝ)
70	69	recnd 11243	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝐴) ∈ ℂ)
71	70	2halvesd 12459	. . . . . . . . . . . 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 11643	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) − (abs‘𝑧)) ∈ ℝ)
73	16, 20	abs2difd 15408	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) − (abs‘𝑧)) ≤ (abs‘(𝐴 − 𝑧)))
74		min1 13171	. . . . . . . . . . . . . . . . . . 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 11216	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 1 ∈ ℝ)
77	64, 76, 57	lemul1d 13062	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 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 5176	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ≤ (1 · ((abs‘𝐴) / 2)))
80	58	recnd 11243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → ((abs‘𝐴) / 2) ∈ ℂ)
81	80	mullidd 11233	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (1 · ((abs‘𝐴) / 2)) = ((abs‘𝐴) / 2))
82	79, 81	breqtrd 5167	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝑇 ≤ ((abs‘𝐴) / 2))
83	46, 48, 58, 54, 82	ltletrd 11375	. . . . . . . . . . . . . 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 11373	. . . . . . . . . . . . 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 11813	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝐴) < ((abs‘𝑧) + ((abs‘𝐴) / 2)))
87	71, 86	eqbrtrd 5163	. . . . . . . . . . 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 11808	. . . . . . . . . . 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 13075	. . . . . . . . 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 15405	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 · 𝑧)) = ((abs‘𝐴) · (abs‘𝑧)))
92	91	oveq1d 7419	. . . . . . . . . 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 11243	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘𝑧) ∈ ℂ)
94	51	recnd 11243	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → 𝐵 ∈ ℂ)
95	70, 93, 94	mul32d 11425	. . . . . . . . . 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 2766	. . . . . . . . 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 5169	. . . . . . . 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 11373	. . . . . . 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 11376	. . . . . 6 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 − 𝑧)) < ((abs‘(𝐴 · 𝑧)) · 𝐵))
100	21, 23	absrpcld 15399	. . . . . . 7 ⊢ (((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑧 ∈ (ℂ ∖ {0}) ∧ (abs‘(𝑧 − 𝐴)) < 𝑇)) → (abs‘(𝐴 · 𝑧)) ∈ ℝ⁺)
101	46, 51, 100	ltdivmuld 13070	. . . . . 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 5163	. . . 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 3140	. 2 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ⁺) → ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑇 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
106		breq2 5145	. . 3 ⊢ (𝑦 = 𝑇 → ((abs‘(𝑧 − 𝐴)) < 𝑦 ↔ (abs‘(𝑧 − 𝐴)) < 𝑇))
107	106	rspceaimv 3612	. 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 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 ∖ cdif 3940 ifcif 4523 {csn 4623 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 < clt 11249 ≤ cle 11250 − cmin 11445 / cdiv 11872 2c2 12268 ℝ⁺crp 12977 abscabs 15185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187

This theorem is referenced by: rlimdiv 15596 divcnOLD 24735 divcn 24737 climrec 44872

W3C validator