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Theorem rlimcnp2 26471

Description: Relate a limit of a real-valued sequence at infinity to the continuity of the function 𝑆(𝑦) = 𝑅(1 / 𝑦) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)

**Hypotheses**
Ref	Expression
rlimcnp2.a	⊢ (𝜑 → 𝐴 ⊆ (0[,)+∞))
rlimcnp2.0	⊢ (𝜑 → 0 ∈ 𝐴)
rlimcnp2.b	⊢ (𝜑 → 𝐵 ⊆ ℝ)
rlimcnp2.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
rlimcnp2.r	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
rlimcnp2.d	⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
rlimcnp2.s	⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)
rlimcnp2.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
rlimcnp2.k	⊢ 𝐾 = (𝐽 ↾_t 𝐴)

**Assertion**
Ref	Expression
rlimcnp2	⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦 𝜑,𝑥,𝑦 𝑦,𝑅 𝑥,𝑆 Allowed substitution hints: 𝑅(𝑥) 𝑆(𝑦) 𝐽(𝑥,𝑦) 𝐾(𝑥,𝑦)

Proof of Theorem rlimcnp2 Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4229	. . . . . . . 8 ⊢ (𝐵 ∩ (1[,)+∞)) ⊆ 𝐵
2		resmpt 6038	. . . . . . . 8 ⊢ ((𝐵 ∩ (1[,)+∞)) ⊆ 𝐵 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
3	1, 2	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
4		0xr 11261	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
5		0lt1 11736	. . . . . . . . . . 11 ⊢ 0 < 1
6		df-ioo 13328	. . . . . . . . . . . 12 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
7		df-ico 13330	. . . . . . . . . . . 12 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
8		xrltletr 13136	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((0 < 1 ∧ 1 ≤ 𝑤) → 0 < 𝑤))
9	6, 7, 8	ixxss1 13342	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ 0 < 1) → (1[,)+∞) ⊆ (0(,)+∞))
10	4, 5, 9	mp2an 691	. . . . . . . . . 10 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
11		ioorp 13402	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
12	10, 11	sseqtri 4019	. . . . . . . . 9 ⊢ (1[,)+∞) ⊆ ℝ⁺
13		sslin 4235	. . . . . . . . 9 ⊢ ((1[,)+∞) ⊆ ℝ⁺ → (𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ (𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺)
15		resmpt 6038	. . . . . . . 8 ⊢ ((𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺) → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
16	14, 15	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
17	3, 16	eqtr4d 2776	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))))
18		resres 5995	. . . . . 6 ⊢ (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞)))
19		resres 5995	. . . . . 6 ⊢ (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞)))
20	17, 18, 19	3eqtr4g 2798	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)))
21		rlimcnp2.r	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
22	21	fmpttd 7115	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝑆):𝐵⟶ℂ)
23	22	ffnd 6719	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝑆) Fn 𝐵)
24		fnresdm 6670	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ↦ 𝑆) Fn 𝐵 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ 𝑆))
25	23, 24	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ 𝑆))
26	25	reseq1d 5981	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)))
27		elinel1 4196	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ ℝ⁺) → 𝑦 ∈ 𝐵)
28	27, 21	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → 𝑆 ∈ ℂ)
29	28	fmpttd 7115	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆):(𝐵 ∩ ℝ⁺)⟶ℂ)
30		frel 6723	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆):(𝐵 ∩ ℝ⁺)⟶ℂ → Rel (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → Rel (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
32		eqid 2733	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆)
33	32, 28	dmmptd 6696	. . . . . . . 8 ⊢ (𝜑 → dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) = (𝐵 ∩ ℝ⁺))
34		inss1 4229	. . . . . . . 8 ⊢ (𝐵 ∩ ℝ⁺) ⊆ 𝐵
35	33, 34	eqsstrdi 4037	. . . . . . 7 ⊢ (𝜑 → dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⊆ 𝐵)
36		relssres 6023	. . . . . . 7 ⊢ ((Rel (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ∧ dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⊆ 𝐵) → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
37	31, 35, 36	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
38	37	reseq1d 5981	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)))
39	20, 26, 38	3eqtr3d 2781	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)))
40	39	breq1d 5159	. . 3 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
41		rlimcnp2.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
42		1red 11215	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
43	22, 41, 42	rlimresb 15509	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
44	34, 41	sstrid 3994	. . . 4 ⊢ (𝜑 → (𝐵 ∩ ℝ⁺) ⊆ ℝ)
45	29, 44, 42	rlimresb 15509	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
46	40, 43, 45	3bitr4d 311	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶))
47		inss2 4230	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℝ⁺) ⊆ ℝ⁺
48	47	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℝ⁺) ⊆ ℝ⁺)
49	48	sselda 3983	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
50	49	rpreccld 13026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / 𝑦) ∈ ℝ⁺)
51	50	rpne0d 13021	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / 𝑦) ≠ 0)
52	51	neneqd 2946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ¬ (1 / 𝑦) = 0)
53	52	iffalsed 4540	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅) = ⦋(1 / 𝑦) / 𝑥⦌𝑅)
54		oveq2 7417	. . . . . . . . . 10 ⊢ (𝑥 = (1 / 𝑦) → (1 / 𝑥) = (1 / (1 / 𝑦)))
55		rpcnne0 12992	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
56		recrec 11911	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (1 / 𝑦)) = 𝑦)
57	49, 55, 56	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / (1 / 𝑦)) = 𝑦)
58	54, 57	sylan9eqr 2795	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → (1 / 𝑥) = 𝑦)
59	58	eqcomd 2739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → 𝑦 = (1 / 𝑥))
60		rlimcnp2.s	. . . . . . . 8 ⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)
61	59, 60	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → 𝑆 = 𝑅)
62	61	eqcomd 2739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → 𝑅 = 𝑆)
63	50, 62	csbied 3932	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 = 𝑆)
64	53, 63	eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅) = 𝑆)
65	64	mpteq2dva 5249	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
66	65	breq1d 5159	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) ⇝_𝑟 𝐶 ↔ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶))
67		rlimcnp2.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (0[,)+∞))
68		rlimcnp2.0	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐴)
69		rlimcnp2.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ)
70	69	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 = 0) → 𝐶 ∈ ℂ)
71	67	sselda 3983	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ (0[,)+∞))
72		0re 11216	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
73		pnfxr 11268	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
74		elico2 13388	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → (𝑤 ∈ (0[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞)))
75	72, 73, 74	mp2an 691	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞))
76	71, 75	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞))
77	76	simp1d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
78	77	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ ℝ)
79	76	simp2d 1144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 0 ≤ 𝑤)
80		leloe 11300	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (0 ≤ 𝑤 ↔ (0 < 𝑤 ∨ 0 = 𝑤)))
81	72, 77, 80	sylancr 588	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (0 ≤ 𝑤 ↔ (0 < 𝑤 ∨ 0 = 𝑤)))
82	79, 81	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (0 < 𝑤 ∨ 0 = 𝑤))
83	82	ord 863	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 0 < 𝑤 → 0 = 𝑤))
84		eqcom 2740	. . . . . . . . . . . 12 ⊢ (0 = 𝑤 ↔ 𝑤 = 0)
85	83, 84	imbitrdi 250	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 0 < 𝑤 → 𝑤 = 0))
86	85	con1d 145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 = 0 → 0 < 𝑤))
87	86	imp 408	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 0 < 𝑤)
88	78, 87	elrpd 13013	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ ℝ⁺)
89		rpcnne0 12992	. . . . . . . . 9 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 ∈ ℂ ∧ 𝑤 ≠ 0))
90		recrec 11911	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℂ ∧ 𝑤 ≠ 0) → (1 / (1 / 𝑤)) = 𝑤)
91	89, 90	syl 17	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ⁺ → (1 / (1 / 𝑤)) = 𝑤)
92	88, 91	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / (1 / 𝑤)) = 𝑤)
93	92	csbeq1d 3898	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 = ⦋𝑤 / 𝑥⦌𝑅)
94		oveq2 7417	. . . . . . . . 9 ⊢ (𝑦 = (1 / 𝑤) → (1 / 𝑦) = (1 / (1 / 𝑤)))
95	94	csbeq1d 3898	. . . . . . . 8 ⊢ (𝑦 = (1 / 𝑤) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 = ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅)
96	95	eleq1d 2819	. . . . . . 7 ⊢ (𝑦 = (1 / 𝑤) → (⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ ↔ ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ))
97	63, 28	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
98	97	ralrimiva 3147	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
99	98	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
100		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ 𝐴)
101		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝜑)
102		eleq1 2822	. . . . . . . . . . . . 13 ⊢ (𝑦 = (1 / 𝑤) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑤) ∈ 𝐵))
103	94	eleq1d 2819	. . . . . . . . . . . . 13 ⊢ (𝑦 = (1 / 𝑤) → ((1 / 𝑦) ∈ 𝐴 ↔ (1 / (1 / 𝑤)) ∈ 𝐴))
104	102, 103	bibi12d 346	. . . . . . . . . . . 12 ⊢ (𝑦 = (1 / 𝑤) → ((𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴) ↔ ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴)))
105		rlimcnp2.d	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
106	105	ralrimiva 3147	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
107	106	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
108		rpreccl 13000	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℝ⁺ → (1 / 𝑤) ∈ ℝ⁺)
109	108	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (1 / 𝑤) ∈ ℝ⁺)
110	104, 107, 109	rspcdva 3614	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴))
111	91	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (1 / (1 / 𝑤)) = 𝑤)
112	111	eleq1d 2819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / (1 / 𝑤)) ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
113	110, 112	bitr2d 280	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ 𝐵))
114	101, 88, 113	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ 𝐵))
115	100, 114	mpbid 231	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ 𝐵)
116	88	rpreccld 13026	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ ℝ⁺)
117	115, 116	elind 4195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺))
118	96, 99, 117	rspcdva 3614	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ)
119	93, 118	eqeltrrd 2835	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋𝑤 / 𝑥⦌𝑅 ∈ ℂ)
120	70, 119	ifclda 4564	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) ∈ ℂ)
121	109	biantrud 533	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / 𝑤) ∈ 𝐵 ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺)))
122	113, 121	bitrd 279	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺)))
123		elin 3965	. . . . 5 ⊢ ((1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺) ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺))
124	122, 123	bitr4di 289	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺)))
125		iftrue 4535	. . . 4 ⊢ (𝑤 = 0 → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) = 𝐶)
126		eqeq1 2737	. . . . 5 ⊢ (𝑤 = (1 / 𝑦) → (𝑤 = 0 ↔ (1 / 𝑦) = 0))
127		csbeq1 3897	. . . . 5 ⊢ (𝑤 = (1 / 𝑦) → ⦋𝑤 / 𝑥⦌𝑅 = ⦋(1 / 𝑦) / 𝑥⦌𝑅)
128	126, 127	ifbieq2d 4555	. . . 4 ⊢ (𝑤 = (1 / 𝑦) → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) = if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅))
129		rlimcnp2.j	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
130		rlimcnp2.k	. . . 4 ⊢ 𝐾 = (𝐽 ↾_t 𝐴)
131	67, 68, 48, 120, 124, 125, 128, 129, 130	rlimcnp 26470	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) ⇝_𝑟 𝐶 ↔ (𝑤 ∈ 𝐴 ↦ if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))
132		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑤if(𝑥 = 0, 𝐶, 𝑅)
133		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 0
134		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥𝐶
135		nfcsb1v 3919	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑅
136	133, 134, 135	nfif 4559	. . . . 5 ⊢ Ⅎ𝑥if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅)
137		eqeq1 2737	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = 0 ↔ 𝑤 = 0))
138		csbeq1a 3908	. . . . . 6 ⊢ (𝑥 = 𝑤 → 𝑅 = ⦋𝑤 / 𝑥⦌𝑅)
139	137, 138	ifbieq2d 4555	. . . . 5 ⊢ (𝑥 = 𝑤 → if(𝑥 = 0, 𝐶, 𝑅) = if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅))
140	132, 136, 139	cbvmpt 5260	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) = (𝑤 ∈ 𝐴 ↦ if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅))
141	140	eleq1i 2825	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0) ↔ (𝑤 ∈ 𝐴 ↦ if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅)) ∈ ((𝐾 CnP 𝐽)‘0))
142	131, 141	bitr4di 289	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))
143	46, 66, 142	3bitr2d 307	1 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ⦋csb 3894 ∩ cin 3948 ⊆ wss 3949 ifcif 4529 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ↾ cres 5679 Rel wrel 5682 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 / cdiv 11871 ℝ⁺crp 12974 (,)cioo 13324 [,)cico 13326 ⇝_𝑟 crli 15429 ↾_t crest 17366 TopOpenctopn 17367 ℂ_fldccnfld 20944 CnP ccnp 22729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-rlim 15433 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-bases 22449 df-cnp 22732

This theorem is referenced by: rlimcnp3 26472

W3C validator